A simple and general framework for the construction of thermodynamically compatible schemes for computational fluid and solid mechanics

Abstract

We introduce a simple and general framework for the construction of thermodynamically compatible schemes for the numerical solution of overdetermined hyperbolic PDE systems that satisfy an extra conservation law. As a particular example in this paper, we consider the general Godunov-Peshkov-Romenski (GPR) model of continuum mechanics that describes the dynamics of nonlinear solids and viscous fluids in one single unified mathematical formalism.
A main peculiarity of the new algorithms presented in this manuscript is that the entropy inequality is solved as a primary evolution equation instead of the usual total energy conservation law, unlike in most traditional schemes for hyperbolic PDE. Instead, total energy conservation is obtained as a mere consequence of the proposed thermodynamically compatible discretization. The approach is based on the general framework introduced in Abgrall (2018) [1]. In order to show the universality of the concept proposed in this paper, we apply our new formalism to the construction of three different numerical methods. First, we construct a thermodynamically compatible finite volume (FV) scheme on collocated Cartesian grids, where discrete thermodynamic compatibility is achieved via an edge/face-based correction that makes the numerical flux thermodynamically compatible. Second, we design a first type of high order accurate and thermodynamically compatible discontinuous Galerkin (DG) schemes that employs the same edge/face-based numerical fluxes that were already used inside the finite volume schemes. And third, we introduce a second type of thermodynamically compatible DG schemes, in which thermodynamic compatibility is achieved via an element-wise correction, instead of the edge/face-based corrections that were used within the compatible numerical fluxes of the former two methods. All methods proposed in this paper can be proven to be nonlinearly stable in the energy norm and they all satisfy a discrete entropy inequality by construction. We present numerical results obtained with the new thermodynamically compatible schemes in one and two space dimensions for a large set of benchmark problems, including inviscid and viscous fluids as well as solids. An interesting finding made in this paper is that, in numerical experiments, one can observe that for smooth isentropic flows the particular formulation of the new schemes in terms of entropy density, instead of total energy density, as primary state variable leads to approximately twice the convergence rate of high order DG schemes for the entropy density.

Full text

ZurichOpenRepositoryandArchiveUniversityofZurichUniversityLibraryStrickhofstrasse39CH-8057Zurichwww.zora.uzh.chYear:2023AsimpleandgeneralframeworkfortheconstructionofthermodynamicallycompatibleschemesforcomputationaluidandsolidmechanicsAbgrall,Remi;Busto,Saray;Dumbser,MichaelAbstract:WeintroduceasimpleandgeneralframeworkfortheconstructionofthermodynamicallycompatibleschemesforthenumericalsolutionofoverdeterminedhyperbolicPDEsystemsthatsatisfyanextraconserva-tionlaw.Asaparticularexampleinthispaper,weconsiderthegeneralGodunov-Peshkov-Romenski(GPR)modelofcontinuummechanicsthatdescribesthedynamicsofnonlinearsolidsandviscousuidsinonesingleuniedmathematicalformalism.Amainpeculiarityofthenewalgorithmspresentedinthismanuscriptisthattheentropyinequalityissolvedasaprimaryevolutionequationinsteadoftheusualtotalenergyconservationlaw,unlikeinmosttraditionalschemesforhyperbolicPDE.Instead,totalenergyconservationisobtainedasamereconsequenceoftheproposedthermodynamicallycompatiblediscretization.eapproachisbasedonthegeneralframeworkintroducedinAbgrall(2018)[1].Inordertoshowtheuniversalityoftheconceptproposedinthispaper,weapplyournewformalismtotheconstructionofthreedierentnumericalmethods.First,weconstructathermodynamicallycompatiblenitevolume(FV)schemeoncollocatedCartesiangrids,wheredis-cretethermodynamiccompatibilityisachievedviaanedge/face-basedcorrectionthatmakesthenumericaluxthermodynamicallycompatible.Second,wedesignarsttypeofhighorderaccurateandthermodynamicallycompatiblediscontinuousGalerkin(DG)schemesthatemploysthesameedge/face-basednumericaluxesthatwerealreadyusedinsidethenitevolumeschemes.Andthird,weintroduceasecondtypeofthermodynami-callycompatibleDGschemes,inwhichthermodynamiccompatibilityisachievedviaanelement-wisecorrection,insteadoftheedge/face-basedcorrectionsthatwereusedwithinthecompatiblenumericaluxesoftheformertwomethods.Allmethodsproposedinthispapercanbeproventobenonlinearlystableintheenergynormandtheyallsatisfyadiscreteentropyinequalitybyconstruction.Wepresentnumericalresultsobtainedwiththenewthermodynamicallycompatibleschemesinoneandtwospacedimensionsforalargesetofbenchmarkproblems,includinginviscidandviscousuidsaswellassolids.Aninterestingndingmadeinthispaperisthat,innumericalexperiments,onecanobservethatforsmoothisentropicowstheparticularformulationofthenewschemesintermsofentropydensity,insteadoftotalenergydensity,asprimarystatevariableleadstoapproximatelytwicetheconvergencerateofhighorderDGschemesfortheentropydensity.DOI:hps://doi.org/10.1016/j.amc.2022.127629PostedattheZurichOpenRepositoryandArchive,UniversityofZurichZORAURL:hps://doi.org/10.5167/uzh-224389JournalArticlePublishedVersion
efollowingworkislicensedunderaCreativeCommons:Aribution-NonCommercial-NoDerivatives4.0In-ternational(CCBY-NC-ND4.0)License.

Originallypublishedat:Abgrall,Remi;Busto,Saray;Dumbser,Michael(2023).Asimpleandgeneralframeworkfortheconstructionofthermodynamicallycompatibleschemesforcomputationaluidandsolidmechanics.AppliedMathematicsandComputation,440:127629.DOI:hps://doi.org/10.1016/j.amc.2022.1276292

Applied Mathematics and Computation 440 (2023) 127629
Contents lists available at ScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
A simple and general framework for the construction of
thermodynamically compatible schemes for computational
uid and solid mechanics
Rémi Abgrall
a
, Saray Busto
b , ∗, Michael Dumbser
c
a
Institute of Mathematics, University of Zürich, Winterthurerstrasse 190, Zürich CH-8057, Switzerland
b
Departamento de Matemática Aplicada I, Universidade de Vigo, Campus As Lagoas Marcosende s/n, Vigo E-36310, Spain
c
Laboratory of Applied Mathematics, DICAM, University of Trento, via Mesiano 77, Trento I-38123, Italy
a r t i c l e i n f o
Article history:
Available online 11 November 2022
Keywords:
Hyperbolic and thermodynamically
compatible (HTC) systems with extra
conservation law
Entropy inequality
Nonlinear stability in the energy norm
Thermodynamically compatible nite
volume schemes
Thermodynamically compatible
discontinuous Galerkin schemes
Unied rst order hyperbolic formulation of
continuum mechanics
a b s t r a c t
We introduce a simple and general framework for the construction of thermodynamically
compatible schemes for the numerical solution of overdetermined hyperbolic PDE systems
that satisfy an extra conservation law . As a particular example in this paper, we consider
the general Godunov-Peshkov-Romenski (GPR) model of continuum mechanics that de-
scribes the dynamics of nonlinear solids and viscous uids in one single unied mathe-
matical formalism.
A main peculiarity of the new algorithms presented in this manuscript is that the en-
tropy inequality is solved as a primary evolution equation instead of the usual total energy
conservation law, unlike in most traditional schemes for hyperbolic PDE. Instead, total en-
ergy conservation is obtained as a mere consequence of the proposed thermodynamically
compatible discretization. The approach is based on the general framework introduced in
Abgrall (2018) [1]. In order to show the universality of the concept proposed in this pa-
per, we apply our new formalism to the construction of three different numerical meth-
ods. First, we construct a thermodynamically compatible nite volume (FV) scheme on
collocated Cartesian grids, where discrete thermodynamic compatibility is achieved via an
edge/face-based correction that makes the numerical ux thermodynamically compatible.
Second, we design a rst type of high order accurate and thermodynamically compatible
discontinuous Galerkin (DG) schemes that employs the same edge/face-based numerical
uxes that were already used inside the nite volume schemes. And third, we introduce a
second type of thermodynamically compatible DG schemes, in which thermodynamic com-
patibility is achieved via an element-wise correction, instead of the edge/face-based correc-
tions that were used within the compatible numerical uxes of the former two methods.
All methods proposed in this paper can be proven to be nonlinearly stable in the energy
norm and they all satisfy a discrete entropy inequality by construction . We present numeri-
cal results obtained with the new thermodynamically compatible schemes in one and two
space dimensions for a large set of benchmark problems, including inviscid and viscous
uids as well as solids. An interesting nding made in this paper is that, in numerical ex-
periments, one can observe that for smooth isentropic ows the particular formulation of
the new schemes in terms of entropy density, instead of total energy density, as primary
∗Corresponding author.
E-mail addresses:
remi.abgrall@math.uzh.ch (R. Abgrall), saray.busto@unitn.it (S. Busto), michael.dumbser@unitn.it (M. Dumbser) .
https://doi.org/10.1016/j.amc.2022.127629
0 096-30 03/© 2022 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license
(
http://creativecommons.org/licenses/by-nc-nd/4.0/ )

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
state variable leads to approximately twice the convergence rate of high order DG schemes
for the entropy density.
© 2022 The Authors. Published by Elsevier Inc.
This is an open access article under the CC BY-NC-ND license
( http://creativecommons.org/licenses/by-nc-nd/4.0/ )
1. Introduction
The seminal paper An interesting class of quasilinear systems published by Godunov in 1961 established for the rst time
the connection between symmetric hyperbolicity in the sense of Friedrichs [2] and thermodynamic compatibility, see [3] .
Godunov showed that, for hyperbolic systems which have an underlying variational formulation, the total energy conserva-
tion law is an extra conservation law that can be obtained as a consequence of all the other equations via their suitable linear
combination at the aid of the so-called thermodynamic dual variables, which are the partial derivatives of the total energy
potential with respect to the conservative variables. The formalism introduced by Godunov was rediscovered independently
10 years later by Friedrichs and Lax [4] . Important contributions to the subject were also made by Boillat [5] and Ruggeri
[6] . In [6] the thermodynamic dual variables were denoted as the so-called main eld , while other works refer to them as
the Godunov variables , see e.g. [7] .
The results obtained in the original paper of Godunov [3] apply to the Euler equations of gas dynamics, to the shallow
water equations and other simple inviscid hyperbolic systems in conservation form and without involution constraints. In
subsequent work Godunov and Romenski extended the theory of symmetric hyperbolic and thermodynamically compatible
(SHTC) systems to a much wider class of mathematical models, including magnetohydrodynamics [8] , nonlinear hyperelas-
ticity [9] , compressible multi-phase ows [10,11] and even relativistic uid and solid mechanics, see [12,13] . An extension to
continuum mechanics with torsion was made in Peshkov et al. [14] , while a connection of SHTC systems with Hamiltonian
continuum mechanics was recently established in [15] . A rather general presentation of the overall formalism can be found
in [16] , and [17] .
Usually in SHTC systems the entropy density is the primary evolution variable , while the conservation law for the total
energy density is the extra conservation law , since it can be obtained as a consequence by a suitable linear combination of
all the other evolution equations. The total energy potential has a privileged role in SHTC systems, because it is used in
the Lagrangian of the underlying variational principle from which all SHTC systems can be derived. For other recent and
very interesting hyperbolic and thermodynamically compatible systems based on an augmented Lagrangian approach for
the derivation of hyperbolic models of dispersive systems via variational principles, see e.g. [18,19] .
Most of the existing entropy preserving and entropy-stable schemes are built on the seminal ideas of Tadmor [20] . They
discretize the total energy conservation directly and obtain a discrete compatibility with the entropy inequality as a con-
sequence, mimicking the ideas of Friedrichs and Lax [4] on the discrete level. Also numerical schemes which are discretely
compatible with kinetic energy preservation for the Euler equations fall into the larger class of schemes that satisfy addi-
tional extra conservation laws of the governing PDE system at the discrete level. Important contributions to the development
of kinetic energy compatible and entropy compatible nite difference schemes for hyperbolic PDE that make use of skew
symmetric forms and/or a discrete summation by parts (SBP) property can be found, for example, in the papers of Ducros
et al. [21,22] , Fisher et al. [23,24] , Carpenter and Nordström et al. [25–28] , Pirozzoli [29,30] , Sjögreen and Yee [31–33] and
in Reiss and Sesterhenn [34] . Without pretending completeness of the following overview, important recent developments
on high order entropy-compatible schemes can be found, for example, in the work of Mishra and collaborators [35–37] ,
Gassner et al., [38–42] , Shu and collaborators [43,44] and in Chandrashekar and Klingenberg [45] , Ray et al. [46] , Ray and
Chandrashekar [47] , Chan and Taylor [48] , Chan et al. [49] , Gaburro et al. [50] , while entropy-compatible schemes for non-
conservative hyperbolic systems were presented in Fjordholm and Mishra [51] .
A simple and general framework for the construction of compatible numerical methods which satisfy extra conservation
laws at the discrete level was very recently put forward by Abgrall and collaborators in [1 , 52–55] and will also be the basis
of the schemes presented in this paper.
In Lagrangian hydrodynamics thermodynamically compatible schemes have been developed in order to obtain the total
energy conservation as a consequence of a compatible discretization of the equations of continuity, momentum and inter-
nal energy, see Caramana and R.Loubère [56] , Bauera et al. [57] , Maire et al. [58] . However, these schemes apply only to
hydrodynamics and not to the GPR model of continuum mechanics treated in this paper.
Up to now, nite volume and discontinuous Galerkin methods that discretize directly the entropy inequality and which
are able to obtain the total energy conservation law as a consequence of the compatible discretization of all the other equa-
tions are still quite rare. First progress in this direction has been recently made in Busto et al. [59–61] , where a novel family
of thermodynamically compatible nite volume schemes was introduced for turbulent shallow water ows, for the Euler
and MHD equations, as well as for the GPR model of continuum mechanics. In [60,61] the entropy density was solved as
primary evolution variable and the total energy conservation law was obtained as a consequence. These ideas have recently
also been extended to high order discontinuous Galerkin nite element schemes, see [62] . A common building block in all
2

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
methods presented in [59–62] is the use of a path integral in order to obtain a thermodynamically compatible numerical
ux for the underlying inviscid Euler or shallow water subsystem. This path integral was discretized at the aid of suitable
numerical quadrature formulas, which, however, can lead to a small source of total energy conservation errors if the num-
ber of quadrature points was not large enough, see [61] for a detailed analysis. It is therefore the main objective of the
present paper to construct thermodynamically compatible nite volume and discontinuous Galerkin schemes for the Euler
equations and for the GPR model of continuum mechanics that do not need the approximate computation of a path inte-
gral in order to obtain a thermodynamically compatible ux. In this paper, we make use of the new concept of a direct
discretization of the entropy inequality with appropriate non-negative production term in order to obtain the discrete total
energy conservation law as a consequence. Our method can therefore be seen as a dual scheme with respect to traditional
entropy-conserving and entropy-stable methods, where total energy is discretized and the entropy inequality is obtained as
a consequence. A major innovation and special property of the new family of schemes presented in this paper is that they
do not need the Godunov parametrization of the ux of the underlying Euler subsystem in terms of a generating potential,
in contrast to the methods presented previously in Busto et al. [59–62] , which were all explicitly relying on the Godunov
parametrization in terms of a generating potential.
The rest of this paper is organized as follows: in Section 2 , we present the governing PDE system treated in this
manuscript, namely the rst order hyperbolic GPR model of continuum mechanics supplemented with a thermodynam-
ically compatible parabolic vanishing viscosity regularization. The next three sections are devoted to the introduction of
three different but related thermodynamically compatible numerical schemes for the discretization of the GPR model. To
facilitate the reader, the presentation of the three different schemes is organized in increasing level of complexity:
1. In Section 3 a new thermodynamically compatible cell centered nite volume method is presented. The readers who are
only interested in FV schemes are invited to focus on this section.
2. In Section 4 we introduce a rst DG scheme (DG scheme of type I) that employs the same thermodynamically compatible
edge/face-based uxes for the Euler subsystem as the one used in the nite volume method presented in Section 3 . As
such, the FV method and the DG schemes of type I are related to each other. The readers who are interested in a
straightforward extension of nite volume schemes to the DG framework are invited to focus on Sections 3 and 4 .
3. In Section 5 we propose a second DG scheme (DG scheme of type II) which is completely different from the previous two
methods as it establishes thermodynamic compatibility directly in a genuinely multi-dimensional fashion via a suitable
element-wise correction.
For all schemes presented in this paper, a cell entropy inequality and nonlinear stability in the energy norm can be proven
for the semi-discrete case. In Section 6 , numerical results are shown for a set of different test cases, ranging from the uid
to the solid limit of the GPR model. The conclusions are presented together with an outlook to future work in Section 7 .
2. Governing partial differential equations
This paper is concerned with a new family of thermodynamically compatible schemes for the unied rst order hyper-
bolic model of continuum mechanics of Godunov, Peshkov and Romenski (GPR model) see [3,9,16,17,63,64] . The governing
PDE system, which in this paper has been regularized via appropriate thermodynamically compatible parabolic vanishing
viscosity terms, reads as follows:
∂ρ∂t
+
∂(ρv
k
)
∂x
k −∂
∂x
m
∂ρ∂x
m
)= 0 , (1a)
∂ρv
i
∂t
+
∂
(
ρv
i
v
k+ p δik
+ σik
+ ω
ik
)
∂x
k −∂
∂x
m
∂ρv
i
∂x
m
)= 0 , (1b)
∂ρS
∂t
+
∂
(
ρSv
k
+ βk
)
∂x
k −∂
∂x
m
∂ρS
∂x
m
)= + αik
αik
θ1
(τ1
) T
+ βi
βi
θ2
(τ2
) T ≥0 , (1c)
∂A
ik
∂t+
∂(A
im
v
m
)
∂x
k
+ v
m
∂A
ik
∂x
m −∂A
im
∂x
k )−∂
∂x
m
∂A
ik
∂x
m
)= −αik
θ1
(τ1
)
, (1d)
∂J
k
∂t
+
∂
(
J
m
v
m
+ T
)
∂xk
+ v
m
∂J
k
∂x
m −∂J
m
∂x
k
)−∂
∂x
m
∂J
k
∂x
m
)= −βk
θ2
(τ2
)
, (1e)
∂E
∂t
+
∂
(
v
k
(
E
1
+E
2
+ E
3
+ E
4
)
+ v
i
(p δik
+ σik
+ ω
ik
) + h
k
)
∂x
k −∂
∂x
m
∂E
∂x
m
)= 0 . (1f)
with the state vector q = { q
i
} = (ρ, ρv
i
, ρS, A
ik
, J
k
)
T and the total energy density E = ρE = E
1
+ E
2
+ E
3
+ E
4 with E
i
= ρE
i
.
Throughout this paper the tensor indices i , k and m run from 1 to 3, i.e. we always consider the full equations written for
3

