Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes for Hyperbolic Conservation Laws

Abstract

This work is focused on the entropy analysis of a semi-discrete nodal discontinuous Galerkin spectral element method (DGSEM) on moving meshes for hyperbolic conservation laws. The DGSEM is constructed with a local tensor-product Lagrange-polynomial basis computed from Legendre–Gauss–Lobatto points. Furthermore, the collocation of interpolation and quadrature nodes is used in the spatial discretization. This approach leads to discrete derivative approximations in space that are summation-by-parts (SBP) operators. On a static mesh, the SBP property and suitable two-point flux functions, which satisfy the entropy condition from Tadmor, allow to mimic results from the continuous entropy analysis, if it is ensured that properties such as positivity preservation (of the water height, density or pressure) are satisfied on the discrete level. In this paper, Tadmor’s condition is extended to the moving mesh framework. We show that the volume terms in the semi-discrete moving mesh DGSEM do not contribute to the discrete entropy evolution when a two-point flux function that satisfies the moving mesh entropy condition is applied in the split form DG framework. The discrete entropy behavior then depends solely on the interface contributions and on the domain boundary contribution. The interface contributions are directly controlled by proper choice of the numerical element interface fluxes. If an entropy conserving two-point flux is chosen, the interface contributions vanish. To increase the robustness of the discretization we use so-called entropy stable two-point fluxes at the interfaces that are guaranteed entropy dissipative and thus give a bound on the interface contributions in the discrete entropy balance. The remaining boundary condition contributions depend on the type of the considered boundary condition. E.g. for periodic boundary conditions that are of entropy conserving type, our methodology with the entropy conserving interface fluxes is fully entropy conservative and with the entropy stable interface fluxes is guaranteed entropy stable. The presented proof does not require any exactness of quadrature in the spatial integrals of the variational forms. As it is the case for static meshes, these results rely on the assumption that additional properties like positivity preservation are satisfied on the discrete level. Besides the entropy stability, the time discretization of the moving mesh DGSEM will be investigated and it will be proven that the moving mesh DGSEM satisfies the free stream preservation property for an arbitrary s-stage Runge–Kutta method, when periodic boundary conditions are used. The theoretical properties of the moving mesh DGSEM will be validated by numerical experiments for the compressible Euler equations with periodic boundary conditions.