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
the three-dimensional case, independently of the actual number of space dimensions considered in the numerical scheme.
The system above belongs to the class of overdetermined hyperbolic systems. The vanishing viscosity parameter is denoted
by > 0 and the entropy production term associated with the vanishing viscosity regularization reads
=T ∂
x
m
q
i
∂
2
q
i
q
j
E ∂
x
m
q
j
≥0 , (2)
where the nonnegativity is obtained by assuming > 0 , T > 0 and that the Hessian of the total energy potential is
at least positive semi-denite, i.e. H
i j
:= ∂
2
q
i
q
j
E ≥0 . In the following, we will make use of the notations ∂
p
= ∂ /∂ pand
∂
2
pq
= ∂
2
/ (∂ p∂ q ) for the rst and second partial derivatives with respect to the quantities pand q . We will also assume
the Einstein summation convention over repeated indices and we will make use bold symbols for vectors and matrix, for
example q = { q
i
} and A = { A
ik
} . In this paper, we write the total energy density as a sum of four contributions given by
E
1
= ργγ−1
e
S/c
v
, E
2
=
1
2
ρv
i
v
i
, E
3
=
1
4
ρc
2
s
˚G
i j
˚G
i j
, E
4
=
1
2
c
2
h
ρJ
i
J
i
, (3)
where G is a metric tensor dened as G
ik
= A
ji
A
jk
. Its trace-free part, or deviator, is denoted by
˚G and reads
˚G
ik
= G
ik
−1
3
G
mm
δik
. The main eld or so-called thermodynamic dual variables are denoted by p = ∂
q
E = { p
i
} =
(
r, v
i
, T , αik
, βk
)
T and
are dened as
r = ∂
ρE, v
i
= ∂
ρv
i
E, T = ∂
ρS
E, αik
= ∂
A
ik
E, βk
= ∂
J
k
E. (4)
The hydrodynamic pressure pis given by p = ρ∂
ρE + ρv
i
∂
ρv
i
E + ρS ∂
ρS
E −E = ρ2
∂
ρE, while the shear stress tensor and the
thermal stress tensor read
σik
= A
ji
∂
A
jk
E = A
ji
αjk
= ρc
2
s
G
i j
˚G
jk
, ω
ik
= J
i
∂
J
k
E = J
i
βk
= ρc
2
h
J
i
J
k
, (5)
respectively. In the above model the heat ux is dened as
h
k
= ∂
ρS
E ∂
J
k
E = T βk
= ρc
2
h
T J
k
. (6)
The total energy ux F
k
in (1f) is the sum of two uxes, F
k
= F
12
k
+ F
34
k
, where F
12
k
is related to the Euler subsystem and F
34
k
contains the work of the stress tensors σik
and ω
ik
as well as the heat ux h
k
, i.e.
F
12
k
= v
k
(
E
1
+ E
2
)
+ v
i
δik
p, F
34
k
= v
k
(
E
3
+ E
4
)
+ v
i
(
σik
+ ω
ik
)
+ h
k
. (7)
Furthermore, θ1
(τ1
) > 0 and θ2
(τ2
) > 0 are functions that depend on q and on the relaxation times τ1
> 0 and τ2
> 0 as
follows:
θ1
=
1
3
ρz
1
τ1
c
2
s
|
A
|
−5
3
, θ2
= ρz
2
τ2
c
2
h
, z
1
= ρ0
ρ, z
2
= ρ0
T
0
ρT
. (8)
Here, ρ0 and T
0 are a reference density and a reference temperature, respectively. After some calculations one can verify
that (1f) is a consequence of (1a) –(1e) since the following identity holds:
(1f) = r ·(1a) + v
i
·(1b) + T ·(1c) + αik
·(1d) + βk
·(1e) . (9)
The above relation is directly related to the Gibbs identity
1 ·dE = r ·dρ+ v
i
·d(ρv
i
) + T ·d(ρS) + αik
·dA
ik
+ βk
·dJ
k
= p ·dq . (10)
Note that in the above identity the factor in front of the total energy differential is simply unity . This not only highlights
the privileged role that the total energy potential has in SHTC systems, but it also substantially eases the calculations when
the PDE for Eis taken as the consequence of all other equations, instead of the PDE for the entropy density. The factor T in
front of the entropy differential can become rather complex for general EOS or for more complicated PDE systems. Instead,
the factor of unity in front of the total energy differential is always trivial and is independent of the equation of state and
even of the mechanical system under consideration. For other HTC systems with the same property, the reader is referred
to Romenski et al. [11] , Romenski [16] , Favrie and Gavrilyuk [18] , Dhaouadi et al. [19] , Busto et al. [59] , Busto and Dumbser
[60] , Dhaouadi and Dumbser [65] .
A formal asymptotic analysis of the model (1) was presented in [64] . It is shown that for small relaxation times, τ1 and
τ2
, the Navier–Stokes–Fourier limit is obtained, i.e. the stress tensor σik
and the heat ux h
k
tend to
σik
= −1
6
ρ0
c
2
s
τ1
∂
k
v
i
+ ∂
i
v
k
−2
3
(
∂
m
v
m
)
δik
, h
k
= −ρ0
T
0
c
2
h
τ2
∂
k
T . (11)
In this case one can relate the viscosity coecient to the relaxation time τ1 and the shear sound speed c
s as μ=
1
6
ρ0
c
2
s
τ1
,
while the thermal conductivity coecient is related to the relaxation time τ2 and the heat wave speed c
h
by κ= ρ0
T
0
c
2
h
τ2
.
In this paper, we will also make use of the following more compact formulation of the above PDE system, casting it into
the general form
∂
t
q + ∂
k
f
k
(q ) + ∂
k
h
k
(q ) + B
k
(q ) ∂
k
q −∂
m
(
∂
m
q)
= P + S (q ) (12)
4

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
with the extra conservation law for the total energy density
∂E
∂t
+ ∂
k
F
k
(q ) −∂
m
(
∂
m
E
)
= 0 . (13)
Here, the ux f
k
(q ) is associated with the inviscid Euler subsystem (black terms), the ux h
k
(q ) and the non-conservative
product B
k
(q ) ∂
k
q include the terms related to the distorsion eld and the thermal impulse (red terms), the parabolic dissi-
pation terms are ∂
m
(
∂
m
q
)
with the associated entropy production P (blue terms) and the algebraic relaxation source terms,
which are potentially stiff, are denoted by S (q ) (green). In the extra conservation law (13) the total energy ux is given by
F
k
(q ) . Since p = ∂
q
Eand therefore p ·∂
t
q = ∂
q
E ·∂
t
q = ∂
t
E, for thermodynamic compatibility of system (12) with the extra
conservation law (13) the following identities must be satised, in particular the compatibility of the conservative ux terms
and of the non-conservative products with the total energy ux,
p ·(
∂
k
f
k
(q ) + ∂
k
h
k
(q ) + B
k
(q ) ∂
k
q
)
= ∂
k
F
k
. (14)
More specically, the ux of the Euler subsystem f
k
must be compatible with the energy ux F
12
k
p ·∂
k
f
k
(q ) = ∂
k
F
12
k
(15)
and the remaining terms must be compatible with the energy ux F
34
k
, i.e.
p ·(
∂
k
h
k
(q ) + B
k
(q ) ∂
k
q
)
= ∂
k
F
34
k
. (16)
Furthermore, one has the compatibility of the entropy production term with the parabolic dissipation terms
p ·P + p ·∂
m
(
∂
m
q
)
= ∂
m
(
∂
m
E
)
, (17)
and the compatibility of the algebraic relaxation source terms
p ·S (q ) = 0 . (18)
As already stated previously, the main peculiarity of the new algorithms presented in this manuscript is that the entropy
inequality is solved as a primary evolution equation instead of the total energy conservation law, unlike in most traditional
schemes for hyperbolic systems of conservation laws. Instead, in our framework, the total energy conservation is obtained
as a consequence of the proposed thermodynamically compatible discretization.
3. Thermodynamically compatible nite volume schemes on collocated meshes
In favor of clarity and in order to ease the reading, we start presenting the construction of our thermodynamically
compatible schemes step by step in one space dimension only, using the different colours in (1) as guidance. We start with
the discretization of the inviscid Euler subsystem (black terms), then including the viscous terms (blue) and nally adding
the distortion eld and the specic thermal impulse (red terms). Throughout this paper, we use lower case subscripts, i, j, k ,
for tensor indices, while lower case superscript,  , refer to the spatial discretization index. We denote the spatial control
volumes in 1D by 
= [ x
 −1
2
, x
 +
1
2
] and x = x
 +
1
2 −x
 −1
2 is the uniform mesh spacing.
3.1. Compatible discretization of the Euler subsystem in 1D
The inviscid Euler subsystem with the related extra conservation law for the total energy density reads
∂
t
q + ∂
x
f
1
(q ) = 0 , (19)
∂
t
E
12
+ ∂
x
F
12
1
= 0 . (20)
A semi-discrete nite volume scheme for (19) is thus given by
d
dt
q

= −F
 +
1
2 −F
 −1
2
x
= −F
 +
1
2 −f

−F
 −1
2 −f

x
= −D
 +
1
2
−+ D
 −1
2
+
x
, (21)
with f

= f
1
(q

) to ease notation, f
1
(q ) = (ρv
1
, ρv
i
v
1
+ pδi 1
, ρSv
1
, 0 , 0 )
T the uxes of the Euler subsystem and the associated
total energy ux of the Euler subsystem F
12
1
= v
1
(E
1
+ E
2
+ p) and the numerical ux F
 +
1
2
. The uctuations D
 +
1
2
−and D
 −1
2
+
are related to the numerical uxes via the relations
D
 +
1
2
−= F
 +
1
2 −f

, D
 −1
2
+
= f

−F
 −1
2
. (22)
To obtain a discrete total energy conservation law as a consequence of the discretization of (19) , see (9), we compute the
dot product of the discrete dual variables, p

= ∂
q
E(q

) , with the semi-discrete scheme (21) ,
p

·d
dt
q

=
d
dt
E
= −p

·F
 +
1
2 −f

+
f

−F
 −1
2
x
= −D
 +
1
2
E, −+ D
 −1
2
E, +
x
= −F
12 , +
1
2
1 −F
12 , −1
2
1
x
. (23)
5

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
We now look for a suitable numerical ux F
 +
1
2 that achieves thermodynamic compatibility of the nite volume scheme
(21) with the discrete form of the total energy conservation law (23) . For that purpose we dene the right and left energy
uctuations as
D
 +
1
2
E, −= p

·D
 +
1
2
−= p

·(F
 +
1
2 −f

) , D
 −1
2
E, +
= p

·D
 −1
2
+
= p

·(f

−F
 −1
2
) , (24)
which must satisfy the consistency property
D
 +
1
2
E, −+ D
 +
1
2
E, +
= p

·D
 +
1
2
−+ p
 +1
·D
 +
1
2
+
= p

·(F
 +
1
2 −f

) + p
 +1
·(f
 +1
−F
 +
1
2
) = F
12 , +1
1 −F
12 ,
1
(25)
in order to obtain a conservative discretization of the extra conservation law (20) . The numerical total energy uxes F
12 , ±1
2
1
appearing in (23) are related to the energy uctuations via
D
 +
1
2
E, −= F
12 , +
1
2
1 −F
12 ,
1
, D
 −1
2
E, +
= F
12 ,
1 −F
12 , −1
2
1
, (26)
with F
12 ,
1
= F
12
1
(q

) the discrete total energy ux related to the Euler subsystem evaluated in cell 
. Following the ideas
outlined in the general framework [1] , in the following we assume the thermodynamically compatible numerical ux F
 +
1
2
to have the rather general form
F
 +
1
2 = ˜
F
 +
1
2 −α +
1
2
p
 +1
−p

=
F
 +
1
2
ρ, F
 +
1
2
ρv
i
, F
 +
1
2
ρS
, 0 , 0
T
, (27)
where ˜
F
 +
1
2 could be in principle any central numerical ux that does not necessarily guarantee discrete thermodynamic
compatibility of the system of conservation laws (19) with the extra conservation law (20) and which is then corrected via
a suitable scalar parameter α +
1
2 in order to achieve discrete thermodynamic compatibility. Imposing the condition (25) on
the ux (27) we obtain
p

·(
˜
F
 +
1
2 −f

) + p
 +1
·(f
 +1
−˜
F
 +
1
2
) −α +
1
2
p

·p
 +1
−p

+ α +
1
2
p
 +1
·p
 +1
−p

= F
12 , +1
1 −F12 ,
1
. (28)
Rearranging terms in the above equation yields
−˜
F
 +
1
2 ·p
 +1
−p

+ p
 +1
·f
 +1
−p

·f

+ α +
1
2
p
 +1
−p

2
= F
12 , +1
1 −F
12 ,
1
, (29)
from which we can obtain the scalar correction factor α +
1
2
α +
1
2 =
F
12 , +1
1 −F
12 ,
1
+
˜
F
 +
1
2 ·p
 +1
−p

−p
 +1
·f
 +1
−p

·f

p
 +1
−p

2
(30)
that guarantees the discrete thermodynamic compatibility of the scheme (21) with the discrete extra conservation law (23) .
In the case of a vanishing denominator, p
 +1
−p

2
= 0 , we set α +
1
2 = 0 . Throughout this paper we simply choose the
dissipationless central ux as underlying numerical ux ˜
F
 +
1
2
, i.e.
˜
F
 +
1
2 =
1
2
f

+ f
 +1
, (31)
since suitable numerical dissipation terms that are compatible with the rst and second law of thermodynamics will be
provided in the next section. We stress that the numerical ux (27) with (30) does not need the Godunov parametrization
of the ux f
1
= ∂
p
(v
1
L ) in terms of a generating potential L , unlike the HTC nite volume schemes presented in [59–61] .
Note that this part of the scheme is only related to the reversible (inviscid) terms of the governing equations, hence it
is sucient to consider the simple central ux (31) in (27) . A proper thermodynamically compatible discretization of the
viscous terms, which mimics the parabolic vanishing viscosity regularization of the governing PDE system, including the
non-negative entropy production term, is done separately and will be presented later in the next section.
3.2. Thermodynamically compatible discretization of the viscous terms in 1D
In the end we are interested in constructing a dissipative scheme for (1) that is thermodynamically compatible. For this
purpose, and since the compatible numerical ux (27) with (31) and (30) developed in the previous section is dissipationless,
we now still need to add a thermodynamically compatible numerical viscosity in order to get a dissipative scheme that
also works in the presence of shock waves and other discontinuities. Hence, the ux (27) is extended by a dissipative
contribution and a corresponding non-negative entropy production term, which mimic the vanishing viscosity regularization
introduced in the governing PDE system (1) at the discrete level:
d
dt
q

+
F
 +
1
2 −F
 −1
2
x
=
G
 +
1
2 −G
 −1
2
x
+ P

. (32)
6

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
The dissipative numerical ux is chosen as
G
 +
1
2 =  +
1
2
q
 +
1
2
x
, q
 +
1
2 = q
 +1
−q

, (33)
with a scalar numerical viscosity coecient that can be either chosen to be constant, i.e.  +
1
2 = , or which is chosen as
follows,
 +
1
2 =
1
2
1 −φ +
1
2
x s
 +
1
2
max
≥0 , (34)
where s
 +
1
2
max
according to the Rusanov or local Lax-Friedrichs ux is chosen as the maximum signal speed arising at the
cell interface and φ+
1
2 is a ux limiter allowing to reduce the numerical viscosity regions where the numerical solution is
smooth. In the following, and if not stated otherwise, we will employ the minbee ux limiter given by
φ +
1
2 = min
φ +
1
2
−, φ +
1
2
+ , with φ +
1
2
±= max
0 , min
1 , h
 +
1
2
±, (35)
where the ratios of total energy slopes, similar to the SLIC scheme of Toro [66] , read
h
 +
1
2
−=
E

−E
−1
E
 +1
−ρ
, and h
 +1
2
+
=
E
 +2
−E
 +1
E
 +1−ρ
. (36)
The nal thermodynamically compatible dissipative Rusanov ux, which includes both the convective and the diffusive
terms, is given by
F
 +
1
2 + G
 +
1
2 =
1
2
f
 +1
+ f

−α +
1
2
p
 +1
−p

− +
1
2
x q
 +1
−q

, (37)
where we have used our choice (31) and where α +
1
2 is given by (30) . Unlike in [59 , 61] the ux (37) does not require
the evaluation of path integrals in phase space and is therefore computationally much more ecient, as shown later by
numerical experiments.
Taking the dot product of Eq. (32) with the dual variables p
 yields
dE

dt
+
1
x
F
12 , +
1
2
1 −F
12 , −1
2
1 =
1
x
p

·G
 +
1
2 −G
 −1
2
+ p

·P

. (38)
The thermodynamic compatibility of the inviscid part on the left hand side of (38) is obvious since it was already shown in
the previous section. Therefore, we can now simply focus on the terms appearing on the right hand side of (38). After some
calculations we obtain
p

·P

+ p

·G
 +
1
2 −G
 −1
2
x
= p

·P

+
1
x
1
2
p

·G
 +
1
2 +
1
2
p
 +1
·G
 +
1
2 +
1
2
p

·G +
1
2 −1
2
p
 +1
·G
 +
1
2
−1
x
1
2
p

·G
 −1
2 +
1
2
p
 −1
·G
 −1
2 +
1
2
p

·G
 −1
2 −1
2
p
 −1
·G
 −1
2
= p

·P

+
1
2
p
 +1
+ p

x
· +
1
2
q
 +
1
2
x −1
2
p

+ p
 −1
x
· −1
2
q
 −1
2
x
−1
2
p
 +1
−p

x
· +
1
2
q
 +
1
2
x −1
2
p

−p
 −1
x
· −1
2
q
 −1
2
x
. (39)
Because of the relation
q
 +1
∫
q

p ·d q =
q
 +1
∫
q
 ∂
q
E ·d q = E
 +1
−E

= E
 +
1
2
, (40)
one may interpret the term
1
2
(p
 +1
+ p

) ·q
 +
1
2 as an approximation of the total energy density difference E
 +
1
2
, with an
approximation of the path integral via the simple trapezoidal rule. Due to (39) and (40) , the energy ux including convective
and diffusive terms is
F
 +
1
2
d
= F
12 , +
1
2
1 −1
2
(p
 +1
+ p

) · +
1
2
q
 +
1
2
x ≈F
12 , +
1
2
1 − +
1
2
E
 +
1
2
x
. (41)
In order to control the sign of the entropy production, we now rewrite the jump terms in the dual p variables as jumps in
the conservative q variables. To this end, we make use of a Roe-type matrix ∂
2
qq
˜
E
 +
1
2 that veries the Roe property
∂
2
qq
˜
E
 +
1
2 ·(q
 +1
−q

) = p
 +1
−p

. (42)
7

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
The simple segment path
˜
ψ in the conservative q variables,
˜
ψ (s ) = q

+ s
q
 +1
−q

, 0 ≤s ≤1 , (43)
allows us to construct the Roe matrix that we are looking for:
∂
2
qq
˜
E
 +
1
2 =
1
∫
0 ∂
2
qq
E
˜
ψ (s )
ds =:
∂
2
pp
˜
L
 +
1
2
−1
, (44)
which satises (42) and allows to rewrite (38) , after substitution of (39) , as
d
dt
E

+
F
 +
1
2
d −F
 −1
2
d
x
= p

·P

−1
2
 +
1
2
q
 +1
−q

x
·∂
2
qq
˜
E
 +
1
2
q
 +1
−q

x −1
2
 −1
2
q

−q
 −1
x
·∂
2
qq
˜
E
 −1
2
q

−q
 −1
x
. (45)
The only equation in the governing PDE system (1) that admits the addition of a non-negative production term on the right
hand side is obviously the entropy inequality given by (1c) . Hence, in order to obtain discrete total energy conservation
(1f) as a consequence of all other equations, we need to balance all contributions due to the discretization of the dissipative
terms by dening the production term P

= (0 , 0 , 
, 0 , 0 )
T as
p

·P

= T


=
1
2
 +
1
2
q
 +
1
2
x
·∂
2
qq
˜
E
 +
1
2
q
 +
1
2
x
+
1
2
 −1
2
q
 −1
2
x·∂
2
qq
˜
E
−1
2
q
 −1
2
x
. (46)
This choice leads to the sought semi-discrete total energy conservation law
d
dt
E

+
F
 +
1
2
d −F
 −1
2
d
x
= 0 . (47)
3.3. Thermodynamically compatible discretization of the remaining terms in 1D
We now take into account also the remaining terms of the governing PDE system (1) , i.e. the red and green terms. The
thermodynamically compatible nite volume scheme in 1D reads
d
dt
q