Full text

Journal of Scientific Computing (2020) 82:69
https://doi.org/10.1007/s10915-020-01171-7
Entropy Stable Discontinuous Galerkin Schemes on Moving
Meshes for Hyperbolic Conservation Laws
Gero Schnücke1·Nico Krais2·Thomas Bolemann2·Gregor J. Gassner3
Received: 21 December 2018 / Revised: 13 February 2020 / Accepted: 20 February 2020 /
Published online: 3 March 2020
© The Author(s) 2020
Abstract
This work is focused on the entropy analysis of a semi-discrete nodal discontinuous Galerkin
spectral element method (DGSEM) on moving meshes for hyperbolic conservation laws.
The DGSEM is constructed with a local tensor-product Lagrange-polynomial basis com-
puted from Legendre–Gauss–Lobatto points. Furthermore, the collocation of interpolation
and quadrature nodes is used in the spatial discretization. This approach leads to discrete
derivative approximations in space that are summation-by-parts (SBP) operators. On a static
mesh, the SBP property and suitable two-point flux functions, which satisfy the entropy
condition from Tadmor, allow to mimic results from the continuous entropy analysis, if it is
ensured that properties such as positivity preservation (of the water height, density or pres-
sure) are satisfied on the discrete level. In this paper, Tadmor’s condition is extended to the
moving mesh framework. We show that the volume terms in the semi-discrete moving mesh
DGSEM do not contribute to the discrete entropy evolution when a two-point flux function
that satisfies the moving mesh entropy condition is applied in the split form DG framework.
The discrete entropy behavior then depends solely on the interface contributions and on the
domain boundary contribution. The interface contributions are directly controlled by proper
choice of the numerical element interface fluxes. If an entropy conserving two-point flux is
chosen, the interface contributions vanish. To increase the robustness of the discretization we
use so-called entropy stable two-point fluxes at the interfaces that are guaranteed entropy dis-
sipative and thus give a bound on the interface contributions in the discrete entropy balance.
The remaining boundary condition contributions depend on the type of the considered bound-
ary condition. E.g. for periodic boundary conditions that are of entropy conserving type, our
methodology with the entropy conserving interface fluxes is fully entropy conservative and
with the entropy stable interface fluxes is guaranteed entropy stable. The presented proof does
not require any exactness of quadrature in the spatial integrals of the variational forms. As it
is the case for static meshes, these results rely on the assumption that additional properties
like positivity preservation are satisfied on the discrete level. Besides the entropy stability,
the time discretization of the moving mesh DGSEM will be investigated and it will be proven
that the moving mesh DGSEM satisfies the free stream preservation property for an arbitrary
s-stage Runge–Kutta method, when periodic boundary conditions are used. The theoretical
BGero Schnücke
gschnuec@math.uni-koeln.de
Extended author information available on the last page of the article
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69 Page 2 of 42 Journal of Scientific Computing (2020) 82 :69
properties of the moving mesh DGSEM will be validated by numerical experiments for the
compressible Euler equations with periodic boundary conditions.
Keywords Discontinuous Galerkin ·Summation-by-parts ·Moving meshes ·Entropy
stability ·Free stream preservation
1 Introduction
A lot of applications in engineering and physics require the approximation of conservation
laws on time-dependent domains, e.g. domains with moving boundaries. For instance, mov-
ing mesh discontinuous Galerkin (DG) methods have been investigated in [5,43,52,54]. In
particular, moving mesh discontinuous Galerkin spectral element methods (DGSEM) have
been constructed and analyzed in [37,48,64]. In the literature, there are also moving mesh
methods with the capability to change the connectivity of the mesh, e.g. with finite volume
(FV) methods [44,57] and with a DG method [60]. Moving mesh finite difference meth-
ods were constructed in [1,51], in [66] a moving mesh collocation method was constructed
and in [27] a moving mesh continuous finite element method was constructed. In general,
moving mesh methods are well suited to preserve motion related properties like the Galilean-
invariance. These properties are necessary to describe physical processes like the formation
of disc galaxies [45].
A common way to approximate conservation laws on time-dependent domains is to use the
Arbitrary Lagrangian–Eulerian (ALE) approach [17]. In this approach the conservation law
is transformed from the time-dependent domain onto a time-independent reference domain.
The motion of the mesh on the physical domain is part of the transformation. Thus, the grid
velocity field appears as a new quantity in the equation on the reference domain. On the one
hand the ALE transformation simplifies the discretization, since a static mesh can be used in
the reference domain. On the other hand, the new quantities in the equation on the reference
domain complicate the discrete stability analysis, even in the linear case [37].
In this work, moving mesh DGSEM to solve non-linear, symmetrizable and hyperbolic
systems of conservation laws are investigated. It is well known that symmetrizable systems
are equipped with an entropy/entropy flux pair [26,49]. For scalar conservation laws, entropy
admissibility criteria provide the unique physically relevant weak solution [15,39]. In gen-
eral, entropy admissibility criteria are not enough to ensure well-posedness for systems of
conservation laws [11]. Nevertheless, the entropy is an essential quantity to analyze systems
of conservation laws. In particular, for gas dynamics a possible mathematical entropy is
the scaled negative thermodynamic entropy which shows that the mathematical model cor-
rectly captures the second law of thermodynamics [2]. The entropy is conserved for smooth
solutions of a conservation law and decays for discontinuous solutions [29,59].
It is reasonable to construct numerical schemes for conservation laws which reflect the
properties of the entropy on the discrete level. Tadmor [58] developed a discrete entropy
criterion to construct a specific class of two-point flux functions for low-order finite differ-
ence (FD) and FV methods. FD/FV methods with these class of two-point flux functions
preserve entropy on the discrete level. Moreover, these FV methods can be modified by
adding dissipation to the numerical fluxes such that the entropy is decreasing for all times.
Therefore, two-point fluxes with Tadmor’s discrete entropy condition are called entropy
conservative fluxes. LeFloch et al. [40] gave a framework to construct high-order entropy
conservative schemes in periodic domains. Fisher and Carpenter [20] combined this approach
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Journal of Scientific Computing (2020) 82 :69 Page 3 of 42 69
with summation-by-parts (SBP) operators and proved that two-point entropy conservative
fluxes can be used to construct high-order schemes when the derivative approximations in
space are SBP operators. A SBP operator provides a discrete analogue of the integration-
by-parts formula [19,22,38]. It is worth to mention that the derivative matrix in the DGSEM
provides a SBP operator, if the tensor-product Lagrange-polynomial basis is computed from
Legendre–Gauss–Lobatto (LGL) points and interpolation and quadrature are collocated.
Gassner et al. [23,24] showed that split forms of the partial differential equations can be
discretely recovered when specific choices of numerical volume fluxes in the flux form vol-
ume integral of Fisher and Carpenter are chosen. Thus, an entropy stable DGSEM can be
constructed by the following building blocks:
(1) The derivative matrix satisfies the SBP property.
(2) There are two-point flux functions with Tadmor’s discrete entropy condition that can be
extended to high-order in a split form DG framework.
This methodology has been used in the construction of high-order entropy stable DGSEM
on quadrilateral/hexahedral elements, e.g. [4,23,61], or on triangular/tetrahedral elements,
e.g. [6,10,13]. All these methods are provably entropy stable and the semi-discrete entropy
analysis for them is based merely on the properties of the SBP operators and the assumptions
that the time integration is exact. Additionally, properties like positivity preservation (of the
water height, density or pressure) must be satisfied on the discrete level. The exactness of
quadrature in the spatial integrals of the variational form is not necessary. Available entropy
stable moving mesh methods are for instance the low order continuous finite element method
by Guermond et al. [27] and a spectral collocation based approach published during the
extended review process of the current paper by Yamaleev et al. [66].
The remainder of the paper is organized as follows: The ALE transformation and continu-
ous entropy analysis is presented in the Sects. 2.1,2.2 and 2.3. The framework for the spectral
element discretization with the SBP operator is given in the Sect. 2.4 and the DG split form
framework is presented in the Sect. 2.5. The moving mesh DGSEM is finally presented in the
Sect. 2.5. A discrete entropy analysis for the moving mesh DGSEM is given in the Sects. 2.6
and 2.7. Furthermore, in Sect. 2.8 it is proven that the moving mesh DGSEM satisfies the free
stream preservation property. In Sect. 4, numerical examples with the compressible Euler
equations are presented to validate our theoretical findings.
2 Entropy Stable DGSEM on Moving Meshes
The main goal of this work is the construction of an entropy stable moving mesh DGSEM. On
static meshes, it is possible to construct high-order entropy stable DGSEM, if the derivative
matrix is an SBP operator and entropy conservative two-point flux functions are available.
This methodology has been used in the construction of high-order entropy stable DGSEM
on quadrilateral/hexahedral elements, e.g. [4,23,61]. In this section, it will be shown that
similar ideas can be used to construct high-order entropy stable moving mesh DGSEM. The
construction of the entropy stable moving mesh DGSEM will be presented for an arbitrary
symmetrizable and hyperbolic system of conservation laws
∂u
∂t+
3

i=1
∂fi
∂xi=0,(2.1)
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69 Page 4 of 42 Journal of Scientific Computing (2020) 82 :69
on a time-dependent domain (t)⊆R3. The vector of conservative variables is uand
fi,i=1,2,3, are the physical flux vectors. The state vectors are of size pdepending on the
number of equations in the system under consideration and the conservation law is subjected
to appropriate initial and boundary conditions (see the comment below Eq. (2.44) in Sect. 2.3).
The block vector nomenclature in [24] simplifies the analysis of the system (2.1)on
curved elements. Thus, we translate the conservation law (2.1) in block vector notation. A
block vector is highlighted by the double arrow
↔
f:= ⎡
⎣
f1
f2
f3⎤
⎦.(2.2)
The dot product of two block vectors is given by
↔
f·↔
g:=
3

i=1
fT
igi.(2.3)
Furthermore, the dot product of a vector vin the three dimensional space and a block vector
is defined by
v·↔
f:=
3

i=1
vifi.(2.4)
We note that the dot product (2.3) is a scalar quantity and the dot product (2.4) is a vector
in a pdimensional space, where the number pcorresponds to the number of conservative
variables in the conservation law(2.1). The interaction between a vector vand the conservative
variables is defined as the block vector
vu:= ⎡
⎣
v1u
v2u
v3u⎤
⎦.(2.5)
Thus, in particular, the spatial gradient of the conservative variables is defined by