+
F
 +
1
2 −F
 −1
2
x
+
R
 +
1
2
−+ R
 −1
2
+
x
=
G
 +
1
2 −G
 −1
2
x
+ P

+ S
q

. (48)
According to the detailed derivation provided in Busto et al. [61] , the thermodynamically compatible discretization of the
uctuations R
 +
1
2
−and R
 +
1
2
+
in 1D is
R
 +
1
2
−=
⎛
⎜
⎜
⎜
⎜
⎝
0
σ +
1
2
i 1 −σ
i 1
+ ω +
1
2
i 1 −ω

i 1
1
2
β +1
1 −β
1
1
2
A
 +
1
2
im v
 +1
m −v

m
n
k+
1
2
˜
u
 +
1
2
A A
 +1
ik −A

ik
1
2
J
 +
1
2
m v
 +1
m −v

m
n
k
+
1
2
˜
u
 +
1
2
J J
 +1
k −J

k
+
1
2
T
 +1
−T

n
k ⎞
⎟
⎟
⎟
⎟
⎠
, (49)
and
R
 +
1
2
+
=
⎛
⎜
⎜
⎜
⎜
⎝
0
σ +1
i 1 −σ +
1
2
i 1
+ ω
 +1
i 1 −ω
 +
1
2
i 1
1
2
β +1
1 −β
1
1
2
A
 +
1
2
im v
 +1
m −v

m
n
k+
1
2
˜
u
 +
1
2
A A
 +1
ik −A

ik
1
2
J
 +
1
2
m v
 +1
m −v

m
n
k
+
1
2
˜
u
 +
1
2
J J
 +1
k −J

k
+
1
2
T
 +1
−T

n
k ⎞
⎟
⎟
⎟
⎟
⎠
, (50)
with the normal n = (1 , 0 , 0) for this one-dimensional case, the compatible discretization of the stress tensors
σ +
1
2
ik
=
1
2
A +1
mi
+ A

mi
1
2
α +1
mk
+ α
mk
, ω
 +
1
2
ik
=
1
2
J
 +1
i
+ J

i
1
2
β +1
k
+ β
k
(51)
and
A
 +
1
2
im
=
1
2
A
 +1
im
+ A

im
, ˜
u
 r
A
=
F
 +
1
2
ρE
 +1
3 −E

3
1
2
α +1
ik
+ α
ik
A
 +1
ik −A

ik
, (52)
J
 +
1
2
i
=
1
2
J

i
+ J
 +1
i , ˜
u
 +
1
2
J
=
F
 +
1
2
ρE
 +1
4 −E

4
1
2
β +1
k
+ β
k
J
 +1
k −J

k
, (53)
8

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
while the discrete algebraic source term simply reads
S
q

=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0
0
α
ik
α
ik
θ
1
(τ1
) T

+ β
k
β
k
θ
2
(τ2
) T

−α
ik
θ
1
(τ1
)
−β
i
θ
2
(τ2
) ⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (54)
Following [61] one can check that the uctuations satisfy the compatibility relation with the total energy ux related to the
red terms in (1) :
p

·R
 +
1
2
−+ p
 +1
·R
 +
1
2
+
= F
34 , +1
1 −F
34 ,
1
. (55)
The thermodynamic compatibility of the algebraic source term,
p

·S
q

= 0 , (56)
is obvious, since it is clearly the pointwise discrete analogue of (18) .
3.4. Thermodynamically compatible nite volume scheme in multiple space dimensions
In multiple space dimensions the construction of the thermodynamically compatible semi-discrete cell centered nite
volume scheme is completely analogous to the one shown for the one-dimensional case illustrated in the previous section,
making use of uxes and uctuations in the normal direction across the cell boundaries. In what follows, we will provide
the precise expressions of the nal scheme and prove the cell entropy inequality and the marginal nonlinear stability in the
energy norm, which is a consequence of the discrete thermodynamic compatibility. We consider the spatial control volume
 with circumcenter x

, one of its neighbors r and the common edge ∂ r
, n
 r
= (n
r
1
, n
 r
2
, n
 r
3
)
T being the outward unit
normal vector to the face ∂ r pointing from element  to r
, with the property n
r 
k
= −n
 r
k
and N
 being the set of
neighbors of cell 
. Note that in two space dimensions n
 r
3
= 0 . The mesh spacing in direction k is denoted by x
k
. The
semi-discrete nite volume scheme in multiple space dimensions reads
∂q

∂t
= −1
|

|
∑
r ∈ N

∣∣∂ r
∣∣F
q

, q
r
·n
 r
+ R
q

, q
r
·n
 r
−G
q

, q
r
·n
 r
−P
q

, q
r
+ S (q

) (57)
with the thermodynamically compatible ux in normal direction
F
 r
= F
q

, q
r
·n
 r
= ˜
F
 r
−α r
p
r
−p

=
F
 r
ρ, F
 r
ρv
i
, F
 r
ρS
, 0 , 0
T
, ˜
F
 r
=
1
2
f

k
+ f
r
k
n
 r
k
, (58)
α r
=
F
12 , r
k −F
12 ,
k
+
1
2
f

k
+ f
r
k
·p
r
−p

−p
r
·f
r
k
−p

·f

k
(
p
r
−p

)
2
·n
 r
k
, (59)
the numerical ux for the viscous terms
G
q

, q
r
=  r
q
r
−q

δ r
=  r
q
 r
δ r
, δ r
=
∥∥x
r
−x

∥∥= x
k
n
 r
k
, (60)
and the uctuations and entropy production term related to the viscosity
R
q

, q
r
·n
 r
=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0
σ r
ik −σ
ik
·n
 r
k
+
ω
 r
ik
−ω

ik
·n
 r
k
1
2
βr
k
−β
k
·n
 r
k
1
2
A
r
im
v
r
m
−v

m
n
 r
k
+
1
2
˜
u
 r
A
A
r
ik
−A

ik
1
2
J
 r
m
v
r
m
−v

m
n
 r
k
+
1
2
˜
u
 r
J J
r
k
−J

k
+
1
2
T
r
−T

n
 r
k ⎞
⎟
⎟
⎟
⎟
⎟
⎠
, P
q

, q
r
=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0
0
 r
0
0 ⎞
⎟
⎟
⎟
⎟
⎟
⎠
, (61)
where
σ r
jk
=
1
2
A

i j
+ A
r
i j
1
2
α
ik
+ αr
ik
, ω
 r
ik
=
1
2
J

i
+ J
r
i
1
2
β
k
+ βr
k
, (62)
A
 r
im
=
1
2
A

im
+ A
r
im
, ˜
u
 r
A
=
F
 r
ρE
r
3
−E

3
1
2
α
ik
+ αr
ik
A
r
ik
−A

ik
, (63)
9

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
J
 r
i
=
1
2
J

i
+ J
r
i
, ˜
u
 r
J
=
F
 r
ρE
r
4
−E

4
1
2
β
k
+ βr
k
J
r
k
−J
k
, (64)
 r
=
1
2
 r
q
 r
T

·∂
2
qq
˜
E
 r
q
 r
δ r
, T

= ρ
γ−1
(
γ−1
)
c
v
e
S

c
v
. (65)
The Roe matrix of the Hessian of the energy potential reads
∂
2
qq
˜
E
 r
=
1
∫
0 ∂
2
qq
E
˜
ψ (s )
ds =:
∂
2
pp
˜
L
 r
−1
, (66)
and is based on the simple straight line segment path in q variables
˜
ψ (s ) = q

+ s
q
r
−q

, 0 ≤s ≤1 . (67)
By construction, the Roe matrix ∂
2
qq
˜
E
 r satises the Roe property
∂
2
qq
˜
E
 r
·q
r
−q

=
p
r
−p

, (68)
which allows to convert jumps in the conservative variables into jumps of the thermodynamic dual variables, i.e. into jumps
in the main eld. Throughout this paper the path integral in (66) is calculated numerically with a Gauss-Legendre quadrature
formula using three quadrature points. However, we stress that in the theoretical analysis of the schemes presented later, we
assume that the quadrature is exact . For a detailed analysis on the inuence of the quadrature error the reader is referred to
[61] . Finally, the algebraic source terms are dened according to (54) . It is easy to check that, by construction, the numerical
ux, the uctuations and the source terms verify the compatibility conditions:
p

·F
 r
−f

k
n
 r
k
+ p
r·f
r
k
n
 r
k −F
 r
=
F
12 , r
k −F
12 ,
k n
 r
k
;(69)
p

·R
q

, q
r
·n
 r
+ p
r
·R
q
r
, q

·n
r 
=
F
34 , r
k −F
34 ,
k n
 r
k
;(70)
p

·S
q

= 0 . (71)
It is also obvious that the following identity holds,
∑
r ∈ N

∣∣∂ r
∣∣n
 r
=0 ,(72)
since the integral of the normal vector over a closed surface vanishes.
Theorem 3.1 (Cell entropy inequality) . The HTC FV scheme (57) satises the following cell entropy inequality:
∂ ρS

∂t
+
1
|

|
∑
r ∈ N

∣∣ r
∣∣F
ρS
q

, q
r
n
 r
+
1
2
βr
k
+ β
k
·n
 r
k −G
ρS
q

, q
r
·n
 r
≥0 . (73)
Proof. Taking the discrete equation for the entropy density from (57) , substituting (61), (65) and (54) , and using the fact
that the integral of the normal vector over a closed surface vanishes
∑
r ∈ N

∣∣∂ r
∣∣n
 r
k
= 0
multiplied by β
k
, we obtain
∂
(
ρS
)

∂t
+
1
|

|
∑
r ∈ N

∣∣∂ r
∣∣F
ρS
q

, q
r
·n
 r
+
1
2
βr
k
+ β
k
·n
 r
k −G
ρS
q

, q
r
·n
 r
=
1
|

|
∑
r ∈ N

∣∣∂ r
∣∣1
2
 r
q
 r
T

·∂
2
qq
˜
E
 r
q
 r
δ r
+ α
ik
α
ik
θ
1
(τ1
) T

+ β
k
β
k
θ
2
(τ2
) T
 ≥0 ,
where the positivity of the right hand side is obtained thanks to θ
1
, θ
2
and T
 being positive and due to the positive
semi-deniteness of the Hessian ∂
2
qq
E. §
10

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Theorem 3.2 (Nonlinear stability in the energy norm) . The scheme (57) with the numerical ux, the viscous and source terms
dened in (58) –(65) is nonlinearly stable in the energy norm in the sense that, for vanishing boundary uxes, we have
∫
∂E
∂t
dx = 0 . (74)
Proof. To demonstrate non-linear stability in the energy norm, we rst compute the semi-discrete energy conservation law
resulting from the dot product of the thermodynamically dual variables p
 with the semi-discrete scheme (57) :
p

·∂q

∂t
= −1
|

|
∑
r ∈ N

∣∣∂ r
∣∣p

·F
q

, q
r
·n
 r
+p

·R
q

, q
r
·n
 r
+
1
|

|
∑
r ∈ N

∣∣∂ r
∣∣p

·G
q

, q
r
·n
 r
+p

·P
q

, q
r
+ p

·S (q

) .
From (71) the source terms in the former equation cancel. Moreover, adding and subtracting the terms corresponding to
1
2
p
r
·R
q
r
, q

·n
r 
,
1
2
p
r
·F
 r and
1
2
p
r
·G
q

, q
r
·n
 r
, we get
∂E

∂t
= −1
|

|
∑
r ∈ N

∣∣∂ r
∣∣1
2
p

+ p
r
·F
 r
−1
|

|
∑
r ∈ N

∣∣∂ r
∣∣1
2
p

−p
r
·F
 r
−1
|

|
∑
r ∈ N

∣∣∂ r
∣∣1
2
p

·R
q

, q
r
·n
 r
+
1
2
p
r
·R
q
r
, q

·n
r 
−1
|

|
∑
r ∈ N

∣∣∂ r
∣∣1
2
p

·R
q

, q
r
·n
 r
−1
2
p
r
·R
q
r
, q

·n
r 
+
1
|

|
∑
r ∈ N

∣∣∂ r
∣∣1
2p

+ p
r
·G
q

, q
r
·n
 r
+
1
2
p

−p
r
·G
q

, q
r
·n
 r
+ p

·P
q

,q
r
.
The compatibility conditions (69) and (70) and n
r 
= −n
 r yield
∂E

∂t
= −1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣F
12 , r
k −F
12 ,
k n
 r
k
−1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣p

·f

k
−p
r
·f
r
k
n
 r
k −1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣p

+ p
r
·F
 r
−1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣F
34 , r
k −F
34 ,
k n
 r
k −1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣p

·R
q

, q
r
+ p
r
·R
q
r
, q

·n
 r
+
1
|

|
∑
r ∈ N

∣∣∂ r
∣∣1
2p

+ p
r
·G
q

, q
r
·n
 r
+
1
2
p

−p
r
·G
q

, q
r
·n
 r
+ p

·P
q

,q
r
.
Adding p

·f

k
multiplied by (72) and using (60) and (65) , we get
∂E

∂t
= −1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣F
r
k
+ F

k
n
 r
k
+
1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣p
r
·f
r
k
+ p

·f

k
n
 r
k
−1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣p

+ p
r
·F
 r
−1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣p

·R
q

, q
r
+ p
r
·R
q
r
, q

·n
 r
+
1
|

|
∑
r ∈ N

∣∣∂ r
∣∣1
2
p

+ p
r
·G
q

, q
r
·n
 r
+
1
2
p

−p
r
· r
q
 r
δ r
+
1
2
 r
q
 r
·∂
2
qq
˜
E
 r
q
 r
δ r ).
Since the last two terms cancel due to the Roe property (68) of the Roe matrix of the Hessian of the total energy potential,
we nally obtain the discrete energy conservation law in terms of a sum of numerical total energy uxes as
∂E

∂t
= −1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣F
r
k
+ F

k
n
 r
k
+
1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣p
r
·f
r
k
+ p

·f

k
n
 r
k
−1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣p

+ p
r
·F
 r
−G
q

, q
r
·n
 r
−1
2
|

|
∑
r ∈ N

∣∣∂ r
∣∣p

·R
q

, q
r
+ p
r
·R
q
r
, q

·n
 r
.
11

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
As a consequence we obtain the sought nonlinear stability in the energy norm
∫
∂E
∂t
dx =
∑
 ∂E

∂t
= 0
by assuming the uxes to be zero at the boundary and after applying the telescopic sum of the uxes at the interfaces. §
4. Thermodynamically compatible discontinuous Galerkin nite element schemes of type I
The rst kind of HTC DG schemes proposed in this manuscript is built by simply using the numerical HTC ux for the
inviscid part developed for the HTC nite volume scheme described previously. Like in the FV case, we will rst detail the
derivation of such DG type I scheme in 1D and we will then move to the multi-dimensional case.
4.1. One dimensional case
To introduce a rst HTC DG scheme for the discretization of (1) , called HTC DG scheme of type I in the following, we
start dening a one dimensional cell as T

= [ x
 −1
2
, x
 +
1
2] . We also introduce a DG approximation space with spatial basis
functions, ϕ
m
(x ) , given by the Lagrange interpolation polynomials of degree Npassing through the N + 1 Gauss-Legendre
quadrature points in each element and which are allowed to jump at the element boundaries. Thus, we assume that the
solution q (x, t) can be expressed as a linear combination of a set of spatial basis functions as
q
h
(x, t) =
N
∑
m =0
ϕ
m
(x )
ˆ
q

m
(t) , (75)
where ˆ
q

m
(t) are the time dependent degrees of freedom and Ndenotes the polynomial approximation degree. Accordingly,
one could also write the thermodynamic dual variables, p , as a linear combination of basis functions as
p
h
(x, t) =
N
∑
m =0
ϕ
m
(x )
ˆ
p
m
(t ) , ˆ
p
m
(t ) = p
ˆ
q

m
(t)
, (76)
as well as the total energy, E,
E
h
(x, t) =
N
∑
m =0
ϕ
m
(x )
ˆ
E

m
(t ) , ˆ
E

m
(t ) = E
ˆ
q

m
(t)
. (77)
Besides, in the following, we will denote
ϕ
 −1
2
k
= ϕ
k
x
 −1
2
+ , ϕ
 +
1
2
k
= ϕ
k
x
 +
1
2
−, ∂
x
ϕ
 −1
2
k
= ∂
x
ϕ
k
x
 −1
2
+ , ∂
x
ϕ
 +
1
2
k
= ∂
x
ϕ
k
x
 +
1
2
−.
We now substitute (75) in (1a) –(1e) multiply by a test function ϕk
, integrate on a cell T
 and apply integration by parts to
the convective and viscous terms obtaining
x
 +
1
2
∫
x
 −1
2 ϕ
k
∂
t
q
h
dx + ϕ
 +
1
2
k
F
 +
1
2 −ϕ
 −1
2
k
F
 −1
2 −x
 +
1
2
−∫
x
 −1
2
+ ∂
x
ϕ
k
f
1
(q
h
) dx
+ ϕ
 +
1
2
k
R
 +
1
2
−+ ϕ
 −1
2
k
R
 −1
2
+
+
x
 +
1
2
−∫
x
 −1
2
+ ϕ
k
(
∂
x
h
1
(q
h
) + B
1
(q
h
) ∂
x
q
h
)
d x =
x
 +
1
2
∫
x
 −1
2 ϕ
k
S (q
h
) d x
+ ϕ
 +
1
2
k
G
 +
1
2 −ϕ
 −1
2
k
G
 −1
2 + ∂
x
ϕ
 +
1
2
k
V
 +
1
2 + ∂
x
ϕ
 −1
2
k
V
 −1
2 −x
 +
1
2
−∫
x
 −1
2
+ ∂
x
ϕ
k
∂
x
q
h
dx + P
k
. (78)
In the former equation, the thermodynamically compatible ux is chosen exactly as in the nite volume case (27) and
therefore reads
F
 +
1
2 = F
q
 +
1
2
−, q
 +
1
2
+ = ˜
F
 +
1
2 −α +
1
2
p
 +1
−−p

+
=
F
 +
1
2
ρ, F
 +
1
2
ρv
i
, F
 +
1
2
ρS
, 0 , 0
, (79)
with the central ux, which is in general not compatible,
˜
F
 +
1
2 =
1
2
f
1
q
 +
1
2
−+ f
1
q
 +
1
2
+ , (80)
12

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
and the scalar correction factor
α +
1
2 =
F
12
1 q
 +
1
2
−−F
12
1 q
 +
1
2
+ +
˜
F
 +
1
2 ·p
 +
1
2
+ −p
 +
1
2
−−p
 +
1
2
+
·f
 +
1
2
+ −p
 +
1
2
−·f
 +
1
2
−p
 +
1
2
+ −p
 +
1
2
−2
. (81)
In the case of vanishing denominator, the correction factor is set to zero. The numerical ux can be rewritten in terms of
uctuations as follows,
D
 +
1
2
−= F
 +
1
2 −f
q
+
1
2
−and D
 +
1
2
+
= f
q
 +
1
2
+ −F
 +
1
2
. (82)
Thanks to the particular construction of the numerical ux, the uctuations satisfy the following compatibility relation,
corresponding to (15) on the discrete level:
p
 +
1
2
−·D
 +
1
2
−+p
 +
1
2
+
·D
 +
1
2
+
= F
12
1 q
 +
1
2
+ −F
12
1 q
 +
1
2
−, (83)
similar to the compatibility relation (25) of the nite volume scheme presented previously. We stress again that our ther-
modynamically compatible numerical ux does not rely on an underlying Godunov parametrization of the physical ux as
f
k
= ∂
p
(v
k
L ) in terms of a generating potential L , unlike the schemes presented in Busto et al. [59–62] . Moreover, we have
the viscous numerical ux G
 +
1
2 =
G
 +
1
2
ρ, G
 +
1
2
ρv
i
, G +
1
2
ρS
, G
 +
1
2
A
ik
, G
 +
1
2
J
k given by
G
 +
1
2 =
1
2
∂
2
pp
˜
L
 +
1
2
∂
x
p
 +
1
2
−+ ∂
x
p
 +
1
2
+ + η +
1
2
q
 +
1
2
+ −q
 +
1
2
−, η +
1
2 =
1
2
s
 +
1
2
max
+
2 N + 1
x (84)
following the seminal ideas of Gassner et al. [67] . The jump terms related to the viscous terms read
V
 +
1
2 =
1
2
q
 +
1
2
+ −q
 +
1
2
−, (85)
and the discrete entropy production term related to the viscous terms P
k
= (0 , 0 , k
, 0 , 0 )
T with
k
=
x
 +
1
2
∫
x
 −1
2 ϕ
k
T
∂
x
q
h
·∂
2
qq
E ∂
x
q
h
dx + ϕ
 −1
2
k η −1
2
2 T
 −1
2
+ q
 −1
2
+ −q
 −1
2
−·∂
2
qq
˜
E
 −1
2
q
 −1
2
+ −q
 −1
2
−+ ϕ
 +
1
2
k η +
1
2
2 T
 +
1
2
−q
 +
1
2
+ −q
 +
1
2
−·∂
2
qq
˜
E
 +
1
2
q
 +
1
2
+ −q
 +
1
2
−. (86)
Alternatively, to ease calculations in the proofs shown later, we can also use a one sided production term, i.e.,
 +
1
2
k
=
x
 +
1
2
∫
x
 −1
2 ϕ
k
T
∂
x
q
h
·∂
2
qq
E ∂
x
q
h
dx + ϕ
 +
1
2
k η +
1
2
T
 +
1
2
−q
 +
1
2
+ −q
 +
1
2
−·∂
2
qq
˜
E
i +
1
2
q
 +
1
2
+ −q
 +
1
2
−. (87)
Finally, the uctuations R
 +
1
2
±=
R
 +
1
2
ρ, R
 +
1
2
ρv
i, ±, R
 +
1
2
ρS
, R
 +
1
2
A
ik
, R
 +
1
2
J
k are given by
R
 +
1
2
−=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0
σ +
1
2
i 1 −σ +
1
2
, −i 1
+ ω
 +
1
2
i 1 −ω
 +
1
2
, −i 1
1
2
β +
1
2
, +
1 −β +
1
2
, −1 1
2
A
 +
1
2
im v
 +
1
2
, +
m −v
 +
1
2
, −m n
k
+
1
2
˜
u
 +
1
2
A A
 +
1
2
, +
ik −A
 +
1
2
, −ik 1
2
J
 +
1
2
m v
 +
1
2
, +
m −v
 +
1
2
, −m n
k
+1
2
˜
u
 +
1
2
J J
 +
1
2
, +
k −J
 +1
2
, −k +
1
2
T
 +
1
2
, +
−T
 +
1
2
, −n
k ⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(88)
and
R
 +
1
2
+
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0
σ +
1
2
, +
i 1 −σ +
1
2
i 1
+ ω
 +
1
2
, +
i 1 −ω
 +
1
2
i 1
1
2
β +
1
2
, +
1 −β +
1
2
, −1 1
2
A
 +
1
2
im v
 +
1
2
, +
m −v
 +
1
2
, −m n
k
+
1
2
˜
u
 +
1
2
A A
 +
1
2
, +
ik −A
 +
1
2
, −ik 1
2
J
 +
1
2
m v
 +
1
2
, +
m −v
 +
1
2
, −m n
k
+1
2
˜
u
 +
1
2
J J
 +
1
2
, +
k −J
 +1
2
, −k +
1
2
T
 +
1
2
, +
−T
 +
1
2
, −n
k ⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(89)
13

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
with the compatible discretization of the stress tensors
σ +
1
2
ik
=
1
2
A +
1
2
, +
mi
+ A
 +
1
2
, −mi 1
2
α +
1
2
, +
mk
+ α +
1
2
, −mk , ω
 +
1
2
ik
=
1
2
J
 +
1
2
, +
i
+ J
 +
1
2
, −i 1
2
β +
1
2
, +
k
+ β +
1
2
, −k , (90)
where
A
 +
1
2
im
=
1
2
A
 +
1
2
, +
im
+ A
 +
1
2
, −im , ˜
u
 +
1
2
A
=
F
 +
1
2
ρE
 +
1
2
, +
3 −E
 +
1
2
, −3 1
2
α +
1
2
, +
ik
+ α +
1
2
, −ik A
 +
1
2
, +
ik −A
 +
1
2
, −ik , (91)
J
 +
1
2
i
=
1
2
J
 +
1
2
, +
i
+ J
 +
1
2
, −i , ˜
u
 +
1
2
J
=
F
 +
1
2
ρE
 +
1
2
, +
4 −E
 +
1
2
, −4 1
2
β +
1
2
, +
k
+ β +
1
2
, −k J
 +
1
2
, +
k −J
 +
1
2
, −k , (92)
and n = (1 , 0 , 0) in the 1D case. As for the nite volume scheme, the uctuations satisfy the compatibility relation (16) at
the discrete level:
p
 +
1
2
−·R
 +
1
2
−+ p
 +
1
2
+
·R
 +
1
2
+
= F
34
1 q
 +
1
2
+ −F
34
1 q
 +
1
2
−. (93)
Theorem 4.1 (Cell entropy inequality) . The HTC DG scheme of type I (78) satises the following cell entropy inequality:
x
 +
1
2
∫
x
 −1
2 ∂
t
(ρS)
h
dx + F
 +
1
2
ρS −F
 −1
2
ρS
+ β +
1
2
1 −β −1
2
2 −G
 +
1
2
ρS
+ G
 −1
2
ρS ≥0 . (94)
Proof. Choosing as test function the constant function ϕ
k
= 1 inside a cell, we obtain the evolution equation for the cell
average of the entropy density ρSaccording to the HTC DG scheme of type I (78) as follows:
x
 +
1
2
∫
x
 −1
2 ∂
t
(ρS)
h
dx + F
 +
1
2
ρS −F
 −1
2
ρS
+ R
 +
1
2
ρS
+ R
 −1
2
ρS
+
x
 +
1
2
−∫x
 −1
2
+ ∂
x
β1
d x =
x
 +
1
2
∫
x
−1
2 πd x
+ G
 +
1
2
ρS −G
 −1
2
ρS
+
x
 +
1
2
∫
x
 −1
2 T
∂
x
q
h
·∂
2
qq
E ∂
x
q
h
dx + η +
1
2
T
 +
1
2
−q
 +
1
2
+ −q
 +
1
2
−·∂
2
qq
˜
E
i +
1
2
q
 +
1
2
+ −q
 +
1
2
−, (95)
where we have introduced the abbreviation
π= αik
αik
θ1
(τ1
) T
+ βi
βi
θ2
(τ2
) T ≥0 . (96)
Since the red terms reduce to a ux difference
R
 +
1
2
ρS
+ R
 −1
2
ρS
+
x
 +
1
2
−∫
x
 −1
2
+ ∂
x
β1
dx =
1
2
β +
1
2
, +
1 −β +
1
2
, −1 +
1
2
β −1
2
, +
1 −β −1
2
, −1 + β +
1
2
, −1 −β −1
2
, +
1
=
1
2
β +
1
2
, +
1
+ β +
1
2
, −1 −1
2
β −1
2
, +
1
+ β −1
2
, −1 := β +
1
2
1 −β −1
2
1
we can rewrite (95) as
x
 +
1
2
∫
x
 −1
2 ∂
t
(ρS)
h
dx + F
 +
1
2
ρS −F
 −1
2
ρS
+ β +
1
2
1 −β −1
2
1 −G
 +
1
2
ρS
+ G
 −1
2
ρS
=
x
 +
1
2
∫
x
 −1
2 πdx +
+
x
 +
1
2
∫
x
 −1
2 T
∂
x
q
h
·∂
2
qq
E ∂
x
q
h
dx + η +
1
2
T
 +
1
2
−q
 +
1
2
+ −q
 +
1
2
−·∂
2
qq
˜
E
i +
1
2
q
 +
1
2
+ −q
 +
1
2
−≥0 , (97)
which concludes the proof, since we are assuming the Hessian of the total energy potential to be at least positive semi-
denite and all the terms on the right hand side of (97) are quadratic forms and are thus non-negative. §
14

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Theorem 4.2 (Nonlinear stability in the energy norm) . The scheme (78) with the numerical ux, the uctuations, the viscous
ux and the source terms dened in (79) –(89) is nonlinearly stable in the energy norm in the sense that, for vanishing boundary
uxes, we have
∫
∂E
∂t
dx = 0 . (98)
Proof. To show that the proposed scheme is thermodynamically compatible, we take the dot product of p
h
with (78) , apply
(77) and integrate by parts the volume integral of the physical ux, getting
x
 +
1
2
∫
x
 −1
2
p
h
·∂
tq
h
dx + p
 +
1
2
−·F
 +
1
2 −p
 −1
2
+
·F
 −1
2 −p
 +
1
2
−·f
 +
1
2
−−p
 −1
2
+
·f
 −1
2
+ + p
 +
1
2
−·R
 +
1
2
−+ p
 −1
2
+
·R
 −1
2
+
+
x
 +
1
2
−∫
x
 −1
2
+
p
h
·(
∂
x
f + ∂
x
h
1
(q
h
) + B
1
(q
h
) ∂
x
q
h
)
dx =
x
 +
1
2
∫
x
 −1
2
p
h
·S (q
h
) dx
+ p
 +
1
2
−·G
 +
1
2 −p
 −1
2
+
·G
 −1
2 + ∂
x
p
 +
1
2
−·V
 +
1
2 + ∂
x
p
 −1
2
+
·V
 −1
2 −x +
1
2
−∫
x
 −1
2
+ ∂
x
p
h
·∂
x
q
h
dx + P
k
·ˆ
p

k
. (99)
Next, adding and subtracting p
 +
1
2
+
·F
 +
1
2
, p
 +
1
2
+
·f
 +
1
2
+
and p
 +
1
2
+
·R
 +
1
2
+
and rearranging terms yields
x
 +
1
2
∫
x
 −1
2 ∂
t
E
h
dx +
p
 +
1
2
+
·F
 +
1
2 −f
 +
1
2
+ −p
 −1
2
+
·F
 −1
2 −f
 −1
2
+ +
p
 −1
2
+
·R
 −1
2
+ −p
 +
1
2
+·R
 +
1
2
+ + p
 +
1
2
−·F
 +
1
2 −f
 +
1
2
−+ p
 +
1
2
+
·f
 +
1
2
+ −F
 +
1
2
+
p
 +
1
2
−·R
 +
1
2
−+ p
 +
1
2
+
·R
 +
1
2
+ +
x
 +
1
2
−∫
x
 −1
2
+
p
h
·(
∂
x
f + ∂
x
h
1
(q
h
) + B
1
(q
h
) ∂
x
q
h
)
dx
= p
 +
1
2
−·G
 +
1
2−p
 −1
2
+
·G
 −1
2 + ∂
x
p
 +
1
2
−·V
 +
1
2 + ∂
x
p
 −1
2
+
·V
 −1
2
−x
 +
1
2
−∫
x
 −1
2
+ ∂
x
p
h
·∂
x
q
h
dx + P
k
·ˆ
p

k
+
x
 +
1
2
∫
x
 −1
2
p
h
·S (q
h
) dx.
Applying the compatibility Eqs. (83) and (93) and using (14) leads to
x
 +
1
2
∫
x
 −1
2 ∂
t
E
h
dx +
p
+1
2
+
·F
 +
1
2 −f
+1
2
+ −p
−1
2
+
·F
 −1
2 −f
−1
2
+ +
p
−1
2
+
·R
−1
2
+ −p
+1
2
+·R
+1
2
+ + F
1
q
 +
1
2
+ −F
1
q
 +
1
2
−+ F
1
q
 +
1
2
−−F
1
q
 −1
2
+ = p
 +
1
2
−·G
 +
1
2−p
 −1
2
+
·G
 −1
2 + ∂
x
p
 +
1
2
−·V
 +
1
2 + ∂
x
p
 −1
2
+
·V
 −1
2
−x
 +
1
2
−∫
x
 −1
2
+ ∂
x
p
h
·∂
x
q
h
dx + P
k
·ˆ
p

k
+
x
 +
1
2
∫
x
 −1
2
p
h
·S (q
h
) dx
which, taking into account (18) , is equivalent to
x
 +
1
2
∫
x
 −1
2 ∂
t
E
h
dx +

F
1
q
 +
1
2
+ + p
 +
1
2
+
·F
 +
1
2 −f
 +
1
2
+ −R
 +
1
2
+ −F
1
q
 −1
2
+ −p
 −1
2
+
·F
 −1
2 −f
 −1
2
+ −R
 −1
2
+ 
15

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
= p
 +
1
2
−·G
 +
1
2 −p
 −1
2
+
·G
 −1
2 + ∂
x
p
 +
1
2
−·V
 +
1
2 + ∂
x
p
 −1
2
+
·V
 −1
2 −x
 +
1
2
−∫
x
 −1
2
+ ∂
x
p
h
·∂
x
q
h
dx + P
k
·ˆ
p

k
.
We now focus on the terms related to dissipation. Adding and subtracting p
 +
1
2
+
G
 +
1
2 and ∂p
 +
1
2
+
V
 +
1
2 we get
x
 +
1
2
∫
x
 −1
2 ∂
t
E
h
dx +

F
1
q
 +
1
2
+ + p
 +
1
2
+
·F
 +
1
2 −f
 +
1
2
+ −R
 +
1
2
+ −F
1
q
 −1
2
+ −p
 −1
2
+
·F
 −1
2 −f
 −1
2
+ −R
 −1
2
+ 
= p
 +
1
2
+
·G
 +
1
2 −p
 −1
2
+
·G
 −1
2 + ∂
x
p
 −1
2
+
·V
 −1
2 −∂
x
p
 +
1
2
+
·V
 +
1
2
+
p
 +
1
2
−−p
 +
1
2
+ ·G
 +
1
2 +
∂
x
p
 +
1
2
−+ ∂
x
p
 +
1
2
+ ·V
 +
1
2 −x
 +
1
2
−∫
x
 −1
2
+ ∂
x
p
h
·∂
x
q
h
dx + P
k
·ˆ
p

k
.
The rst four terms related to the viscosity are the sought viscous terms that lead to the numerical viscosity ux in the
energy equation, while the remaining terms vanish:
p
 +
1
2
−−p
 +
1
2
+ ·G
 +
1
2 +
∂
x
p
 +
1
2
−+ ∂
x
p
 +
1
2
+ ·V
 +
1
2 −x
 +
1
2
−∫
x
 −1
2
+ ∂
x
p
h
·∂
x
q
h
dx + P
k
·ˆ
p

k
= −p
 +
1
2
+ −p
 +
1
2
−·1
2
∂
2
pp
˜
L
 +
1
2
∂
x
p
 +
1
2
−+ ∂
x
p
 +
1
2
+ + η +
1
2
q
 +
1
2
+ −q
 +
1
2
−+
∂
x
p
 +
1
2
−+ ∂
x
p
 +
1
2
+ ·1
2
q
 +
1
2
+ −q
 +
1
2
−−x
 +
1
2
−∫
x
 −1
2
+ ∂
x
p
h
·∂
x
q
h
dx + P
k
·ˆ
p

k
= 0
since due to the Roe property of matrix ∂
2
pp
˜
L
 +
1
2 and by the denition of the production term, we have
−p
 +
1
2
+ −p
 +
1
2
−·η +
1
2
q
 +
1
2
+ −q
 +
1
2
−−x
 +
1
2
−∫
x
 −1
2
+ ∂
x
p
h
·∂
x
q
h
dx +
x
 +
1
2
∫
x
 −1
2 ∂
x
q
h
·∂
2
qq
E ∂
x
q
h
dx
+ η +
1
2
q
 +
1
2
+ −q
 +
1
2
−·∂
2
qq
˜
E
i +
1
2
q
 +
1
2
+ −q
 +
1
2
−= 0 .
Finally, we get the discrete total energy conservation law:
x
 +
1
2
∫
x
 −1
2
∂
t
E
h
dx +

F
1
q
 +
1
2
+ + p
 +
1
2
+
·F
 +
1
2 −f
 +
1
2
+ −R
 +
1
2
+ −F
1
q
 −1
2
+ −p
 −1
2
+
·F
 −1
2 −f
 −1
2
+ −R
 −1
2
+ 
= p
 +
1
2
+
·G
 +
1
2 −p
 −1
2
+
·G
 −1
2 + ∂
xp
 −1
2
+
·V
 −1
2 −∂
x
p
 +
1
2
+
·V
 +
1
2
.
Summing over all elements and assuming the boundary uxes to vanish, we get the sought result
∫
∂
t
E
h
dx =
∑

x
 +
1
2
∫
x
 −1
2 ∂
t
E
h
dx = 0 ,
since the internal uxes cancel. §
4.2. HTC DG scheme of type I in multiple space dimensions
For the 2D case we dene a Cartesian cell as T
i
= [ x
i
1
−1
2
1
, x
i
1
+
1
2
1
] ×[ x
i
2
−1
2
2
, x
i
2
+
1
2
2
] , while in 3D it reads T
i
= [ x
i
1
−1
2
1
, x
i
1
+
1
2
1
] ×
[ x
i
2
−1
2
2
, x
i
2
+
1
2
2
] ×[ x
i
3
−1
2
3
, x
i
3
+
1
2
3
] with multi-index i and we assume that the solution q (x , t) can be expressed as a linear com-
16

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
bination of a set of spatial basis functions ϕ
m as
q
h
(x , t) =
N
∑
m =0
ϕ
m
(x )
ˆ
q
i
m
(t) ,
where ˆ
q
i
m
(t) are the time dependent degrees of freedom and N = (N + 1)
d is the total number of degrees of freedom in
dspace dimensions, related to the polynomial approximation degree. Moreover, we consider nodal basis functions com-
puted from the Lagrange interpolation polynomials passing through the Gauss-Legendre quadrature points. Like in the one-
dimensional case, we now compute the product of (1a) –(1e) by a test function ϕ
k
, we integrate on the spacial control
volume T
i
and we apply integration by parts obtaining
∫
Ti ϕ
k
∂
t
q dx +
∫
∂T
i ϕ
k
F(q

h
, q
r
h
) ·n dS −∫
T
◦i ∂
m
ϕ
k
f
m
(q
h
) dx
+
∫
∂T
i ϕ
k
R (q

h
, q
r
h
) ·n dS +
∫
T
◦i ϕ
k
(
∂
m
h
m
(q
h
) +B
m
(q
h
) ∂
m
q
h
)
d x
=
∫
∂T
i ϕ
k
G(q

h
, q
r
h
) ·n dS +
∫
∂T
i ∂
m
ϕ
k
V(q

h
, q
r
h
) ·n
m
dS −∫
T
◦i ∂
m
ϕ
k
(
∂
m
q
h
)
dx + P
k
+
∫
T
i ϕ
k
S (q
h
) dx . (100)
Let us note that, within this section, q

h
, q
r
h
denote the boundary extrapolated values of the discrete solution at the left
and the right sides of the element boundary, respectively. Then, in the DG scheme of type I we can directly employ the
edge/face-based numerical uxes and uctuations already introduced for the nite volume case (57) –(59) and (61) –(65) . On
the other hand, the viscous uxes are computed according to
G
q