∇xu:= ⎡
⎢
⎣
∂u
∂x1
∂u
∂x2
∂u
∂x3
⎤
⎥
⎦.(2.6)
The gradient of a vector valued function g=[g1,g2,g3]Tis a second order tensor, written
in matrix form as

∇x⊗gT=⎡
⎢
⎣
∂g1
∂x1
∂g1
∂x2
∂g1
∂x3
∂g2
∂x1
∂g2
∂x2
∂g2
∂x3
∂g3
∂x1
∂g3
∂x2
∂g3
∂x3
⎤
⎥
⎦,(2.7)
where ⊗is the outer product of two vectors in a three dimensional space. The dot product
(2.3) and the spatial gradient (2.6) are used to define the divergence of a block vector flux as

∇x·↔
f:=
3

i=1
∂fi
∂xi
.(2.8)
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Journal of Scientific Computing (2020) 82 :69 Page 5 of 42 69
Moreover, for a vector valued function gand the conservative variables, we have the product
rule

∇x·(gu)=
∇x·gu+g·
∇xu(2.9)
with respect to the dot products (2.3)and(2.4). These notations allow to write the conservation
law (2.1) in the compact form
∂u
∂t+
∇x·↔
f=0.(2.10)
2.1 Building Blocks of the ALE Transformation for Hexahedral Curved Meshes
In order to set up the moving mesh DGSEM in the Sect. 2.5,wemakeforallt∈[0,T]the
assumptions:
(A1) For a fixed number K∈Nthe physical domain (t)can be subdivided into K
time-dependent, non-overlapping and conforming hexahedral elements, eκ(t),κ=
1,...,K. These elements can have curved faces.
(A2) The time-dependent elements eκ(t)are mapped into the spatial computational domain
E=[−1,1]3with a bijective isoparametric transfinite mapping
eκ(t)x(t)=χ
ξ,τ,
ξ∈E,τ∈[0,T].(2.11)
Winters constructed in his PHD thesis [65] a mapping for this set up. Like in [65], it is
assumed that the curved faces satisfy for all τ∈[0,T]

1−1,ξ3,τ=
6−1,ξ3,τ,
2−1,ξ3,τ=
61,ξ3,τ,

3−1,ξ2,τ=
6ξ2,−1,τ,

11,ξ3,τ=
4−1,ξ3,τ,
21,ξ3,τ=
41,ξ3,τ,

31,ξ2,τ=
4ξ2,−1,τ,

1ξ1,−1,τ=
3ξ1,−1,τ,
2ξ1,−1,τ=
3ξ1,1,τ,

5−1,ξ2,τ=
6ξ2,1,τ,

1ξ1,1,τ=
5ξ1,−1,τ,
2ξ1,1,τ=
5ξ1,1,τ,

51,ξ2,τ=
4ξ2,1,τ.(2.12)
The location of the curved faces is sketched in Fig. 1. The curved faces of an element
eκ(t)are approximated as interpolation polynomials up to degree Nsuch that
IN
i(η, ζ, τ ):=
N

j,k=0
iηj,ζ
k,τj(η)k(ζ),i=1,2,3,4,5,6,(2.13)
where jN
j=0,{k}N
k=0are the Lagrange polynomials associated with the interpolation
points ηjN
j=0and {ζk}N
k=0.
(A3) The determinant Jof the Jacobian matrix 
∇
ξ⊗χT
satisfies
J:= det 
∇
ξ⊗χ>0,∀τ∈[0,T].(2.14)
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Fig. 1 Left the reference element E=[−1,1]3and on the right a general hexahedral element eκ(t)
with the curved faces 
1ξ1,ξ3,τ,
2ξ1,ξ3,τ,
3ξ1,ξ2,τ,
4ξ2,ξ3,τ,
5ξ1,ξ2,τ,and

6ξ2,ξ3,τ. The mapping x(t)=χ
ξ,τconnects Eand eκ(t)
Mesh curving techniques are discussed by Hindenlang et al. [30] and methodologies to
construct a moving mesh with the properties (A1)–(A3) are given in the literature e.g. the
book of Huang and Russell [32, Chapter 6, Chapter 7]. In many situations the moving mesh
methodology depends on the underlying problem, e.g. [45].
The mapping provides the grid velocity field
ν=[ν1,ν
2,ν
3]T:= ∂χ1
∂τ ,∂χ2
∂τ ,∂χ3
∂τ T
=∂χ
∂τ .(2.15)
It is desirable that the grid velocity is continuous, since the mesh should be conforming
and watertight at each time level. The next statement provides conditions on the element
boundaries to guarantee that the grid velocity becomes continuous.
Lemma 2.1 Let e1(t)and e2(t)be two neighboring elements which share one of the faces

1
1=
2
2,
1
3=
2
5,
1
4=
2
6,
2
1=
1
2,
2
3=
1
5,
2
4=
1
6,(2.16)
where 
l
i,l =1,2, and i =1,2,3,4,5,6, are the faces of the element el(t). Furthermore,
suppose that the faces 
l
i(·,·,τ)are continuously differentiable in the time interval [0,T].
Then the grid velocity field is continuous in the points which belong to the face that the
elements share.
In Appendix Athe Lemma 2.1 is proven in two dimensions. The three dimensional proof
can be done by the same argumentation.
2.2 Transformation of the Conservation Law onto a Reference Element
In the following, we show that the system (2.10) can be transformed from a time-dependent
element eκ(t)on the reference element E. The mapping (A.1) provides the covariant basis
vectors
ai:= ∂χ
∂ξi,i=1,2,3,(2.17)
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Journal of Scientific Computing (2020) 82 :69 Page 7 of 42 69
and the volume weighted contravariant vectors
Jai=aj×ak,(i,j,k)cyclic.(2.18)
The quantity 
ξ=ξ1,ξ2,ξ3Tis a vector in the reference element E=[−1,1]3.Thecovari-
ant and the volume weighted contravariant vectors represent the Jacobian matrix 
∇
ξ⊗χT
and its adjoint matrix