, q
r
·n=
1
2
∂
2
pp
˜
L
 r
∂
k
p

+ ∂
k
p
r
n
k
+ η r
q
r
−q

, η r
=
1
2
s
 r
max
+
2 N + 1
δ r  r
, (101)
the jump terms read
V
q

, q
r
=
1
2
 r
q
r
−q

, (102)
and the discretization of the non-negative entropy production term, P
k
= (0 , 0 , k
, 0 , 0 )
T
, is
k
=
∫
T
◦i ϕ
k
T
∂
k
q
h
·∂
2
qq
E ∂
k
q
h
d x +
∫
∂T
i ϕ
k η r
2 T

q
r
−q

·∂
2
qq
˜
E
 r
q
r
−q

d S. (103)
Theorem 4.3 (Cell entropy inequality) . The HTC DG scheme of type I
(100) satises the following cell entropy inequality:
∫
T
i ∂
t
(ρS)
h
dx +
∫
∂T
i
F
ρS
(q

h
, q
r
h
) ·n dS +
∫
∂T
i
1
2
β
m
+ βr
m
·n
m
dS −∫
∂T
i
G
ρS
(q

h
, q
r
h
) ·n dS ≥0 . (104)
Proof. Setting the test function ϕ
k
= 1 in (100) and notation (96) together with (103) , we have
∫
T
i ∂
t
ρSdx +
∫
∂T
i
F
ρS
(q

h
, q
r
h
) ·n dS +
∫
∂T
i
RρS
(q

h
, q
r
h
) ·n dS
+
∫
T
◦i
(
∂
m
h
m
(q
h
) + B
m
(q
h
) ∂
m
q
h
)
dx −∫
∂T
i
G
ρS
(q

h
, q
r
h
) ·n dS =
∫
T
i πdx
+
∫
T
◦i T
∂
k
q
h
·∂
2
qq
E ∂
k
q
h
dx +
∫
∂T
i η r
2 T

q
r
−q

·∂
2
qq
˜
E
 r
q
r
−q

dS.
Using (61) , we get
17

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
∫
T
i ∂
t
ρSdx +
∫
∂T
i
F
ρS
(q

h
, q
r
h
) ·n dS +
∫
∂T
i
1
2
βr
k
−β
k
·n
k
dS +
∫
T
◦i ∂
m
βm
dx −∫
∂T
i
G
ρS
(q

h
, q
r
h
) ·n dS
=
∫
T
i πdx +
∫
T
◦i T
∂
k
q
h
·∂
2
qq
E ∂
k
q
h
d x +
∫
∂T
i η r
2 T

q
r
−q

·∂
2
qq
˜
E
 r
q
r
−q

d S.
Finally, applying Gauss’ theorem results in
∫
T
i ∂
t
ρSdx +
∫
∂T
i
F
ρS
(q

h
, q
r
h
) ·n dS +
∫
∂T
i
1
2
βr
k
+ β
k
·n
k
dS −∫
∂T
i
G
ρS
(q

h
, q
r
h
) ·n dS
=
∫
T
i πdx +
∫
T
◦i T
∂
k
q
h
·∂
2
qq
E ∂
k
q
h
d x +
∫
∂T
i η r
2 T

q
r
−q

·∂
2
qq
˜
E
 r
q
r
−q

d S ≥0 ,
where the positivity of the right hand side comes from π≥0 and ∂
2
qq
E≥0 . So we have obtained the sought cell entropy
inequality. §
Theorem 4.4 (Nonlinear stability in the energy norm) . The scheme (100) with the ux, viscous and source terms dened in
(57) –(59) , (61) –(65) and (101) –(103) is nonlinearly stable in the energy norm in the sense that, for vanishing boundary uxes,
we have
∫
∂E
∂t
dx = 0 . (105)
Proof. Similarly to what has been done in the one dimensional case we multiply scheme (100) by ˆ
p
i
m
and sum up all
equations, leading to ∫
T
i
p
h
·∂
t
q
h
dx +
∫
∂T
i
p

h
·F(q

h
, q
r
h
) ·n dS −∫
T
◦i ∂
m
p
h
·f
m
(q
h
) dx
+
∫
∂T
i
p

h
·R (q

h
,q
r
h
) ·n dS +
∫
T
◦i
p
h
·(
∂
m
h
m
(q
h
) + B
m
(q
h
) ∂
m
q
h
)
dx
=
∫
∂T
i
p

h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i
∂
m
p

h
·V(q

h
, q
r
h
) ·n
m
dS −∫
T
◦i ∂
m
p
h
·(
∂
m
q
h
)
d x + P
k
·ˆ
p
i
k
+
∫
T
i
p
h
·S (q
h
) d x .
Applying integration by parts, we get ∫
T
i
p
h
·∂
t
q
h
dx +
∫
∂T
i
p

h
·F(q

h
, q
r
h
) ·n −f
m
(q

h
) n
m
dS
+
∫
∂T
i
p

h
·R (q

h
, q
r
h
) ·n dS +
∫
T
◦i
p
h
·(
∂
m
f
m
(q
h
) + ∂
m
h
m
(q
h
) + B
m
(q
h
) ∂
m
q
h
)
dx
=
∫
∂T
i
p

h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i ∂
m
p

h
·V(q

h
, q
r
h
) ·n
m
dS −∫
T
◦i ∂
m
p
h
·(
∂
m
q
h
)
dx + P
k
·ˆ
p
i
k
.
Using (14) , the notation F
 r
= F(q

h
, q
r
h
) ·n and adding and subtracting
1
2 
∂T
i
p
r
h
·f
m
(q
r
h
) n
m
dS, we obtain
∫
T
i ∂
t
E
h
dx +
1
2
∫
∂T
i
p

h
·F
 r
−f
m
(q

h
) n
m
dS +
1
2
∫
∂T
i
p

h
·F
 r
−f
m
(q

h
) n
m
dS
+
1
2
∫
∂T
i
p
r
h
·f
m
(q
r
h
) n
m
−F
 r
dS −1
2
∫∂T
i
p
r
h
·f
m
(q
r
h
) n
m
−F
 r
dS
+
∫
∂T
i
p

h
·R (q

h
, q
r
h
) ·n dS +
∫
T
◦i ∂
m
F
m
dx
=
∫
∂T
i
p

h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i ∂
m
p

h
·V(q

h
, q
r
h
) ·n
m
dS −∫
T
◦i ∂
m
p
h
·(
∂
m
q
h
)
dx + P
k
·ˆ
p
i
k
.
18

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Rearranging terms and adding and subtracting 
∂T
i
1
2
p
r
h
·R (q
r
h
, q

h
) ·(−n ) dS, yields
∫
T
i ∂
t
E
h
dx +
1
2
∫
∂T
i p

h
+ p
r
h
·F
 r
dS −1
2
∫
∂T
i p
r
h
·f
m
(q
r
h
) + p

h
·f
m
(q

h
)
n
m
dS
+
∫
T
◦i ∂
m
F
m
dx +
1
2
∫
∂T
i F
12 , r
m −F
12 ,
m n
m
dS
+
∫
∂T
i
1
2
p

h
·R (q

h
, q
r
h
) −p
r
h
·R (q
r
h
, q

h
)
·n d S +
∫
∂T
i
1
2
p

h
·R (q

h
, q
r
h
) + p
r
h
·R (q
r
h
, q

h
)
·n d S
=
∫
∂T
i
p

h
·G(q

h
, q
r
h
) ·ndS +
∫
∂T
i ∂
m
p
h
·V(q

h
, q
r
h
) ·n
m
dS −∫
T
◦i ∂
m
p
h
·(
∂
m
q
h
)
dx + P
k
·ˆ
p
i
k
.
Applying Gauss’ theorem and taking into account (70) with n
r 
= −n
 r
, we have
∫
T
i ∂
t
E
h
dx +
1
2
∫
∂T
i [p

h
+ p
r
h·F
 r
−p
r
h
·f
m
(q
r
h
) + p

h
·f
m
(q

h
)
n
m
]dS +
∫
∂T
i
F

m
n
m
dS
+
1
2
∫
∂Ti F
12 , r
m −F
12 ,
m n
m
d S +
1
2
∫
∂T
i F
34 , r
m −F
34 ,
m n
m
d S +
∫
∂T
i
1
2
p

h
·R (q

h
, q
r
h
) + p
r
h
·R (q
r
h
, q

h
)
·n d S
=
∫
∂T
i
p

h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i ∂
m
p

h
·V(q

h
, q
r
h
) ·n
m
dS −∫
T
◦i ∂
m
p
h
·(
∂
m
q
h
)
dx + P
k
·ˆ
p
i
k
.
We now put together the contribution relations to the uxes F :
∫
T
i ∂
t
E
h
dx +
1
2
∫
∂T
i [p

h
+ p
r
h
·F
 r
−p
r
h
·f
m
(q
r
h
) + p

h
·f
m
(q

h
)
n
m
]dS +
1
2
∫
∂T
i F
r
m
+ F

m
n
m
dS
+
∫
∂T
i
1
2
p

h
·R (q

h
, q
r
h
) + p
r
h
·R (q
r
h
, q

h
)
·n dS
=
∫
∂T
i
p

h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i ∂
m
p

h
·V(q

h
, q
r
h
) ·n
m
dS −∫
T
◦i∂
m
p
h
·(
∂
m
q
h
)
dx + P
k
·ˆ
p
i
k
. (106)
Finally, we analyse the terms related to dissipation. To this end, we add and subtract
1
2 
∂T
i
p
r
h
·G(q

h
, q
r
h
) ·n dS and
1
2 
∂T
i
∂
m
p
r
h
·V(q

h
, q
r
h
) ·n
m
dS, obtaining
∫
∂T
i
p

h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i ∂
m
p

h
·V(q

h
, q
r
h
) ·n
m
dS −∫
T
◦i ∂
m
p
h
·(
∂
m
q
h
)
dx+ P
k
·ˆ
p
i
k
=
∫
∂T
i
1
2
p

h
+ p
r
h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i
1
2
p

h
−p
r
h
·G(q

h
, q
r
h
) ·n dS
+
∫
∂T
i
1
2
∂
m
p

h
+∂
m
p
r
h
·V(q

h
, q
r
h
) ·n
m
dS +
∫
∂T
i
1
2
∂
m
p

h
−∂
m
p
r
h
·V(q

h
, q
r
h
) ·n
m
dS
−∫
T
◦i ∂
m
ph
·(
∂
m
q
h
)
dx + P
k
·ˆ
p
i
k
.
Taking into account the denition of the dissipative terms (101) and (102) , yields
∫
∂T
i
p

h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i ∂
m
p

h
·V(q

h
, q
r
h
) ·n
m
dS −∫
T
◦i ∂
m
p
h
·(
∂
m
q
h
)
dx + P
k
·ˆ
p
i
k
=
∫
∂T
i
1
2
p

h
+ p
r
h
·G(q

h
, q
r
h
) ·n dS
19

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
−∫
∂T
i
1
2
p
r
h
−p

h·
1
2
 r
∂
2
pp
˜
L
 r
∂
k
p

h
+ ∂
k
p
r
h
n
k
+ η r
q
r
h
−q

h

dS
+
∫
∂T
i
1
2
∂
m
p

h
+ ∂
m
p
r
h
·1
2
 r
q
r
h
−q

h·n
m
dS +
∫
∂T
i
1
2
∂
m
p

h
−∂
m
p
r
h
·V(q

h
, q
r
h
) ·n
m
dS
−∫
T
◦i ∂
m
p
h
·(
∂
m
q
h
)
dx + P
k
·ˆ
p
i
k
.
Introducing the production term, (103) , and using the denition of Roe matrix of the Hessian and which makes the rst
term in the second integral and the third one cancel, we get
∫
∂T
i
p

h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i ∂
m
p

h
·V(q

h
, q
r
h
) ·n
m
dS −∫
T
◦i ∂
m
p
h
·(
∂
m
q
h
)
dx + P
k
·ˆ
p
i
k
=
∫
∂T
i
1
2
p

h
+ p
r
h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i
1
2
∂
m
p

h
−∂
m
p
r
h
·V(q

h
, q
r
h
) ·n
m
dS
−∫
∂T
i
1
2
p
r
h
−p

h
·[η r
q
r
h
−q

h
]dS −∫
T
◦i ∂
m
p
h
·(
∂
m
q
h
)
dx
+
∫
T
◦i ∂
k
q
h
·∂
2
qq
E ∂
k
q
h
dx +
∫
∂T
i
1
2
η r
q
r
−q

·∂
2
qq
˜
E
 r
q
r
−q

dS
=
∫
∂T
i
1
2
p

h
+ p
r
h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i
1
2
∂
m
p

h
−∂
m
p
r
h
·V(q

h
, q
r
h
) ·n
m
dS.
Substituting the uxes obtained into (106) yields the discrete total energy conservation law
∫
T
i ∂
t
E
h
dx +
∫
∂T
i
1
2
[p

h
+ p
r
h
·F
 r
−p
r
h
·f
m
(q
r
h
) + p

h
·f
m
(q

h
)
n
m
]dS
+
∫
∂T
i
1
2
F
r
m
+ F

m
n
m
d S +
∫
∂T
i
1
2
p

h
·R (q

h
, q
r
h
) + p
r
h
·R (q
r
h
, q

h
)
·n d S
=
∫
∂T
i
1
2
p

h
+ p
r
h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i
1
2
∂
m
p

h
−∂
m
p
r
h
·V(q

h
, q
r
h
) ·n
m
dS.
Let us note that, when the states coincide then the second, fourth and sixth integrals above cancel remaining only the third
and fth integrals which correspond to the central part of the total energy ux.
Finally, integrating in the whole domain and assuming vanishing boundary uxes, we obtain nonlinear stability in the
energy norm:
∫
∂E
h
∂t
dx =
∑
T
i ∫
T
i ∂t
E
h
dx = 0 ,
since the sum of all internal uxes cancels. §
5. Thermodynamically compatible discontinuous Galerkin nite element schemes of type II
The second type of HTC DG schemes proposed in this paper ensures the HTC compatibility of the entire inviscid part
of the GPR model (black and red terms) using a element-wise correction, thus making the approach genuinely multi-
dimensional. As before, we start presenting the derivation of the HTC DG type II scheme in 1D and then we provide the
numerical scheme for the 2D case together with the theoretical results proving the nonlinear stability of the scheme in the
energy norm and the verication of a discrete cell entropy inequality.
5.1. One dimensional case
Here we use a genuinely multi-dimensional correction relying entirely on uctuations, similar to the one developed in
the framework of residual distribution (RD) schemes, Abgrall [1] , Abgrall et al. [53] . Again, we assume the discrete solution
20

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
in cell T
i
= [ x
i −1
2
, x
i +
1
2
] to be dened as a sum of spatial basis functions ϕ
l
(x ) and time dependent degrees of freedom ˆ
q
i
l
(t)
q (x, t) =
N
∑
l=0
ϕ
l
(x )
ˆ
q
i
l
(t) , (107)
with Nthe polynomial approximation degree and the nodal basis functions ϕ
l
(x ) given by the Lagrange interpolation poly-
nomials passing through the Gauss–Legendre quadrature points. Hence, by construction, the chosen basis functions ϕ
l
(x )
satisfy the partition of unity property. The semi-discrete DG scheme applied to (12) can be derived by multiplication of
(12) with a spatial test function ϕ
k
(x ) by integrating the term including the divergence of the uxes f and h by parts and
by adding a boundary jump term related to the red terms contained in the non-conservative product,
x
i +
1
2
∫
x
i −1
2 ϕ
k
∂
t
q
h
dx + ϕ
i +
1
2
k ˜
F
i +
1
2 + H
i +
1
2
−ϕ
i −1
2
k ˜
F
i −1
2 + H
i −1
2
−x
i +
1
2
−∫
x
i −1
2
+ ∂
x
ϕ
k
(
f
1
(q
h
) + h
1
(q
h
)
)
dx
+ ϕ
i +
1
2
k
D
i +
1
2 + ϕ
i −1
2
k
D
i −1
2 +
x
i +
1
2
−∫
x
i −1
2
+ ϕ
k
(
B
1
(q
h
) ∂
x
q
h
)
d x =
x
i +
1
2
∫
x
i −1
2 ϕ
k
S (q
h
) d x +
x
i +
1
2
−∫
x
i −1
2
+ ϕ
k
∂
x
(
∂
x
q
h
)
d x + P
k
, (108)
where
˜
F
i +
1
2 = ˜
F
i +
1
2
q
i +
1
2
−, q
i +
1
2
+ =
1
2
f
1
q
i +
1
2
−+ f
1
q
i +
1
2
+ (109)
and
H
i +
1
2 = H
i +
1
2
q
i +
1
2
−, q
i +
1
2
+ =
1
2
h
1
q
i +
1
2
−+ h
1
q
i +
1
2
+ (110)
are two simple central numerical uxes related to the conservative part of the system and
D
i +
1
2 = D
i +
1
2
q
i +
1
2
−, q
i +
1
2
+ =
1
2
B
1
¯
q
i +
1
2
q
i +
1
2
+ −q
i +
1
2
−, ¯
q
i +
1
2 =
1
2
q
i +
1
2
+
+ q
i +
1
2
−, (111)
is a simple central approximation of the jump term related to the non-conservative product. The discretization of the viscous
terms is identical to the one of HTC DG schemes of type I discussed before, hence
xi +
1
2
∫
x
i −1
2 ϕ
k
∂
t
q
h
dx + ϕ
k
(x
i +
1
2
)
˜
F
i +
1
2 + H
i +
1
2
−ϕ
k
(x
i −1
2
)
˜
F
i −1
2 + H
i −1
2
+
+ ϕ
i +
1
2
k
D
i +
1
2 + ϕ
i −1
2
k
D
i −1
2 −x
i +
1
2
−∫
x
i −1
2
+ ∂
x
ϕ
k
(
f
1
(q
h
) + h
1
(q
h
)
)
dx +
xi +
1
2
−∫
x
i −1
2
+ ϕ
k
(
B
1
(q
h
) ∂
x
q
h
)
dx
= ϕ
i +
1
2
k
G
i +
1
2 −ϕ
i −1
2
k
G
i −1
2 + ∂
x
ϕ
i +
1
2
k
V
i +
1
2 + ∂
x
ϕ
i −1
2
k
V
i −1
2 −x
i +
1
2
∫
x
i −1
2 ∂
x
ϕ
k
∂
x
q
h
dx
+ P
k
+
x
i +
1
2
∫
x
i −1
2 ϕ
k
S (q
h
) dx, (112)
with the thermodynamically compatible numerical viscosity ux
G
i +
1
2 =
1
2
∂
2
pp
˜
L
i +
1
2
∂
x
p
i +
1
2
−+ ∂
x
p
i +
1
2
+ −ηi +
1
2
q
i +
1
2
+ −q
i+
1
2
−, ηi +
1
2 =
1
2
s
i +
1
2
max
+
2 N + 1
x , (113)
the jump terms
V
i +
1
2 =
1
2
q
i +
1
2
+ −q
i +
1
2
−, (114)
and the discrete entropy production term related to the viscous terms P
k
= (0 , 0 , k
, 0 , 0 )
T with
k
=
x
i +
1
2
∫
x
i −1
2 ϕ
k
T
∂
x
q
h
·∂
2
qq
E ∂
x
q
h
dx + ϕ
k
(x
i −1
2
) ηi −1
2
2 T
i −1
2
+ q
i −1
2
+ −q
i −1
2
−·∂
2
qq
˜
E
i −1
2
q
i −1
2
+ −q
i −1
2
−21

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
+ ϕ
k
(x
i +
1
2
) ηi +
1
2
2 T
i +
1
2
−q
i +
1
2
+ −q
i +
1
2
−·∂
2
qq
˜
E
i +
1
2
q
i +
1
2
+ −q
i +
1
2
−. (115)
Dening the total uctuation of the DG discretization of the inviscid part of the PDE system (12) in cell T
i
= [ x
i −1
2
, x
i +
1
2
] as
˜
i
k
= ϕ
i +
1
2
k F
i +
1
2 + H
i +
1
2
−ϕ
i −1
2
k F
i −1
2 + H
i −1
2
+ ϕ
i +
1
2
k
D
i +
1
2 + ϕ
i −1
2
k
D
i −1
2
−x
i +
1
2
∫
x
i −1
2 ∂
x
ϕ
k
(
f
1
(q ) + h
1
(q )
)
dx +
x
i +
1
2
−∫
x
i −1
2
+ ϕ
k
(
B
1
(q ) ∂
x
q
)
dx, (116)
and with the thermodynamically compatible discretization of the viscous terms from the DG scheme type I
i
k
= ϕ
i +
1
2
k
G
i +
1
2 −ϕ
i −1
2
k
G
i −1
2 + ∂
x
ϕ
i +
1
2
k
V
i +
1
2 + ∂
x
ϕ
i −1
2
k
V
i −1
2
−x
i +
1
2
∫
x
i −1
2 ∂
x
ϕ
k
∂
x
q
h
dx + P
k
+
x
i +
1
2
∫
x
i −1
2 ϕ
k
S (q
h
) dx, (117)
the DG scheme (108) for cell T
i
can be rewritten in a more compact way as
⎛
⎝
x
i +
1
2
∫
x
i −1
2 ϕ
k
ϕ
l
dx
⎞
⎠
∂
ˆ
q
i
l
∂t
+
˜
i
k
=
˜
i
k
. (118)
The total uctuations related to the inviscid terms
˜
k
are not necessarily compatible with the extra conservation law (20) ,
since no special care was taken in a proper compatible discretization of the numerical uxes and of the jump terms, unlike
in the previous DG schemes of type I. The total uctuations are now corrected according to the following ansatz:
i
k
=
˜
i
k
+ αi
ˆ
p
i
k
−¯
p
i
, ¯
p
i
=
1
N + 1
N
∑
l=0
ˆ
p
i
l
. (119)
It is easy to check that the proposed correction is conservative, i.e. that
N
∑
k =0
i
k
=
N
∑
k =0
˜
i
k
(120)
since obviously
N
∑
k =0
ˆ
p
i
k
−¯
p
i
= 0 . (121)
The scalar correction factor αi for each element T
i
is now simply computed by imposing thermodynamic compatibility with
the extra conservation law as follows:
N
∑
k =0
ˆ
p
i
k
·i
k
=
N
∑
k =0
ˆ
p
i
k
·˜
i
k
+ αi
ˆ
p
i
k
·ˆ
p
i
k
−¯
p
i
= F
i +
1
2
1 −F
i −1
2
1
. (122)
It is easy to check that
N
∑
k =0
ˆ
p
i
k
·ˆ
p
i
k
−¯
p
i
=
N
∑
k =0
ˆ
p
i
k
−¯
p
i
2
≥0 , (123)
hence the discrete compatibility condition (122) allows to calculate the scalar correction factor αi as
αi
=
F
i +
1
2
1 −F
i −1
2
1 −N

k =0
ˆ
p
i
k
·˜
i
k
N

k =0
ˆ
p
i
k
−¯
p
i
2
. (124)
In those cases where the denominator vanishes, we simply set αi
= 0 . The thermodynamically compatible DG scheme of
type II therefore becomes
⎛
⎝
x
i +
1
2
∫
x
i −1
2 ϕ
k
ϕ
l
dx
⎞
⎠
∂
ˆ
q
i
l
∂t
= −˜
i
k
−αi
ˆ
p
i
k
−¯
p
i
+ i
k
, (125)
22

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
with αi given by (124) . We stress that the proposed correction only applies to the inviscid part of the system. The parabolic
viscous terms and the entropy production term must still be discretized in a thermodynamically compatible manner, exactly
as in the case of the DG schemes of type I. Since the correction factor in DG schemes of type II is element-wise, the schemes
are formally identical in one and multiple space dimensions. Also the proofs of entropy inequality and energy conservation
are the same. For this reason, the proofs will be presented only in the multi-dimensional case in the next section.
5.2. Multi-dimensional case
The multi-dimensional extension of the DG schemes of type II is straightforward. The computational domain is sup-
posed to be divided into non-overlapping cells T
i
and the discrete solution reads as
q
h
(x , t) =
N
∑
l=1
ϕ
l
(x )
ˆ
q
i
l
(t) , (126)
where Nis again the polynomial approximation degree, the basis functions ϕ
l
(x ) are assumed to be nodal basis functions
that satisfy the partition of unity property 
l
ϕ
l
(x ) = 1 and N is the number of degrees of freedom per cell. Multiplying
(12) with ϕ
k
(x ) , integration by parts of the ux divergence terms and introducing a numerical ux and jump terms on the
boundary of T
i
leads to ∫
T
i ϕ
k
∂
t
q dx +
∫
∂T
i ϕ
k
F(q

h
, q
r
h
) + H(q

h
, q
r
h
)
·n dS +
∫
∂T
i ϕ
k
D(q

h
, q
r
h
) ·n dS
−∫
T
i ∂
m
ϕ
k
(
f
m
(q
h
) + h
m
(q
h
)
)
dx +
∫
T
i ϕ
k
B
m
(q
h
) ∂
m
q
h
dx
=
∫
∂T
i ϕ
k
G(q

h
, q
r
h
) ·n dS +
∫
∂T
i ∂
m
ϕ
k
V(q

h
, q
r
h
) ·n
m
dS −∫
T
i∂
m
ϕ
k
(
∂
m
q
h
)
dx + P
k
+
∫
T
i ϕ
k
S (q
h
) dx , (127)
with the central uxes in normal direction
F(q

h
, q
r
h
) ·n=
1
2
f
m
q

h
+ f
m
q
r
h
n
m (128)
and
H(q

h
, q
r
h
) ·n =
1
2
h
m
q

h
+ h
m
q
r
h
n
m
, (129)
as in the one-dimensional case, the uctuation
D(q

h
, q
r
h
) ·n =
1
2
(
B
m
n
m
) (
¯
q
)
q
r
h
−q

h
, ¯
q =
1
2
q

h
+ q
r
h
, (130)
the viscous ux
G(q

h
, q
r
h
) ·n =
1
2
∂
2
pp
˜
L
∂
m
p

h
+ ∂
m
p
r
h
n
m
−ηq
r
h
−q

h
, η=
1
2
s
max
+
2 N + 1
δ r , (131)
with s
max
= max (| λk
(q

h
) | , | λk
(q
r
h
) | ) the maximum signal speed at the interface and the jump term
V(q

h
, q
r
h
) =
1
2
q
r
h
−q

h
. (132)
The non-negative entropy production term reads P
k
= (0 , 0 , k
, 0 , 0 )
T with
k
=
∫
Ti ϕ
k
T
∂
m
q
h
·∂
2
qq
E ∂
m
q
h
dx +
∫
∂T
i ϕ
k η2 T

h
q
r
h
−q

h
·∂
2
qq
˜
E
 r
q
r
h
−q

h
dS ≥0 . (133)
Also in the multi-dimensional case, we introduce the total uctuation of the DG discretization related to the inviscid part in
cell Ti
as
˜
i
k
=
∫
∂T
i ϕ
k
F(q

h
, q
r
h
) + H(q

h
, q
r
h
)
·n dS +
∫
∂T
i ϕ
k
D(q

h
, q
r
h
) ·n dS
−∫
T
i ∂
m
ϕ
k
·(
f
m
(q ) + h
m
(q )
)
dx +
∫
T
i ϕ
k
(
B
m
(q ) ∂
m
q
)
dx (134)
and the thermodynamically compatible viscous uctuations
i
k
=
∫
∂T
i ϕ
k
G(q

h
, q
r
h
) ·n dS +
∫
∂T
i ∂
m
ϕ
k
V(q

h
, q
r
h
) ·n
m
dS −∫
T
i ∂
m
ϕ
k
(
∂
m
q
h
)
dx + P
k
+
∫
T
i ϕ
k
S (q
h
) dx , (135)
23

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
which allow to rewrite the previous DG scheme for cell T
i
as
⎛
⎝
∫
T
i ϕ
k
ϕ
l
dx
⎞
⎠
∂
ˆ
q
i
l
∂t
+
˜
i
k
= i
k
. (136)
Note that the inviscid part of scheme (136) is not thermodynamically compatible yet. The total uctuations of the inviscid
part are therefore corrected , as in the one-dimensional case, as
i
k
=
˜
i
k
+ αi
ˆ
p
i
k
−¯
p
i
, ¯
p
i
=
1
N
N
∑
l=1
ˆ
p
i
l
. (137)
In multiple space dimensions the thermodynamic compatibility property with the extra conservation law reads
N
∑
k =0
ˆ
p
i
k·i
k
=
N
∑
k =0
ˆ
p
i
k·˜
i
k
+ αi
ˆ
p
i
k·ˆ
p
i
k−¯
p
i
=
∫
∂T
i
1
2
F

k+ F
r
k
n
k
dS, (138)
which allows to compute the scalar correction factor αi in multiple space dimensions in a way that is totally analogous to
the 1D case:
αi
= ∫
∂T
i
1
2
F

k
+ F
r
k
n
k
dS −N
∑
k =1
ˆ
p
i
k
·˜
i
k
N

k =1
ˆ
p
i
k
−¯
p
i
2
. (139)
We again set αi
= 0 when the denominator vanishes. The nal thermodynamically compatible DG scheme of type II in
multiple space dimensions becomes
⎛
⎝
∫
T
i ϕ
k
ϕ
l
dx
⎞
⎠
∂
ˆ
q
i
l
∂t
= −˜
i
k
−αi
ˆ
p
i
k
−¯
p
i
+ i
k
, (140)
with αi computed according to (139) . Note that the scalar correction factor is computed in a genuinely multi-dimensional
manner, since it couples all degrees of freedom and all thermodynamically dual variables with each other in one single
scalar.
Theorem 5.1 (Cell entropy inequality) . The HTC DG scheme of type II (127) satises the following cell entropy inequality:
∫
T
i ∂
t
(ρS)
h
dx +
∫
∂T
i
F
ρS
(q

h
, q
r
h
) ·n dS +
∫
∂T
i
H
ρS
(q

h
, q
r
h
) ·n dS −∫
∂T
i
G
ρS
(q

h
, q
r
h
) ·n dS ≥0 . (141)
Proof. We start by summing up (140) over k . Since we use a nodal basis that must satisfy the partition of unity property,
N
∑
m =1
ϕ
m
(x ) = 1 , (142)
this corresponds to taking the test function ϕ
k
= 1 . We now consider only the discretization for the entropy density for
which the non-conservative term B
m
(q
h
) ∂
m
q
h
disappears and consider the correction term given in (140) with (139) . We
then have
∫
T
i ∂
t
(ρS) dx +
∫
∂T
i
F
ρS
(q

h
, q
r
h
) + H
ρS
(q

h
, q
r
h
)
·n dS +
∫
∂T
i
D
ρS
(q

h
, q
r
h
) ·n d S =
∫
T
i πd x
+
∫
∂T
i
G(q

h
, q
r
h
) ·n dS +
∫
T
◦i T
∂
k
q
h
·∂
2
qq
E ∂
k
q
h
dx +
∫
∂T
i η r
2 T

q
r
−q

·∂
2
qq
˜
E
 r
q
r
−q

dS
−N
∑
k =1
αi

T
i
k
−¯
T
i
.
Since D
ρS
(q

h
, q
r
h
) ·n = 0 and
N
∑
k =1
αi

T
i
k
−¯
T
i
= αi
N
∑
k =1

T
i
k
−αi
N
∑
k =1
¯
T
i
= αi
N
∑
k =1

T
i
k
−αi
N
∑
k=1
(
1
N
N
∑
l=1
ˆ
T
i
l

= 0
24

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
we get the sought cell entropy inequality (141)
∫
T
i ∂
t
(ρS) dx +
∫
∂T
i
F
ρS
(q

h
, q
r
h
) + H
ρS
(q

h
, q
r
h
)
·n dS −∫
∂T
i
G(q

h
, q
r
h
) ·n dS
=
∫
T
i πdx +
∫
T
◦i T
∂
k
q
h
·∂
2
qq
E ∂
k
q
h
dx +
∫
∂T
i η r
2 T

q
r
−q

·∂
2
qq
˜
E
 r
q
r
−q

dS ≥0 ,
where the right hand side is positive due to π≥0 and because the Hessian of the energy potential is at least positive
semi-denite. §
Theorem 5.2 (Nonlinear stability in the energy norm) . The scheme (127) which takes the form (140)
with the uxes, uctuations, jump, production and source terms given by (54) , (128) –(133) , is nonlinearly stable in the energy
norm in the sense that, for vanishing boundary uxes, we have
∫
∂E
∂t
dx = 0 . (143)
Proof. Multiplying (140) by ˆ
p
i
k
and using (137) yields
ˆ
p
i
k
·⎛
⎝
∫
Ti ϕ
k
ϕ
l
dx
⎞
⎠
∂
ˆ
q
i
l
∂t
= −ˆ
p
i
k
·i
k
+
ˆ
p
i
k
·i
k
,
Using the denition of the discrete solution and the property (138) , we get
∫
T
i
p
h
·∂q
h
∂t
dx +
∫
∂T
i
1
2
F

k
+ F
r
k
n
k
dS = ˆ
p
i
k
·i
k
. (144)
Besides, from the DG schemes of type I, we have
ˆ
p
i
k
·i
k
=
∫
∂T
i
p

h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i ∂
m
p

h
·V(q

h
, q
r
h
) ·n
m
dS −∫
T
◦i ∂
m
p
h
·(
∂
m
q
h
)
dx + P
k
·ˆ
p
i
k
−∫
T
i
p
h
·S (q
h
) dx =
∫
∂T
i
1
2
p

h
+ p
r
h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i
1
2
∂
m
p

h
−∂
m
p
r
h
·V(q

h
, q
r
h
) ·n
m
dS. (145)
Gathering (144) and (145) , we obtain ∫
T
i ∂
t
E
h
dx +
1
2
∫
∂T
i F
r
m
+ F

m
n
m
dS
=
∫
∂T
i
1
2
p

h
+ p
r
h
·G(q

h
, q
r
h
) ·n dS +
∫
∂T
i
1
2
∂
m
p

h
−∂
m
p
r
h
·V(q

h
, q
r
h
) ·n
m
dS.
Finally, integrating over the whole domain and assuming vanishing boundary uxes, we obtain nonlinear stability in the
energy norm also for the DG schemes of type II, (143) . §
We would like to conclude this section with the following important remark. In all proofs above we have so far assumed
the calculation of all integrals to be exact, see also [68] . However, a very important difference between DG schemes of type
I and II is the following: in the DG schemes of type II, the numerical scheme is aware of the quadrature errors in the sense
that all quadrature errors are automatically absorbed into the element-wise correction factor αi
, so that the thermodynamic
compatibility condition (138) always holds, also in the presence of numerical quadrature errors.
6. Numerical results
If not specied otherwise, in all test cases for uids the relaxation time τ1 is computed from the dynamic viscosity μand the shear sound speed c
s
according to the relation μ=
1
6
ρ0
c
2
s
τ1
. If not explicitly stated otherwise we set = 0 , γ= 1 . 4 ,
c
v
= 1 and ρ0
= 1 for all test cases presented. The classical fourth order Runge–Kutta method is used as time integrator for
all test problems, but also any other high order time integrator could be used, such as SSP Runge–Kutta schemes [69] . In all
cases the time step size was chosen small enough to allow for an explicit discretization of the viscous terms and to allow
the semi-discrete framework used in this paper to hold, with Courant numbers ranging from 0.1 to 0.5. For the numerical
results presented in this section, we restrict ourselves to one and two space dimensions only. However, we always considers
all variables of the PDE system for the general three-dimensional case, i.e. three components for the velocity v
i
and the
thermal impulse J
i
and nine components of the distorsion eld A
ik
.
25

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Table 1
Numerical convergence results at time t = 0 . 25 in L
2
norm for density
ρ, momentum
density
ρv
1
and entropy density
ρSusing HTC cell centered FV and HTC DG schemes
of type I applied to the Euler subsystem (black terms in
(1) ).
N
x ρρv
1 ρSO(ρ) O(ρv
1
) O(ρS)
HTC CC FV
64 5.1870E −03 1.8351E −03 3.1355E −04
128 1.2747E −03 4.6020E −04 8.0154E −05 2.0 2.0 2.0
256 3.1726E −04 1.1516E −04 2.0144E −05 2.0 2.0 2.0
512 7.9226E −05 2.8847E −05 5.0427E −06 2.0 2.0 2.0
HTC DG type I - N = 1
32 5.1870E −03 6.1093E −03 7.9857E −05
64 1.2747E −03 1.4483E −03 1.1197E −05 2.0 2.1 2.8
128 3.1726E −04 3.5728E −04 1.4880E −06 2.0 2.0 2.9
256 7.9226E −05 8.9128E −05 1.9172E −07 2.0 2.0 3.0
HTC DG type I - N = 2
16 2.9753E −03 4.7954E −03 1.6074E −05
32 5.0543E −04 8.5266E −04 7.7748E −07 2.6 2.5 4.4
64 7.2395E −05 1.5411E −04 3.0162E −08 2.8 2.5 4.7
128 9.4557E −06 2.7237E −05 1.0471E −09 2.9 2.5 4.8
HTC DG type I - N = 3
8 3.6067E −03 6.2320E −03 2.3026E −05
16 2.4324E −04 6.9778E −04 4.6458E −07 3.9 3.2 5.6
32 1.2749E −05 2.7087E −05 2.7667E −09 4.3 4.7 7.4
48 2.3053E −06 6.7687E −06 1.4757E −10 4.2 3.4 7.2
HTC DG type I - N = 4
8 6.8664E −04 1.8162E −03 3.9238E −06
12 1.0909E −04 1.6206E −04 7.6967E −08 4.5 6.0 9.7
16 2.2145E −05 3.9053E −05 4.4581E −09 5.5 4.9 9.9
20 7.4773E −06 1.8647E −05 9.0447E −10 4.9 3.3 7.1
HTC DG type I - N = 5
6 8.5757E −04 1.5182E −03 1.9402E −06
8 1.3355E −04 3.0442E −04 1.3112E −07 6.5 5.6 9.4
12 1.6464E −05 6.1433E −05 3.7147E −09 5.2 3.9 8.8
16 2.6318E −06 1.5658E −05 1.9870E −10 6.4 4.8 10.2
6.1. Numerical convergence study
In order to verify the order of accuracy of the new HTC schemes proposed in this paper, we simulate the isentropic
vortex problem, see [70] , of the pure inviscid Euler equations, i.e. we apply the numerical schemes only to the black terms
in (1) . The model parameters are γ= 1 . 4 , c
s
= 0 , c
h
= 0 and the articial viscosity is set to = 0 . The computational domain
is the square = [0 , 10]
2
. All boundary conditions are periodic. The analytical expression of the initial condition in terms
of primitive variables reads
(ρ, v
1
, v
2
, v
3
, p) = (δρ, δu, δv , 0 , δp) (146)
with the radius r
2
= (x −5)
2
+ (y −5)
2
, the vortex strength ε = 5 , the entropy uctuation δS = 0 and the velocity, temper-
ature, density and pressure proles given by
δu
δv )= ε
2 πe
1 −r
2
2 5 −y
x −5 ), δT = −(γ−1) ε
2
8 γπ2
e
1 −r
2
, δρ= (1 + δT )
1
γ−1
, δp = (1 + δT ) γγ−1
. (147)
Since the vortex is stationary, the exact solution is the initial condition for all times. Simulations are run with all HTC
schemes until time t = 0 . 25 using an equidistant Cartesian mesh composed of N
x
×N
y elements. It is important to highlight
that in this test case the articial viscosity is set to = 0 . The L
2 errors together with the corresponding convergence rates
obtained for the density ρ, the momentum density ρv
1
and the entropy density ρSare reported at the nal time in Tables 1
and 2 . One can observe that all proposed HTC schemes reach their nominal order of accuracy. More precisely, the HTC nite
volume scheme is second order accurate in all variables, while the HTC DG schemes of type I and II in general reach their
designed order of accuracy of N + 1 in density, and momentum density, while the entropy density reaches orders between
2 N + 1 and 2 N + 2 . This can be explained by the fact that the present test problem is isentropic and since for = 0 the
only entropy generation mechanism in both HTC DG schemes is the jump term
1
2
s
max
(q
r
h
−q

h
) in the numerical viscosity
ux G. But because the jumps tend to zero with order between N +1
2
to N + 1 and the related entropy production term
k
in the entropy inequality is quadratic in the jump, we indeed expect twice the convergence order for entropy in this
test case. For isentropic ows this seems to be indeed a very interesting feature of our new HTC DG schemes that are
based on the direct discretization of the entropy inequality in contrast to standard DG schemes which discretize the total
energy conservation law. One remarkable difference between the HTC schemes proposed in this paper with respect to the
original HTC schemes proposed in Busto et al. [61] , Busto and Dumbser [62] is that we avoid the use of path integrals in
26

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Table 2
Numerical convergence results at time t = 0 . 25 in L
2
norm for density
ρ, momentum
density
ρv
1
and entropy density
ρSusing HTC cell centered FV and HTC DG schemes
of type II applied to the Euler subsystem (black terms in
(1) ).
N
x ρρv
1 ρSO(ρ) O(ρv
1
) O(ρS)
HTC CC FV
64 5.1870E −03 1.8351E −03 3.1355E −04
128 1.2747E −03 4.6020E −04 8.0154E −05 2.0 2.0 2.0
256 3.1726E −04 1.1516E −04 2.0144E −05 2.0 2.0 2.0
512 7.9226E −05 2.8847E −05 5.0427E −06 2.0 2.0 2.0
HTC DG type II - N = 1
32 5.1979E −03 6.1038E −03 1.4077E −04
64 1.2754E −03 1.4475E −03 2.0503E −05 2.0 2.1 2.8
128 3.1731E −04 3.5722E −04 2.7459E −06 2.0 2.0 2.9
256 7.9228E −05 8.9124E −05 3.5363E −07 2.0 2.0 3.0
HTC DG type II - N = 2
16 2.9760E −03 4.8004E −03 1.9130E −05
32 5.0550E −04 8.5273E −04 8.0526E −07 2.6 2.5 4.4
64 7.2396E −05 1.5411E −04 3.0200E −08 2.8 2.5 4.7
128 9.4557E −06 2.7238E −05 1.0519E −09 2.9 2.5 4.8
HTC DG type II - N = 3
8 3.6075E −03 6.2317E −03 2.5859E −05
16 2.4324E −04 6.9776E −04 5.2706E −07 3.9 3.2 5.6
32 1.2747E −05 2.7086E −05 7.7266E −09 4.3 4.7 6.1
48 2.3053E −06 6.7687E −06 6.4886E −10 4.2 3.4 6.1
HTC DG type II - N = 4
8 6.8652E −04 1.8158E −03 5.6567E −06
12 1.0909E −04 1.6204E −04 1.3708E −07 4.5 6.0 9.2
16 2.2144E −05 3.9052E −05 7.5439E −09 5.5 4.9 10.1
20 7.4773E −06 1.8647E −05 1.5860E −09 4.9 3.3 7.0
HTC DG type II - N = 5
6 8.5757E −04 1.5181E −03 3.0778E −06
8 1.3355E −04 3.0442E −04 1.4965E −07 6.5 5.6 10.5
12 1.6465E −05 6.1433E −05 3.9483E −09 5.2 3.9 9.0
16 2.6318E −06 1.5658E −05 2.0594E −10 6.4 4.8 10.3
Table 3
Computational time (s) employed to complete the
isentropic vortex problem up to time t = 0 . 25 using
the new HTC FV method proposed in this article (HTC
CC FV) and the HTC FV method
[61] based on path-
integrals (HTC PI FV) applied to the Euler subsystem
(black terms in
(1) ).
N
x HTC CC FV HTC PI FV Ratio (t
PI
/t
CC
)
16 0.00697 0.01264 1.81
32 0.02049 0.05664 2.77
64 0.09889 0.35564 3.60
128 0.68748 2.73660 3.98
256 5.33787 21.45535 4.02
512 41.71814 169.38102 4.06
the compatible numerical ux of the underlying Euler subsystem. As a consequence, the computational cost of the resulting
algorithms is greatly reduced as it can be observed in Table 3 , where both HTC FV schemes are analysed in terms of the
CPU time needed to run the isentropic vortex test case up to t = 0 . 25 with four processes on an Intel® Core
TM i9-10980XE.
6.2. Shear motion in solids and uids
In the following, we consider the evolution of an isolated shear layer in the domain = [ −0 . 5 , +0 . 5] in one space
dimension. The initial conditions are set as v
1
= v
3
= 0 , J = 0 , ρ= 1 , p = 1 , A = I and v
2
= −v
0 for x < 0 and v
2
= + v
0 for
x ≥0 with v
0
= 0 . 1 . We furthermore set γ= 1 . 4 , c
v
= 1 , ρ0
= 1 , c
s
= 1 and c
h
= 1 . The numerical simulations are run until
t = 0 . 4 for different relaxation times using the new thermodynamically compatible HTC schemes developed in this paper,
i.e. the compatible cell centered nite volume scheme as well as the thermodynamically compatible discontinuous Galerkin
schemes of type I and II, for which we choose a polynomial approximation degree of N = 5 . In order to obtain the same
number of degrees of freedom we use 1024 elements for the DG schemes with N = 5 and 6144 control volumes in case of
the nite volume method. For uids we furthermore set κ= μ. In the stiff relaxation limit of the GPR model, i.e. for τ1
1 ,
the exact solution of the rst problem of Stokes for the incompressible Navier–Stokes equations, see e.g. [64,71–73] , can be
27