∇
ξ⊗χT=a1a2a3,adj 
∇
ξ⊗χT=⎡
⎢
⎣Ja1T
Ja2T
Ja3T⎤
⎥
⎦.(2.19)
Furthermore, the contravariant vectors satisfy the metric identities
3

i=1
∂Jai
∂ξi=0.(2.20)
In particular, the covariant and the contravariant vectors allow to transform differential oper-
ators on the time-independent reference element E. On the reference element the gradient of
a function fis given by

∇xf=1
J3

i=1
Jai∂f
∂ξi=1
Jadj 
∇
ξ⊗χ 
∇ξf(2.21)
and the divergence of a vector valued function gis given by

∇x·g=1
J
3

i=1
∂
∂ξiJai·g=1
J
∇ξ·
˜g,(2.22)
where we used the contravariant flux

˜g:= ⎡
⎣
Ja1·g
Ja2·g
Ja3·g⎤
⎦=adj 
∇
ξ⊗χTg.(2.23)
In [24], the following block matrix has been introduced to combine the transformations (2.21)
and (2.22) with the block vector notation
M=⎡
⎣
Ja1
1IpJa2
1IpJa3
1Ip
Ja1
2IpJa2
2IpJa3
2Ip
Ja1
3IpJa2
3IpJa3
3Ip⎤
⎦,(2.24)
where the matrix Ipis the p×pidentity matrix and Jai
jis the component of Jaiin the j-th
Cartesian coordinate direction. The transformation of the gradient becomes

∇xu=1
JM
∇ξu.(2.25)
We note that for a vector valued function gthe following identity holds
g·
∇xu=1
Jg·M
∇ξu=1
J
˜g·
∇ξu.(2.26)
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69 Page 8 of 42 Journal of Scientific Computing (2020) 82 :69
Moreover, by applying the metric identities (2.20), the transformation of the divergence can
be written as

∇x·↔
f=1
J
∇ξ·MT↔
f.(2.27)
Hence, the contravariant block vector flux is given by
↔
˜
f:= ⎡
⎢
⎢
⎣
Ja1·↔
f
Ja2·↔
f
Ja3·↔
f
⎤
⎥
⎥
⎦=MT↔
f.(2.28)
Since the elements {ek(t)}K
k=1are time-dependent, the time evolution of the quantity Jneeds
to be analyzed. Thus, we apply Jacobi’s formula (cf. e.g. Bellman [3]) and obtain by (2.15),
(2.19)
∂J
∂τ =tr adj 
∇
ξ⊗χT∂
∂τ 
∇⊗χT=
3

i=1
Jai·∂ai
∂τ =
3

i=1
Jai·∂ν
∂ξi,
(2.29)
where tr [·]denotes the trace of a matrix. The metric identities (2.20) allow to write the
Eq. (2.29) in conservation form
∂J
∂τ =
3

i=1
∂
∂ξiJai·ν=
∇ξ·
˜ν. (2.30)
The chain rule formula and the identity (2.26) provide
∂u
∂τ =∂u
∂t+1
J
˜ν·
∇ξu.(2.31)
Next,weplug(2.30) into Eq. (2.31), apply the product rule (2.9) and rearrange. This provides
the equation
J∂u
∂t=∂(Ju)
∂τ −
∇ξ·
˜νu.(2.32)
Finally, we combine the identities (2.27)and(2.32) to write the the conservation law (2.10)
in the following form
∂(Ju)
∂τ +
∇ξ·↔
˜
g=0,(2.33)
where
↔
g=⎡
⎣
g1
g2
g3⎤
⎦:= ⎡
⎣
f1−ν1u
f2−ν2u
f3−ν3u⎤
⎦=↔
f−νu.(2.34)
The formulation (2.33) is the representation of the system (2.10) on the time-independent
reference element Efor a time-dependent element eκ(t).
Remark 2.2 The metric identities (2.20) and the Eq. (2.30) provide the geometric conser-
vation law (GCL) [18,28,41,42,46]. A numerical method to solve (2.10)onmovingand
deforming grids needs to satisfy both equations, otherwise the conservation properties of the
conservation law (2.10) are not preserved. Farhat et al. [18,28,41] proved that the absence of
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Journal of Scientific Computing (2020) 82 :69 Page 9 of 42 69
these equations has a critical effect on the accuracy and stability of a moving mesh method.
In particular, the preservation of constant states is no longer guaranteed, if the GCL is not
satisfied on the discrete level.
2.3 Entropy Analysis in Three Dimensions
The system (2.1) is assumed to be symmetrizable. Thus, in particular, it is equipped with
entropy/entropy flux pairs s,fs
i,i=1,2,3, (cf. Godunov [26] and Mock [49]). The
strictly convex function sis the entropy function. The entropy function sprovides the entropy
variables
w:= ∂s
∂u,(2.35)
and it follows by the chain rule
∂s
∂t=wT∂u
∂t,∂s
∂xi=wT∂u
∂xi
,i=1,2,3.(2.36)
The entropy flux functions and the flux functions in the conservation law are related and
satisfy
wT∂fi
∂xi=∂fs
i
∂xi
,i=1,2,3.(2.37)
The identities (2.26)and(2.36)give
wT
˜ν·
∇ξu=JwTν·
∇xu=Jν·
∇xs=
˜ν·
∇ξs.(2.38)
Hence, we obtain with the identity (2.31) and the chain rule
JwT∂u
∂τ =J∂s
∂t+
˜ν·
∇ξs=J∂s
∂τ .(2.39)
Therefore, the product rule provides the identity
wT∂(Ju)
∂τ =J∂s
∂τ +∂J
∂τ wTu
=∂(Js)
∂τ +∂J
∂τ wTu−s
=∂(Js)
∂τ +
∇ξ·
˜νwTu−s,
(2.40)
whereweusedtheGCL(2.30) in the last step. Next, we apply the relation (2.37)forthe
entropy flux functions and obtain
wT∂gi
∂xi=∂
∂xfs
i−νis−∂vi
∂xiwTu−s,i=1,2,3.(2.41)
Next, we apply the vector notation 
fs:= fs
1,fs
2,fs
3T.Then(2.41) and the transformation
formulas for the gradient and divergence in the Sect. 2.2 give
wT
∇ξ·↔
˜
g=
∇ξ·˜
fs−
˜νs−
∇ξ·
˜νwTu−s.(2.42)
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69 Page 10 of 42 Journal of Scientific Computing (2020) 82 :69
Finally, the identities (2.40)and(2.42) provide the balance law
0=wT∂(Ju)
∂τ +
∇ξ·↔
˜
g=∂(Js)
∂τ +
∇ξ·˜
fs−
˜νs.(2.43)
We integrate the Eq. (2.43) over the domain E×[0,T]and obtain
∂
∂τ 
E
Jsd
ξdτ=−T
0∂E˜
fs−
˜νsTˆndSdτ. (2.44)
Boundary conditions then need to be specified so that the bound on the entropy depends only
on the boundary data. We will assume here that boundary data is given in a way that the right
hand side in Eq. (2.44) is non-positive so that the entropy will not increase in time.
For discontinuous solutions the Eq. (2.44) is not satisfied, but under further assumptions
it is possible to proof that a weak solution of (2.33) satisfies the inequality
∂(Js)
∂τ +
∇ξ·˜
fs−
˜νs≤0 (2.45)
in the sense of distributions on E×(0,T)(see Godlewski and Raviart [25, Chapter 1,
Theorem 3.3]). The inequality (2.45) means that it holds the inequality
T
0E
Js∂φ
∂τ d
ξdτ≥−T
0E˜
fs−
˜νsT
∇ξφd
ξdτ, ∀φ∈C∞
0(E×(0,T)) ,φ≥0.
(2.46)
2.4 Building Blocks for the Spectral Element Approximation
A nodal approach is used for the spectral element approximation. The Lagrange basis func-
tions are given by
j(ξ):=
N