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Fig. 1. Simple shear motion in an ideal elastic solid and in viscous uids. Numerical solution of the GPR model at time t =0 . 4 obtained with the new HTC
nite volume scheme (6144 cells) and the new HTC DG schemes of type I and II (1024 elements, polynomial approximation degree N = 5 ). Top left: results
for the solid setting
τ1
=
τ2
= 10
20
. From top right to bottom right: results for uids with viscosities:
μ= 10
−2
,
μ= 10
−3
and
μ= 10
−4
, respectively.
used as reference solution for v
2
:
v
2
(x, t) = v
0
erf
1
2
x
√
νt
), (148)
with ν= μ/ρ0
. For the solid limit of the model, a reference solution is obtained by using a second order TVD nite volume
scheme of the MUSCL-Hancock type, see [66] , on 10,0 0 0 cells. In the solid limit of the GPR model the articial viscosity
is set to = 10
−6
. The obtained numerical results are depicted in Fig. 1 , where an excellent agreement of all numerical
solutions with the reference solutions can be observed.
6.3. Riemann problems
Next, we analyse a set of ve Riemann problems with the left and right initial states and the location of the initial
discontinuity x
c given in Table 4 . The one-dimensional domain is = [ −0 . 5 , +0 . 5] . We consider the Euler subsystem (i.e.
only the black terms in (1) ), as well as the full GPR model (1) . The exact solution of the Riemann problem for the Euler
equations can be found in the textbook of Toro [66] , while for the full GPR model a numerical reference solution is obtained
by solving (1) at the aid of a second order MUSCL-Hancock method on a very ne mesh of 128,0 0 0 control volumes, solving
the total energy Eq. (1f) instead of the entropy inequality (1c) . The Riemann problems contain three test cases for the pure
Euler equations, RP1, RP2, RP3, and two test problems for the uid and solid limit of the GPR model, RP4 and RP5. For
28

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Table 4
Left (L) and right (R) initial states for density, velocity, with v = (u,
v , 0) , and pressure pand location of
the initial discontinuity x
c
for ve Riemann problems solved with the new HTC nite volume and discon-
tinuous Galerkin schemes of type I and II proposed in this paper.
RP ρL u
L
v
L p
L ρR u
R
v
R p
R x
c
RP1 1.0 0.0 0.0 1.0 0.125 0.0 0.0 0.1 0.0
RP2 5.99924 19.5975 0.0 460.894 5.99242 −6.19633 0.0 46.095 −0.2
RP3 1.0 −2.0 0.0 0.4 1.0 + 2.0 0.0 0.4 0.0
RP4 1.0 0.0 −0.2 1.0 0.5 0.0 + 0.2 0.5 0.0
RP5 1.0 0.0 −0.2 1.0 0.5 0.0 + 0.2 0.5 0.0
Fig. 2. Results of the density for Riemann problems RP1 (left) and RP2 (right) at times t = 0 . 2 and t = 0 . 035 , respectively, obtained using the new thermo-
dynamically compatible cell-centered HTC FV scheme (1536 cells for RP1, 6144 cells for RP2) and the HTC DG schemes of type I and II ( N = 5 , 256 elements
for RP1, 1024 elements for RP2) applied to the compressible Euler equations. The exact solution, see
[66] , is represented with a black solid line.
RP4 and RP5 we set the initial conditions for A and J to A = I and J = 0 . Furthermore, we choose c
s
= c
h
= 1 and γ= 1 . 4 .
For RP4 we choose the relaxation times so that μ= κ= 10
−5 and for RP5 we set τ1
= τ2
= 10
20
. In all cases the articial
viscosity is set to a constant value of = 10
−5
. The numerical results obtained with the new HTC schemes proposed in this
paper are compared against the reference solution in Figs. 2–5 . The employed mesh resolution is provided for each test case
in the corresponding gure caption and the polynomial approximation degree for the HTC DG schemes of type I and II is
set to N = 5 . In all cases an excellent agreement between numerical solution and reference solution can be observed. In
particular, we can observe that there is no spurious glitch in the temperature of the 123 problem (RP3) for both types of
HTC DG schemes, unlike in the corresponding HTC FV method.
6.4. Viscous shock wave
The next test case is a stationary viscous shock with a characteristic shock Mach number of M
s
= 2 and a Prandtl number
in the uid of Pr = 0 . 75 , so that an exact solution of the compressible Navier–Stokes equations exists, see e.g. [74,75] and
[64] for the details on the computation of the exact solution. The problem is solved in the one-dimensional domain =
[ −0 . 5 , +0 . 5] with the shock wave centered at x = 0 and the uid moving into the shock from the left to the right. The data
in front of the shock are ρ0
= 1 , v
0
1
= 2 , v0
2
= v
3
= 0 and p
0
= 1 /γ, hence the sound speed in front of the shock is c
0
= 1 .
The Reynolds number based on the reference length L = 1 is Re
s
= ρ0
c
0
M
s
L μ−1
. The remaining parameters of the GPR
model are chosen as γ= 1 . 4 , c
v
= 2 . 5 , c
h
= c
s
= 50 , μ= 2 ×10
−2 and κ= 9
1
3
×10
−2
, hence the shock Reynolds number is
Re
s
= 100 . The initial condition for J and A is J = 0 and A =
3
√
ρI . The comparison of the solution obtained with the novel
thermodynamically compatible HTC schemes applied to (1) and the exact solution obtained for the compressible Navier–
Stokes equations is depicted in Fig. 6 . For the HTC nite volume scheme we employ 1024 equidistant cells, while the HTC
DG schemes of type I and II use 256 cells with polynomial approximation degree N = 9 . For all quantities an excellent
agreement is achieved.
29

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Fig. 3. Results for the 123 problem, RP3, at time t = 0 . 15 , obtained using the new thermodynamically compatible cell-centered HTC FV scheme (6144 cells)
and the HTC DG schemes of type I and II ( N = 5 , 1024 elements) applied to the compressible Euler equations. Density (top panel), velocity (central panel),
temperature (bottom panel). The exact solution, see
[66] , is represented with a black solid line.
30

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Fig. 4. Results for the density for Riemann problem RP4 at time t = 0 . 2 , obtained using the new thermodynamically compatible cell-centered HTC FV
scheme (6144 cells) and the HTC DG schemes of type I and II ( N = 5 , 1024 elements) applied to the GPR model in the stiff relaxation limit (
μ= κ= 10
−5
).
The exact solution of the compressible Euler equations, see
[66] , is represented with a black solid line as a reference solution.
6.5. Double shear layer
The double shear layer test was proposed in Bell et al. [76] and was subsequently used in Busto et al. [61] , Dumbser
et al. [64] , Boscheri et al. [71] , Bermúdez et al. [72] , Busto et al. [73] , Tavelli and Dumbser [77] , 78 ] to assess the behaviour
of compressible ow solvers in the weakly compressible regime, including applications to the GPR model (1) . The two-
dimensional domain is the unit square = [0 , 1]
2 with periodic boundaries everywhere. The initial condition is given by
v
1
=
{tanh
˜ ρ(y −1
4
)
, y ≤1
2
,
tanh
˜ ρ(
3
4
−y )
, y >
1
2
,
v
2
= δsin (2 πx ) ,
v
3
= 0 , ρ= ρ0
= 1 , p = 10
2
/γ, A = I and J = 0 , with δ= 0 . 05 and ˜ ρ= 30 . The remaining parameters of the model are ν=
μ/ρ0
= 2 ×10
−3
, ρ0
= 1 , c
v
= 1 , c
s
= 8 , c
h
= 2 and τ2
= 4 ×10
−3
. The characteristic Mach number of the ow resulting
from this setup is M = 0 . 1 . The numerical simulations are carried out with all new HTC schemes proposed in this paper
until t = 1 . 8 . The HTC nite volume scheme is run on a computational grid of 40 0 0 ×40 0 0 elements, while the HTC DG
schemes of type I and II use a coarser mesh of 1024 ×1024 control volumes with a polynomial approximation degree of
N = 3 . The numerical viscosity is set to = 1 ×10
−6in all cases. In Fig. 7 the results obtained with the new HTC FV scheme
for the time evolution of the distortion eld component A
12 are shown. The results agree very well with those reported in
Busto et al. [61] , where also a validation against an incompressible Navier–Stokes solver [72,79] was provided. The initial
shear layers develop into several vortices, as already described in more detail in Dumbser et al. [64] , Boscheri et al. [71] ,
Bermúdez et al. [72] , Busto et al. [73] , Bell et al. [76] . Almost identical results are obtained for this test problem also with
the HTC DG schemes of type I and II, see Figs. 8 and 9 . Overall, we can therefore conclude that the methods proposed in
this paper allow the reliable simulation of complex ows in the uid limit of the GPR model, leading to numerical results
that are essentially independent of the underlying mesh and numerical method, once a suciently ne mesh has been used.
6.6. Solid rotor
Here, we apply our new thermodynamically compatible schemes to the solid rotor problem proposed in Busto et al. [61] ,
Boscheri et al. [71] . Choosing τ1
= τ2
= 10
20
the governing PDE system (1) describes the dynamics of a nonlinear hyperelastic
solid. We dene the two-dimensional domain as = [ −1 , +1]
2
with transmissive boundary conditions everywhere. As initial
conditions for density, pressure, distortion eld and thermal impulse we set ρ= 1 , p = 1 , A = I and J = 0 , the initial velocity
eld is given by v
1
= −y/R , v
2
= + x/R and v
3
= 0 within the circle
‖
x
‖
≤R of radius R = 0 . 2 , while for r > R the velocity is
v = 0 . The remaining parameters of the model are γ= 1 . 4 , c
s
= 1 . 0 and c
h
= 1 . 0 . The nal simulation time is set to t = 0 . 3 in
all cases. We run this test using the HTC nite volume scheme on 512 ×512 control volumes as well as the HTC DG schemes
of type I and II with 128 ×128 elements and a polynomial approximation degree of N = 5 . In all cases the articial viscosity
is set to = 5 ×10
−4
. The reference solution is provided by a second order MUSCL-Hancock scheme on 512 ×512 control
volumes, solving the total energy conservation law (1f) rather than the entropy inequality (1c) , as also done in Dumbser
31

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Fig. 5. Results for the density for Riemann problem RP5 ( x
c
= 0 ) at time t = 0 . 2 , obtained using the new thermodynamically compatible cell-centered HTC
FV scheme (6144 cells) and the HTC DG schemes of type I and II ( N = 5 , 1024 elements) applied to the GPR model in the solid limit (
τ1
=
τ2
= 10
20
).
Fig. 6. Viscous shock at Re
s
= 100 , M
s
= 2 and Pr = 0 . 75 . Numerical solution obtained with the new thermodynamically compatible cell-centered HTC FV
scheme (1536 cells) and the new HTC DG schemes of type I and II (256 elements, N = 5 ) applied to the GPR model. Comparison with the exact solution
from the compressible Navier–Stokes equations. Density (left), stress
σ11
(center) and heat ux h
1
(right) at time t = 0 . 25 .
32

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Fig. 7. Numerical solution obtained for the double shear layer problem with the new thermodynamically compatible cell-centered HTC nite volume
scheme applied to the GPR model with
μ= 2 ·10
−3
. Distortion eld component A
12
at times t = 0 . 4 (top left), t = 0 . 8 (top right), t = 1 . 2 (bottom left) and
t = 1 . 8 (bottom right).
et al. [64] . In Fig. 10 the computational results of the HTC schemes are compared with each other and with the reference
solution, obtaining an excellent agreement among all of them.
6.7. Lid-driven cavity
The nal test problem is the lid-driven cavity benchmark, see [80] , which is also well-suited to validate numerical
schemes in the low Mach number limit of the compressible Navier–Stokes equations, see e.g. [72,73,78] . This test case was
already successfully solved with different numerical schemes applied to the GPR model in Busto et al. [61] , Dumbser et al.
[64] , Boscheri et al. [71] . The two-dimensional computational domain is the unit square = [0 , 1] ×[0 , 1] and the initial
condition reads ρ= 1 , v = 0 , p = 10
2
, A = I and J = 0 . According to Busto et al. [61] the remaining model parameters are
γ= 1 . 4 , c
v
= 1 , c
s
= 8 , ρ0
= 1 and c
h
= 2 , τ2
= 10
−2 and μ= 10
−2
, hence the characteristic Reynolds number of the ow
based on the lid velocity v = (1 , 0 , 0) is Re = 100 . Apart from the lid, all boundaries are no-slip wall boundaries with zero
velocity. The characteristic Mach number of this test problem based on the lid velocity is about M = 0 . 08 . All HTC schemes
use a mesh composed of 256 ×256 elements and the HTC DG schemes of type I and II employ a polynomial approximation
degree of N = 3 . The nal time for all simulations is t = 10 . We have always used a constant articial viscosity of =10
−3
,
apart from the DG scheme of type II, for which we set = 1 . 5 ×10
−3
. All numerical results are summarized in Fig. 11 and
compared against the reference solution available in Ghia et al. [80] . We can observe an excellent agreement between the
33

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Fig. 8. Numerical solution obtained for the double shear layer problem with the new HTC DG scheme of type I applied to the GPR model with μ= 2 ·10
−3
.
Distortion eld component A
12
at times t = 0 . 4 (top left), t = 0 . 8 (top right), t = 1 . 2 (bottom left) and t = 1 . 8 (bottom right).
solution approximated by solving the GPR model with the new HTC schemes and the reference solution for the incompress-
ible Navier–Stokes equations.
In order to obtain stable results for the DG schemes of type II the articial viscosity needed to be increased in this
test problem compared to the other two schemes. The authors conjecture that the problem is related to the no-slip wall
boundary conditions, which are non-trivial for the lid driven cavity in general, due to the discontinuous velocity eld on the
boundary, and for the GPR model in particular, see [71] for a detailed discussion. The lid-driven cavity generates pressure
peaks in the upper corners of the domain, which may require more limiting in the case of DG schemes of type II compared
to the other two methods. Further and more detailed investigations on the behaviour of DG schemes of type II in the
presence of no-slip wall boundaries are needed in the future.
7. Conclusions
In this paper we have presented three new semi-discrete thermodynamically compatible schemes for the GPR model
of continuum mechanics: one scheme of the cell-centered nite volume type and two different high order DG schemes.
All three methods have in common that they discretize the entropy density as a primary evolution quantity, in contrast to
standard methods for hyperbolic systems, while total energy conservation is obtained as a mere consequence of the ther-
modynamically compatible discretization. All methods satisfy a discrete cell entropy inequality by construction and can be
proven to be nonlinearly stable in the energy norm. For all schemes we have shown numerical results for several bench-
marks in both the solid and uid limit of the GPR model. In all cases a very good agreement with exact or numerical
34

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Fig. 9. Numerical solution obtained for the double shear layer problem with the new HTC DG scheme of type II applied to the GPR model with μ= 2 ·10
−3
.
Distortion eld component A
12
at times t = 0 . 4 (top left), t = 0 . 8 (top right), t = 1 . 2 (bottom left) and t = 1 . 8 (bottom right).
reference solutions was obtained. Compared to previous work on similar schemes presented in Busto et al. [59] , Busto and
Dumbser [60] , Busto et al. [61] , Busto and Dumbser [62] the new numerical methods introduced in this paper were shown
to be computationally more ecient and simpler to implement, since no path-integrals need to be computed in order to
obtain a thermodynamically compatible ux for the inviscid Euler subsystem. Furthermore, the new family of numerical
schemes discussed in this paper does not require an underlying Godunov parametrization of the physical ux in terms of a
generating potential, unlike the thermodynamically compatible schemes forwarded in Busto et al. [59] , Busto and Dumbser
[60] , Busto et al. [61] , Busto and Dumbser [62] . The nite volume schemes are clearly the most simple schemes presented
in this paper. They only only require the calculation of compatible uxes and jump terms. DG methods of type I are simple
straightforward extensions of the former to the DG framework. The new DG schemes of type II proposed in this paper do
not need a special compatible discretization of terms related to the distortion eld A and to the specic thermal impulse
J . Simple arithmetic averages are enough to construct a baseline scheme and subsequently the entire correction leading
to thermodynamic compatibility, including numerical quadrature errors, is achieved in the calculation of the element-wise
scalar correction factor αi
.
Future work will consider an extension of the approach proposed in this paper to the MHD equations, to turbulent shal-
low water ows [81,82] and to the conservative SHTC system of compressible two-uid ows proposed by Romenski et al. in
Romenski et al. [10] , 11 ], which was already studied numerically and analytically in Lukácová-Medvidóvá et al. [83] , Thein
et al. [84] . In the future, we also plan an extension of the methodology outlined in this paper to the fully discrete case
as well as to staggered Cartesian and unstructured meshes, in order to combine it with involution-preserving semi-implicit
discretizations [71,85] and semi-implicit hybrid nite-volume / nite-element schemes [72,73,79] on staggered meshes. All
35

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Fig. 10. Contour colours of the velocity component v
1
for the solid rotor problem obtained solving the GPR model in the solid limit up to time t = 0 . 3 .
Thermodynamically compatible HTC FV scheme (top left), HTC DG scheme of type I (top right), HTC DG scheme of type II (bottom left) and reference
solution obtained with a classical second order MUSCL-Hancock scheme (bottom right).
schemes presented in this paper have been analyzed in the semi discrete setting. For a possible extension to the fully dis-
crete case, at least for the Euler subsystem, see e.g. [61,86–88] . We will also consider the use of conservative and symplectic
time integrators, such as those forwarded in Brugnano and Iavernaro [89] , 90 ], in order to preserve the conservation of
total energy of our semi-discrete schemes exactly also in a fully discrete setting. The incorporation of limiters in the DG
scheme was out of scope of this work. We will therefore also consider proper limiters for DG schemes in the future, such
as slope and moment limiters [91,92] , positivity preserving limiters [93] , or the use of the cell-centered thermodynamically
compatible nite volume schemes presented in this paper as a posteriori subcell FV limiter of the DG schemes of type I and
II, following the ideas on subcell limiting for DG schemes outlined in Rueda-Ramírez et al. [41] , Sonntag and Munz [94] ,
Dumbser et al. [95] , Sonntag and Munz [96] .
Dedication
The new numerical methods introduced in this paper are dedicated to Prof. Eleuterio Francisco Toro at the occasion of his
75th birthday and in honor of his groundbreaking scientic contributions to the eld of numerical methods for hyperbolic
PDE.
36

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Fig. 11. Results for the lid-driven cavity test at Re = 100 using the rst order hyperbolic GPR model in entropy formulation. Results obtained at time t = 10
on 256 ×256 elements using the new thermodynamically compatible cell-centered HTC nite volume scheme (top row), the HTC DG scheme of type I
(middle row) and the HTC DG scheme of type II (bottom row). Left column: colour contours of the velocity component
v
1
. Right column: comparison of
v
1
and
v
2
along the 1D cuts x = 0 . 5 and y = 0 . 5 with the reference solution provided in Ghia et al.
[80] .
37

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
Data Availability
Data will be made available on request.
Acknowledgements
S.B. and M.D. are members of the INdAM GNCS group and acknowledge the nancial support received from the Italian
Ministry of Education, University and Research (MIUR) in the frame of the PRIN 2017 project Innovative numerical methods
for evolutionary partial differential equations and applications and from the Spanish Ministry of Science and Innovation, grant
number PID2021-122625OB-I00. S.B. was also funded by INdAM via a GNCS grant for young researchers and by an UniTN
starting grant of the University of Trento. R.A. was partially funded by SNFS grant #20 0 020_204917. The authors would like
to acknowledge support from the Leibniz Rechenzentrum (LRZ) in Garching, Germany, for granting access to the SuperMUC-
NG supercomputer under project number pr63qo and to the CESGA in Santiago de Compostela, Spain, for the access to the
FT3 supercomputer. The Authors also would like to thank the Universidade de Vigo/CISUG for the funding of the open access
charge.
The authors would like to thank the anonymous referee for the very useful and constructive comments, which helped to
improve the quality and clarity of this paper.
References
[1] R. Abgrall, A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy
dissipative schemes, J. Comput. Phys. 372 (2018) 640–666 .
[2] K. Friedrichs, Symmetric positive linear differential equations, Commun. Pure Appl. Math. 11 (1958) 333–418 .
[3] S. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR 139 (3) (1961) 521–523 .
[4] K. Friedrichs, P. Lax, Systems of conservation equations with a convex extension, Proc. Natl. Acad. Sci. USA 68 (1971) 1686–1688 .
[5] G. Boillat, Sur l’existence et la recherche d’équations de conservation supplémentaires pour les systèmes hyperboliques, C. R. Acad. Sci. Paris Sér. A
278 (1974) 909–912 .
[6] T. Ruggeri, A. Strumia, Main eld and convex covariant density for quasilinear hyperbolic systems. Relativistic uid dynamics, Ann. Inst. H. Poincaré
Sect. A (N.S.) 34 (1981) 65–84.
[7] H. Freistühler, Relativistic barotropic uids: a Godunov–Boillat formulation for their dynamics and a discussion of two special classes, Arch. Ration.
Mech. Anal. 232 (2019) 473–488 .
[8] S. Godunov, Symmetric form of the equations of magnetohydrodynamics, Numer. Methods Mech. Contin. Media 3 (1) (1972) 26–31 .
[9] S. Godunov, E. Romenski, Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates, J. Appl. Mech. Tech. Phys. 13 (1972)
868–885 .
[10] E. Romenski, A. Resnyansky, E. Toro, Conservative hyperbolic formulation for compressible two-phase ow with different phase pressures and temper-
atures, Q. Appl. Math. 65 (2007) 259–279 .
[11] E. Romenski, D. Drikakis, E. Toro, Conservative models and numerical methods for compressible two-phase ow, J. Sci. Comput. 42 (2010) 68–95 .
[12] S.K. Godunov, Thermodynamic formalization of the uid dynamics equations for a charged dielectric in an electromagnetic eld, Comput. Math. Math.
Phys. 52 (2012) 787–799 .
[13] E. Romenski, I. Peshkov, M. Dumbser, F. Fambri, A new continuum model for general relativistic viscous heat-conducting media, Philos. Trans. R. Soc.
A 378 (2020) 20190175 .
[14] I. Peshkov, E. Romenski, M. Dumbser, Continuum mechanics with torsion, Contin Mech. Thermodyn. 31 (2019) 1517–1541 .
[15] I. Peshkov, M. Pavelka, E. Romenski, M. Grmela, Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations,
Contin. Mech. Thermodyn. 30 (6) (2018) 1343–1378.
[16] E. Romenski, Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics, Math. Comput. Model. 28 (10) (1998)
115–130 .
[17] S. Godunov, E. Romenski, Elements of Continuum Mechanics and Conservation Laws, Kluwer Academic/Plenum Publishers, 2003 .
[18] N. Favrie, S. Gavrilyuk, A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves, Nonlinearity
30 (2017) 2718–2736 .
[19] F. Dhaouadi, N. Favrie, S. Gavrilyuk, Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation, Stud. Appl. Math. 142 (2019)
336–358 .
[20] E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws I, Math. Comput. 49 (1987) 91–103 .
[21] F. Ducros, V. Ferrand, F. Nicoud, C. Weber, D. Darracq, C. Gacherieu, T. Poinsot, Large-eddy simulation of the shock/turbulence interaction, J. Comput.
Phys. 152 (1999) 517–549 .
[22] F. Ducros, F. Laporte, T. Soulères, V. Guinot, P. Moinat, B. Caruelle, High-order uxes for conservative skew-symmetric-like schemes in structured
meshes: application to compressible ows, J. Comput. Phys. 161 (20 0 0) 114–139 .
[23] T. Fisher, M. Carpenter, J. Nordström, N. Yamaleev, C. Swanson, Discretely conservative nite-difference formulations for nonlinear conservation laws
in split form: theory and boundary conditions, J. Comput. Phys. 234 (2013) 353–375 .
[24] T. Fisher, M. Carpenter, High-order entropy stable nite difference schemes for nonlinear conservation laws: nite domains, J. Comput. Phys. 252
(2013) 518–557 .
[25] M. Carpenter, T. Fisher, E. Nielsen, S. Frankel, Entropy stable spectral collocation schemes for the Navier–Stokes equations: discontinuous interfaces,
SIAM J. Sci. Comput. 36 (2014) B835–B867 .
[26] M. Carpenter, D. Gottlieb, S. Abarbanel, The stability of numerical boundary treatments for compact high–order nite–difference schemes, J. Comput.
Phys. 108 (1993) 272–295 .
[27] M. Carpenter, J. Nordström, D. Gottlieb, Revisiting and extending interface penalties for multi–domain summation–by–parts operators, J. Sci. Comput.
45 (2010) 118–150 .
[28] J. Nordström, M.H. Carpenter, Boundary and interface conditions for high–order nite–difference methods applied to the Euler and Navier–Stokes
equations, J. Comput. Phys. 148 (1999) 621–645 .
[29] S. Pirozzoli, Generalized conservative approximations of split convective derivative operators, J. Comput. Phys. 229 (2010) 7180–7190 .
[30] S. Pirozzoli, Stabilized non–dissipative approximations of Euler equations in generalized curvilinear coordinates, J. Comput. Phys. 230 (2011)
2997–3014 .
38

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
[31] B. Sjögreen, H. Yee, High order entropy conservative central schemes for wide ranges of compressible gas dynamics and MHD ows, J. Comput. Phys.
364 (2018) 153–185 .
[32] B. Sjögreen, H. Yee, Entropy stable method for the Euler equations revisited: central differencing via entropy splitting and SBP, J. Sci. Comput. 81
(2019) 1359–1385 .
[33] H. Yee, B. Sjögreen, Recent advancement of entropy split methods for compressible gas dynamics and MHD, Appl. Math. Comput. (2022) .
[34] J. Reiss, J. Sesterhenn, A conservative, skew–symmetric nite difference scheme for the compressible Navier–Stokes equations, Comput. Fluids 101
(2014) 208–219 .
[35] U.S. Fjordholm, S. Mishra, E. Tadmor, Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation
laws, SIAM J. Numer. Anal. 50 (2) (2012) 544–573 .
[36] M.J. Castro, U.S. Fjordholm, S. Mishra, C. Parés, Entropy conservative and entropy stable schemes for nonconservative hyperbolic systems, SIAM J.
Numer. Anal. 51 (3) (2013) 1371–1391 .
[37] A. Hiltebrand, S. Mishra, Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws, Numer. Math.
126 (1) (2014) 103–151 .
[38] G. Gassner, A. Winters, D. Kopriva, A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water
equations, Appl. Math. Comput. 272 (2016) 291–308 .
[39] D. Derigs, A.R. Winters, G. Gassner, S. Walch, M. Bohm, Ideal GLM-MHD: about the entropy consistent nine-wave magnetic eld divergence diminishing
ideal magnetohydrodynamics equations, J. Comput. Phys. 364 (2018) 420–467 .
[40] G. Schnücke, N. Krais, T. Bolemann, G.J. Gassner, Entropy stable discontinuous Galerkin schemes on moving meshes for hyperbolic conservation laws,
J. Sci. Comput. 82 (3) (2020) .
[41] A.M. Rueda-Ramírez, S. Hennemann, F.J. Hindenlang, A.R. Winters, G.J. Gassner, An entropy stable nodal discontinuous Galerkin method for the resistive
MHD equations. Part II: subcell nite volume shock capturing, J. Comput. Phys. 4 4 4 (2021) 109935 .
[42] S. Hennemann, A.M. Rueda-Ramírez, F.J. Hindenlang, G.J. Gassner, A provably entropy stable subcell shock capturing approach for high order split form
DG for the compressible euler equations, J. Comput. Phys. 426 (2021) .
[43] T. Cheng, C. Shu, Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws, J. Comput.
Phys. 345 (2017) 427–461 .
[44] Y. Liu, C. Shu, M. Zhang, Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes, J. Comput.
Phys. 354 (2018) 163–178 .
[45] P. Chandrashekar, C. Klingenberg, Entropy stable nite volume scheme for ideal compressible MHD on 2-D Cartesian meshes, SIAM J. Numer. Anal. 54
(2) (2016) 1313–1340 .
[46] D. Ray, P. Chandrashekar, U.S. Fjordholm, S. Mishra, Entropy stable scheme on two-dimensional unstructured grids for Euler equations, Commun.
Comput. Phys. 19 (5) (2016) 1111–1140 .
[47] D. Ray, P. Chandrashekar, An entropy stable nite volume scheme for the two dimensional Navier–Stokes equations on triangular grids, Appl. Math.
Comput. 314 (2017) 257–286 .
[48] J. Chan, C.G. Taylor, Ecient computation of Jacobian matrices for entropy stable summation-by-parts schemes, J. Comput. Phys. 448 (2022) 110701 .
[49] J. Chan, Y. Lin, T. Warburton, Entropy stable modal discontinuous Galerkin schemes and wall boundary conditions for the compressible Navier–Stokes
equations, J. Comput. Phys. 448 (2022) 110723 .
[50] E. Gaburro, P. Öffner, M. Ricchiuto, D. Torlo, High order entropy preserving ADER-DG schemes, Appl. Math. Comput. (2022) .
[51] U. Fjordholm, S. Mishra, Accurate numerical discretizations of non-conservative hyperbolic systems, ESAIM Math. Model. Numer. Anal. 46 (1) (2012)
187–206 .
[52] R. Abgrall, P. Bacigaluppi, S. Tokareva, A high-order nonconservative approach for hyperbolic equations in uid dynamics, Comput. Fluids 169 (2018)
10–22 .
[53] R. Abgrall, P. Öffner, H. Ranocha, Reinterpretation and extension of entropy correction terms for residual distribution and discontinuous Galerkin
schemes: application to structure preserving discretization, J. Comput. Phys. 453 (2022) .
[54] R. Abgrall, J. Nordström, P. Öffner, S. Tokareva, Analysis of the SBP-SAT stabilization for nite element methods. I: linear problems, J. Sci. Comput. 85
(2) (2020) 28, doi:
10.1007/s10915- 020- 01349- z .
[55] R. Abgrall, R. Nordström, P. Öffner, S. Tokareva, Analysis of the SBP-SAT stabilization for nite element methods part II: entropy stability, Commun.
Appl. Math. Comput. (2021), doi:
10.1007/s42967- 020- 00086- 2 .
[56] E. Caramana, R. Loubère, The force/work differencing of exceptional points in the discrete, compatible formulation of Lagrangian hydrodynamics, J.
Comput. Phys. 216 (2006) 1–18 .
[57] A. Bauera, D.E. Burton, E. Caramana, R. Loubère, M. Shashkov, P. Whalen, The internal consistency, stability, and accuracy of the discrete, compatible
formulation of Lagrangian hydrodynamics, J. Comput. Phys. 218 (2006) 572–593 .
[58] P. Maire, I. Bertron, R. Chauvin, B. Rebourcet, Thermodynamic consistency of cell-centered Lagrangian schemes, Comput. Fluids 203 (2020) 104527 .
[59] S. Busto, M. Dumbser, S. Gavrilyuk, K. Ivanova, On thermodynamically compatible nite volume methods and path-conservative ADER discontinuous
Galerkin schemes for turbulent shallow water ows, J. Sci. Comput. 88 (2021) 28 .
[60] S. Busto, M. Dumbser, A new thermodynamically compatible nite volume scheme for magnetohydrodynamics, SIAM J. Numer. Anal. (2022) . in press
[61] S. Busto, M. Dumbser, I. Peshkov, E. Romenski, On thermodynamically compatible nite volume schemes for continuum mechanics, SIAM J. Sci. Com-
put. 44 (2022) A1723–A1751 .
[62] S. Busto, M. Dumbser, On thermodynamically compatible discontinuous Galerkin methods for continuum mechanics and turbulent shallow water
ows, J. Sci. Comput. 93 (56) (2022), doi:
10.1007/s10915- 022- 02017- 0 .
[63] I. Peshkov, E. Romenski, A hyperbolic model for viscous Newtonian ows, Contin. Mech. Thermodyn. 28 (2016) 85–104 .
[64] M. Dumbser, I. Peshkov, E. Romenski, O. Zanotti, High order ADER schemes for a unied rst order hyperbolic formulation of continuum mechanics:
viscous heat–conducting uids and elastic solids, J. Comput. Phys. 314 (2016) 824–862 .
[65] F. Dhaouadi, M. Dumbser, A rst order hyperbolic reformulation of the Navier–Stokes–Korteweg system based on the GPR model and an augmented
Lagrangian approach, J. Comput. Phys. 470 (2022) 111544 .
[66] E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 2009 .
[67] G. Gassner, F. Lörcher, C. Munz, A contribution to the construction of diffusion uxes for nite volume and discontinuous Galerkin schemes, J. Comput.
Phys. 224 (2007) 1049–1063 .
[68] G. Jiang, C. Shu, On a cell entropy inequality for discontinuous Galerkin methods, Math. Comput. 62 (1994) 531–538 .
[69] S. Gottlieb, C. Shu, Total variation diminishing Runge–Kutta schemes, Math. Comput. 67 (1998) 73–85 .
[70] C. Hu, C. Shu, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys. 150 (1999) 97–127 .
[71] W. Boscheri, M. Dumbser, M. Ioriatti, I. Peshkov, E. Romenski, A structure-preserving staggered semi-implicit nite volume scheme for continuum
mechanics, J. Comput. Phys. 424 (2021) 109866 .
[72] A. Bermúdez, S. Busto, M. Dumbser, J. Ferrín, L. Saavedra, M. Vázquez-Cendón, A staggered semi-implicit hybrid FV/FE projection method for weakly
compressible ows, J. Comput. Phys. 421 (2020) 109743 .
[73] S. Busto, L.D. Rio, M. Vázquez-Cendón, M. Dumbser, A semi-implicit hybrid nite volume / nite element scheme for all Mach number ows on
staggered unstructured meshes, Appl. Math. Comput. 402 (2021) 126117 .
[74] R. Becker, Stosswelle und detonation, Physik 8 (1923) 321 .
[75] A. Bonnet, J. Luneau, Aérodynamique. Théories de la Dynamique des Fluides, Cepadues Editions, Toulouse, 1989 .
[76] J.B. Bell, P. Coletta, H.M. Glaz, A second-order projection method for the incompressible Navier–Stokes equations, J. Comput. Phys. 85 (1989) 257–283 .
39

R. Abgrall, S. Busto and M. Dumbser Applied Mathematics and Computation 440 (2023) 127629
[77] M. Tavelli, M. Dumbser, A staggered space–time discontinuous Galerkin method for the incompressible Navier–Stokes equations on two–dimensional
triangular meshes, Comput. Fluids 119 (2015) 235–249 .
[78] M. Tavelli, M. Dumbser, A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution
of the compressible Navier–Stokes equations at all Mach numbers, J. Comput. Phys. 341 (2017) 341–376 .
[79] S. Busto, J. Ferrín, E. Toro, M. Vázquez-Cendón, A projection hybrid high order nite volume/nite element method for incompressible turbulent ows,
J. Comput. Phys. 353 (2018) 169–192 .
[80] U. Ghia, K.N. Ghia, C.T. Shin, High-Re solutions for incompressible ow using Navier–Stokes equations and multigrid method, J. Comput. Phys. 48
(1982) 387–411 .
[81] S. Gavrilyuk, K. Ivanova, N. Favrie, Multi-dimensional shear shallow water ows: problems and solutions, J. Comput. Phys. 366 (2018) 252–280 .
[82] K. Ivanova, S. Gavrilyuk, Structure of the hydraulic jump in convergent radial ows, J. Fluid Mech. 860 (2019) 441–464 .
[83] M. Lukácová-Medvidóvá, G. Puppo, A. Thomann, An all Mach number nite volume method for isentropic two-phase ow, J. Numer. Math. (2022),
doi:
10.1515/jnma- 2022- 0015 . In press
[84] F. Thein, E. Romenski, M. Dumbser, Exact and numerical solutions of the Riemann problem for a conservative model of compressible two-phase ows,
J. Sci. Comput. (2022)
. in press
[85] M. Dumbser, D. Balsara, M. Tavelli, F. Fambri, A divergence-free semi-implicit nite volume scheme for ideal, viscous and resistive magnetohydrody-
namics, Int. J. Numer. Methods Fluids 89 (2019) 16–42 .
[86] H. Ranocha, L. Dalcin, M. Parsani, Fully discrete explicit locally entropy-stable schemes for the compressible Euler and Navier–Stokes equations, Com-
put. Math. Appl. 80 (5) (2020a) 1343–1359 .
[87] H. Ranocha, M. Sayyari, L. Dalcin, M. Parsani, D.I. Ketcheson, Relaxation Runge–Kutta methods: fully discrete explicit entropy-stable schemes for the
compressible Euler and navier-Stokes equations, SIAM J. Sci. Comput. 42 (2020b) A612–A638 .
[88] D. Mitsotakis, H. Ranocha, D. Ketcheson, E. Suli, A conservative fully discrete numerical method for the regularized shallow water wave equations,
SIAM J. Sci. Comput. 43 (2) (2021) B508–B537 .
[89] L. Brugnano, F. Iavernaro, Line Integral Methods for Conservative Problems, Chapman et Hall/CRC, Boca Raton, 2016 .
[90] L. Brugnano, F. Iavernaro, Line integral solution of differential problems, Axioms 7 (2) (2018) 36 .
[91] L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, J. Flaherty, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic
conservation laws, Appl. Numer. Math. 48 (3–4) (2004) 323–338 .
[92] D. Kuzmin, Hierarchical slope limiting in explicit and implicit discontinuous Galerkin methods, J. Comput. Phys. 257 (Part B) (2014) 1140–1162 .
[93] X. Zhang, C. Shu, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J.
Comput. Phys. 229 (23) (2010) 8918–8934 .
[94] M. Sonntag, C. Munz, Shock Capturing for discontinuous Galerkin methods using Finite Volume Subcells, in: J. Fuhrmann, M. Ohlberger, C. Rohde
(Eds.), Finite Volumes for Complex Applications VII, Springer, 2014, pp. 945–953 .
[95] M. Dumbser, O. Zanotti, R. Loubère, S. Diot, A posteriori subcell limiting of the discontinuous Galerkin nite element method for hyperbolic conserva-
tion laws, J. Comput. Phys. 278 (2014) 47–75 .
[96] M. Sonntag, C. Munz, Ecient parallelization of a shock capturing for discontinuous Galerkin methods using nite volume sub-cells, J. Sci. Comput. 70
(2017) 1262–1289 .
40