i=0,j=i
ξ−ξi
ξj−ξi
,j=0,...N,(2.47)
where the nodal points {ξi}N
i=0are the LGL points. We note that ξ0=−1andξN=1. The
Lagrange basis functions satisfy the cardinal property
iξj=δji,(2.48)
where δji is the Kronecker delta. On the reference element E=[−1,1]3the solution and
fluxes of the system (2.33) are approximated by tensor product Lagrange polynomials of
degree N, e.g.
uξ1,ξ2,ξ3,t≈Uξ1,ξ2,ξ3,t:=
N

i,j,k=0
Uijk (t)iξ1jξ2kξ3.(2.49)
In the following, polynomial approximations are highlighted by capital letters, e.g. Uis an
approximation for the state vector uand Fl,l=1,2,3, are approximations for the fluxes fl,
l=1,2,3. The determinant Jof the Jacobian matrix 
∇
ξχis also approximated by tensor
product Lagrange polynomials
Jξ1,ξ2,ξ3,t≈Jξ1,ξ2,ξ3,t:=
N

i,j,k=0
Jijk (t)iξ1jξ2kξ3.(2.50)
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In particular, the interpolation operator for a function gis given by
IN(g)ξ1,ξ2,ξ3=
N

i,j,k=0
gijkiξ1jξ2kξ3,(2.51)
where gijk := gξ1
i,ξ2
j,ξ3
kand ξ1
iN
i=0,ξ2
iN
i=0,ξ3
iN
i=0are sets of LGL points. Deriva-
tives are approximated by exact differentiation of the polynomial interpolants. In general we
have IN(g)= IN−1g(cf. e.g. [7,36]), as differentiation and interpolation only commute
if there are no discretization errors. However, the contravariant coordinate vectors need to be
discretized in such a way that the metric identities (2.20) are satisfied on the discrete level,
too. Kopriva [35] introduced the conservative curl form that computes
Jaα
β:= −ˆxα·
∇ξ×INχγ
∇ξχδ,α=1,2,3,β=1,2,3,(β, γ, δ)cyclic,
(2.52)
to approximate the metric terms. Here χ=[χ1,χ
2,χ
3]Trepresents the mapping from the
element to the reference element and ˆxiis the unit vector in the i-th Cartesian coordinate
direction. The representation (2.52) ensures that
3

α=1
∂INJaα
β
∂ξα=0,β=1,2,3.(2.53)
From now on, the discrete contravariant coordinate vectors are denoted by Jaα
β, when the
curl form (2.52) has been used to compute these quantities.
Integrals are approximated by a tensor product extension of a 2N−1 accurate LGL
quadrature formula. Hence, interpolation and quadrature nodes are collocated. In one spatial
dimension the LGL quadrature formula is given by
1

−1
g(ξ)dξ≈
N

i=0
ωig(ξi)=
N

i=0
ωigi,(2.54)
where ωi,i=0,...,N, are the quadrature weights and ξi,i=0,..., N,aretheLGL
quadrature points. The formula (2.54) motivates the definition of the discrete quantity
f,gN:=
N

i=0
N

j=0
N

k=0
ωiωjωkfT
ijkgijk =
N

i,j,k=0
ωijkfT
ijkgijk (2.55)
for two functions fand g. We note that (2.55) satisfies
IN(g),ϕN=g,ϕN,∀ϕ∈PNE,Rp.(2.56)
Furthermore, for a block vector ↔
fand test functions ϕ∈PN(E,Rp),wedefinethediscrete
surface integral
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69 Page 12 of 42 Journal of Scientific Computing (2020) 82 :69

∂E,N
ϕT↔
f·ˆndS :=
N

j,k=0
ωjωkϕT
Njk(F1)Njk −ϕT
0jk (F1)0jk
+
N

i,k=0
ωiωkϕT
iNk (F2)iNk −ϕT
i0k(F2)i0k
+
N

i,j=0
ωiωjϕT
ijN (F3)ijN −ϕT
ij0(F3)ij0,
(2.57)
where ˆnis the unit outward normal at the faces of the reference element E.
The spectral element approximation with LGL points for interpolation and quadrature
provides a SBP operator Q=MD with the mass matrix Mand the derivative matrix D.
The mass matrix and the derivative matrix are given by
Mij =ωiδij,Dij =j(ξi)i,j=0,...,N.(2.58)
The important characteristic of this SBP operator is the property
Q+QT=B,(2.59)
where B=diag (−1,0,...,0,1). A SBP operator provides a discrete analogue of the
integration-by-parts formula [19,22,38].
Finally, we note that in the LGL points ξ1
i,ξ2
j,ξ3
k,i,j,k=0,...,N,theEq.(2.53)gives
N

m=0Dim Ja1
βmjk +Djm Ja2
βimk +Dkm Ja3
βijm=0,β=1,2,3.(2.60)
2.5 The Semi-discrete Discontinuous Galerkin Method
Now, we apply the notation introduced in Sect. 2.4 and construct a moving mesh DGSEM.
We discretize the Eqs. (2.30)and(2.33) simultaneously. In this way, it is ensured that the
Eq. (2.30) is satisfied on the discrete level [37,48,64]. First, we replace the solution uby
(2.49), the Jacobian Jby (2.50) and approximate the fluxes by the interpolation operator
(2.51). Next, we multiply the GCL (2.30) by test functions ϕ∈PN(E),theEq.(2.33) with
ϕ∈PN(E,Rp), integrate the resulting equations and use integration-by-parts to separate
boundary and volume contributions. The volume integrals in the variational form are approx-
imated with the LGL quadrature. Then, we insert numerical surface fluxes 
˜ν∗and ↔
˜
G∗at
the spatial element interfaces. Afterwards, we use the SBP property (2.59) for the volume
contribution to get the standard DGSEM in strong form:
∂J
∂τ ,ϕ
N=
∇ξ·IN
˜ν,ϕ
N+
∂E,N
ϕ˜ν∗
ˆn−˜νˆndS,∀ϕ∈PN(E),(2.61a)
∂(JU)
∂τ ,ϕ N=−
∇ξ·IN↔
˜
g,ϕ N−
∂E,N
ϕT˜
G∗
ˆn−˜
GˆndS,∀ϕ∈PNE,Rp,
(2.61b)
where we used the notation (2.55) and the notation (2.57) for the discrete surface integral.
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The approximation of 
˜νand the nonlinear flux ↔
˜
gby the interpolation operator (2.51)
causes aliasing errors in the standard strong form. The aliasing errors cannot be bounded
and the errors are independent of the choice of the numerical surface flux. In Gassner [22]
a detailed explanation and analysis of the aliasing problem is given. Furthermore, a spe-
cific reformulation of the volume integrals by using the skew-symmetry strategy has been
developed to fix the aliasing problem. This approach has been enhanced and generalized by
Gassner et al. in [23,24] with a technique developed for high-order FD schemes (LeFloch
et al. [40]orFisherandCarpenter[20]). The generalized approach is called split form DG
framework. Here, we proceed similar as in [24] and replace the interpolation operators in
the discrete volume integrals by derivative projection operators. The interpolation operator
in the discrete equation for the GCL (2.30) is replaced by

DN·
˜νijk :=
N

m=0
2Dim{{ ν}}(i,m)jk ·{{Ja1}}(i,m)jk
+2Djm{{ ν}}i(j,m)k·{{Ja2}}i(j,m)k
+2Dkm{{ ν}}ij(k,m)·{{Ja3}}ij(k,m)
(2.62)
with the volume averages of the metric terms, e.g.
{{·}} (i,m)jk := 1
2(·)ijk +(·)mjk.(2.63)
The derivative projection operator in the discrete equation for (2.33) is computed as in [24].
Thus, the operator is given by

DN·↔
˜
GEC
ijk :=
N

m=0
2Dim ↔
GEC νijk,νmjk,Uijk,Umjk·{{Ja1}}(i,m)jk
+2Djm ↔
GEC νijk,νimk,Uijk,Uimk·{{Ja2}}i(j,m)k
+2Dkm ↔
GEC νijk,νijm,Uijk,Uijm·{{Ja3}}ij(k,m).
(2.64)
We note that the discrete volume weighted contravariant vectors Jal,l=1,2,3, in the
derivative projection operator (2.62)and(2.64) are computed by the conservative curl form
(2.52). The flux ↔
GEC is consistent and symmetric such that, e.g.
↔
GEC νijk,νmjk,U,U=↔
F(U)−{{v}} (i,m)jkU,(2.65)
and
↔
GEC νijk,νmjk,Uijk,Umjk=↔
GEC νmjk,νijk,Umjk,Uijk,(2.66)
for i,j,k,m=0,...,N. Furthermore, the flux functions GEC
l,l=1,2,3, satisfy for
i,j,k,m=0,...,N, the following discrete entropy conditions
[[W]]T
(i,m)jkGEC
lνijk,νmjk,Uijk,Umjk=[[l]] (i,m)jk −{{νl}}(i,m)jk [[]](i,m)jk,
[[W]]T
i(j,m)kGEC
lνijk,νimk,Uijk,Uimk=[[l]]i(j,m)k−{{νl}}i(j,m)k[[ ]]i(j,m)k,
[[W]]T
ij(k,m)GEC
lνijk,νijm,Uijk,Uijm=[[l]]ij(k,m)−{{νl}}ij(k,m)[[]]ij(k,m).
(2.67)
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The quantities and lare polynomial approximations which satisfy in the LGL points
ijk =WT
ijkUijk −Sijk,(l)ijk := WT
ijk (Fl)ijk −Fs
lijk ,l=1,2,3,(2.68)
where Wijk,Sijk and Fs
lijk are the nodal values of the polynomials
W:= IN(w),S:= IN(s),Fs
l:= INfs
l,l=1,2,3.(2.69)
Here, srepresents an entropy for the system (2.1) with the corresponding entropy flux func-
tions fs
l,l=1,2,3, and entropy variables w. Furthermore, the volume jumps in (2.67)are,
e.g.
[[·]] (i,m)jk := (·)ijk −(·)mjk .(2.70)
In Appendix B, flux functions with these properties are presented for the Euler equations.
Finally, for each element eκ(t)the semi-discrete moving mesh DGSEM can be written in
the following form:
∂J
∂τ ,ϕ
N=
DN·
˜ν, ϕN+
∂E,N
ϕ˜ν∗
ˆn−˜νˆndS,∀ϕ∈PN(E),(2.71a)
∂(JU)
∂τ ,ϕ N=−
DN·↔
˜
GEC,ϕ N−
∂E,N
ϕT˜
G∗
ˆn−˜
GˆndS,∀ϕ∈PNE,Rp.
(2.71b)
The unit outward facing normal vector and surface element on the element side are con-
structed from the element metrics by
n:= 1
ˆs
3

l=1Jalˆnl,ˆs:= !!!!!
3

l=1Jalˆnl!!!!!
.(2.72)
Thus, the quantity ˜νˆnin (2.71a) and the flux ˜
Gˆnin (2.71b) are defined by
˜νˆn=ˆsn·ν=
3

l=1ˆnlJal
1ν1+Jal
2ν2+Jal
3ν3,(2.73)
˜
Gˆn=ˆsn·↔
G=
3

l=1ˆnlJal
1G1+Jal
2G2+Jal
3G3=M↔
G·ˆn.(2.74)
To define the numerical surface fluxes in (2.71a)and(2.71b), we introduce notation for states
at the LGL nodes along an interface between two spatial elements to be a primary “−”and
complement the notation with a secondary “+” to denote the value at the LGL nodes on the
opposite side. Then the orientated jump and the arithmetic mean at the interfaces are defined
by
[[·]] : = (·)+−(·)−,and {{·}} := 1
2(·)++(·)−.(2.75)
When applied to vectors, the average and jump operators are evaluated separately for each
vector component. Then the normal vector nis defined unique to point from the “−”to
the “+” side. This notation allows to compute the contravariant surface numerical fluxes in
(2.71a)as
˜ν∗
ˆn=ˆs(n1{{v1}} + n2{{v2}} + n3{{ v3}}).(2.76)
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Journal of Scientific Computing (2020) 82 :69 Page 15 of 42 69
We note that due to the assumptions made in Sect. 2.1, the mesh velocity is a continuous func-
tion and the averages reduce to the uniquely defined values on the surface. The contravariant
surface numerical fluxes in (2.71b)aregivenby
˜
G∗
ˆn=ˆsn1GEC
1+n2GEC
2+n3GEC
3,(2.77)
where the Cartesian fluxes GEC
l,l=1,2,3, satisfy (2.65), (2.66), (2.67). We note that these
fluxes are the baseline choices without interface dissipation, to get a baseline scheme that is
entropy conservative.
Remark 2.3 (i) The discrete volume weighted contravariant vectors Jaα,α=1,2,3, do
not dependent on the solution J of (2.71a), since these vectors are computed by the
conservative curl form (2.52). Thus, the discrete metric identities (2.53) are satisfied
and the normal computation (2.72) is watertight. This means the normal vector and the
surface element are continuous across element interfaces.
(ii) Since the discrete volume weighted contravariant vectors Jaα,α=1,2,3, are com-
puted by the conservative curl form (2.52), the Eq. (2.20) is satisfied on the discrete
level.
(iii) The Eqs. (2.71a)and(2.71a) ensure that the Eq. (2.30) is satisfied on the discrete level.
2.6 Semi-discrete Entropy Conservation
The spatial integral of the entropy is bounded in time on the continuous level. Thus, it is
desirable that a numerical method is stable in the sense that a discrete version of this integral
is bounded in time, too. In the context of the moving mesh semi-discrete DGSEM (2.71), we
are interested to find an upper bound for the quantity
¯
S(τ):=
K

k=1S(τ),J(τ)N,∀τ∈[0,T],(2.78)
where S=IN(s)is a polynomial approximation for the entropy s. Next, we prove the
following statement for the semi-discrete moving mesh DGSEM.
Theorem 2.4 Suppose the flux functions ↔
GEC in the derivative projection operator (2.64),
the numerical surface fluxes ˜
G∗
ˆnare computed by Cartesian fluxes GEC
l,l =1,2,3, with
the properties (2.65),(2.66),(2.67)and periodic boundary conditions are used. Then the
semi-discrete moving mesh DGSEM (2.71)satisfies the discrete entropy equation
¯
S(τ)=¯
S(0),∀τ∈[0,T],(2.79)
where ¯
S(τ)is given by (2.78).
Proof We proceed similar as in the continuous entropy analysis, use the polynomial approx-
imation ϕ=IN(w)=Was test function in the Eq. (2.71b) and obtain
∂(JU)
∂τ ,W N=−
DN·↔
˜
GEC,W N−
∂E,N
WT˜
G∗
ˆn−˜
GˆndS.(2.80)
First, we consider the left hand side in the Eq. (2.80). Since interpolation and quadrature
nodes are collocated, the nodal values can be analyzed by the same arguments as in the
continuous computations (2.38)and(2.39). Hence, we obtain for all i,j,k=0,...,N
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69 Page 16 of 42 Journal of Scientific Computing (2020) 82 :69
JijkWT
ijk ∂
∂τ Uijk=JijkWT
ijk
∂Uijk
∂t+WT
ijk 
˜νijk ·
∇ξUijk
=Jijk
∂Sijk
∂t+
˜ν·
∇ξSijk =Jijk ∂
∂τ Sijk.
(2.81)
Next we multiply the Eq. (2.81)byωijk and compute the sum over all LGL nodes. This gives
J∂U
∂τ ,W N=∂S
∂τ ,J N
.(2.82)
Since we assume time continuity for our semi-discrete analysis, we apply the product rule in
time and obtain by (2.82)
∂(JU)
∂τ ,W N=∂S
∂τ ,J N+∂J
∂τ ,WTU N
=∂
∂τ S,JN+∂J
∂τ ,
N
=∂
∂τ S,JN+
DN·
˜ν, N+
∂E,N˜ν∗
ˆn−˜νˆndS,
(2.83)
where we used in the last step the Eq. (2.71a) with the test function ϕ=. We note that the
quantity is defined as a polynomial with the nodal values (2.68). In the Appendix C.1,the
following equation is proven

DN·↔
˜
GEC,W N=
∂E,N˜
Fs
ˆn−˜νˆnSdS −
DN·
˜ν, N,(2.84)
where ˜
Fs
ˆn=ˆsn·
Fswith 
Fs=Fs
1,Fs
2,Fs
3T. Here the polynomials Fs
l,l=1,2,3, are
given by (2.69). Moreover, we obtain by (2.73)and(2.74)
−WT˜
G∗
ˆn−˜
Gˆn−˜
Fs
ˆn−˜νˆnS
=
3

l=1"ˆsnlWTFl−Fs
l−ˆsnlWTU−S#−WT˜
G∗
ˆn
=˜
ˆn−˜νˆn−WT˜
G∗
ˆn,
(2.85)
where as well as l,l=1,2,3, are polynomials with nodal values (2.68)and ˜
ˆn:=
ˆsn·
with 
=[1,
2,
3]T. Next, we plug the Eqs. (2.83), (2.84), (2.85)in(2.80)and
rearrange. This results in the equation
∂
∂τ S,JN=− 
∂E,N"WT˜
G∗
ˆn−˜
Gˆn+˜
Fs
ˆn−˜νˆnS−˜ν∗
ˆn−˜νˆn#dS
=
∂E,N˜
ˆn−˜ν∗
ˆn−WT˜
G∗
ˆndS.
(2.86)
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Journal of Scientific Computing (2020) 82 :69 Page 17 of 42 69
Then, we sum the Eq. (2.86) over all elements and use that the normal computation (2.72)is
watertight. This provides the equation
∂
∂τ ¯
S(τ)=
Boundary
faces

∂E,N˜
ˆn−ν∗
ˆn˜
−WT˜
G∗
ˆndS
−
Interior
faces 
∂E,N[[ ˜
ˆn]]−{{˜νˆn}}[[]]−[[W]] T˜
G∗
ˆndS.
(2.87)
Since the numerical surface fluxes ˜
G∗
ˆnare computed by Cartesian fluxes GEC
l,l=1,2,3,
with the properties (2.67), it follows
[[ ˜
ˆn]]−{{˜νˆn}}[[]]−[[W]] T˜
G∗
ˆn=
3

l=1ˆsnl[[l]]−{{νl}}[[]] − [[W]] TG∗
l=0.(2.88)
Hence, we obtain the equation
∂
∂τ ¯
S(τ)=
Boundary
faces

∂E,N˜
ˆn−˜ν∗
ˆn−WT˜
G∗
ˆndS.(2.89)
Since the method is investigated with periodic boundary conditions, we obtain the desired
entropy equation by integrating the Eq.(2.89) over the time interval [0,T]. This completes
the proof of Theorem 2.4.
Remark 2.5 The proof of Theorem 2.4 requires the assumptions that the time integration is
exact and that properties like positivity preservation (of the water height, density or pressure)
are satisfied on the discrete level.
For non-periodic boundary conditions, a proper choice of discrete boundary condition is
necessary to bound the term

Boundary
faces

∂E,N˜
ˆn−˜ν∗
ˆn−WT˜
G∗
ˆndS (2.90)
in Eq. (2.89) such that the method becomes entropy conservative/stable. Thus, we obtain
from Theorem 2.4 the following Corollary.
Corollary 2.6 Suppose the flux functions ↔
GEC in the derivative projection operator (2.64)and
the numerical surface fluxes ˜
G∗
ˆn(2.77)are computed by Cartesian fluxes GEC
l,l =1,2,3,
with the properties (2.65),(2.66),(2.67)and a proper dissipative boundary condition, e.g.,
[14,31,53], is applied. Then the semi-discrete moving mesh DGSEM (2.71)satisfies the
discrete entropy inequality
¯
S(τ)≤¯
S(0),∀τ∈[0,T].(2.91)
2.7 Semi-discrete Entropy Stability
Entropy conservation can be merely expected when a reversible process is described by a
system of PDEs. In general, conservation laws are describing irreversible processes with
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69 Page 18 of 42 Journal of Scientific Computing (2020) 82 :69
discontinuous solutions. Hence, it cannot be expected that the entropy conservative moving
mesh DGSEM provides a physical meaningful discretization for the system (2.1). However,
the entropy conservative flux at the element interfaces can be augmented by an artificial
dissipation term.
In the literature, there are different strategies to add dissipation to an entropy conservative
flux. Here, dissipation is added via a matrix operator. This approach, for instance, has been
used in the context of gas dynamics by Chandrashekar [8]orWintersetal.[63].
The conservative variables ucan be written in dependence of the entropy variables w.
Differentiation of the conservative variables u=u(w)provides the symmetric positive
definite matrix ∂u
∂w, since the system (2.1) is assumed to be symmetrizable (cf. e.g. [29]).
Thus, it follows by a Taylor expansion up to first order
[[u]] = ∂u
∂w[[w]] + O|[[w]] |2,(2.92)
where the jump operator is defined by (2.75) at the interfaces. Furthermore, the system (2.1)
is hyperbolic. Thus, the flux Jacobian matrices ∂fl
∂u,l=1,2,3, are diagonalizable and have
real eigenvalues λl
i(u)p
i=1⊆R. The corresponding right eigenvector matrices are Rl.For
the Euler equations Merriam [47] has shown that there are block diagonal scaling matrices
such that the Hessian matrix of the entropy can be represented by scaled right eigenvector
matrices. This result has been generalized and is known as the eigenvector scaling theorem
(cf. Barth [2, Theorem 4]). Hence, according to the eigenvector scaling theorem, there are
symmetric block diagonal scaling matrices Tlwith
∂fl
∂u=˜
Rll(u)˜
R−1
l,∂u
∂w=˜
Rl˜
RT
l,˜
Rl=RlTl,l=1,2,3,(2.93)
where l(u):= diag λl
1(u),...,λ
l
p(u). The flux Jacobian matrices ∂gl
∂u=∂fl
∂u−νlIp
have the real eigenvalues λl
i(u)−νlp
i=1and the same right eigenvectors as the flux Jacobian
∂fl
∂u. We note that Ipis the p×pidentity matrix. Hence, it follows
∂gl
∂u=˜
Rll(ν, u)˜
R−1
l,
l(ν, u):= diag λl
1(u)−νl,...,λ
l
p(u)−νl,
l=1,2,3.(2.94)
Furthermore, we obtain by (2.93)
∂gl
∂w=∂gl
∂u∂u
∂w=˜
Rll(ν, u)˜
RT
l,l=1,2,3.(2.95)
The Eq. (2.95) motivates the definition of the following matrix dissipation operators
Hl=ˆ
Rl|l|ˆ
RT
l,ˆ
Rl=R
lT
l,l=1,2,3.(2.96)
where the matrices R
l,T
l, depend on some averaged values of the states U−,U+and they
are consistent with the right eigenvector matrix Rland the scaling matrix Tl. The matrix |l|
depends on the values λl
iU−−ν−
lp
i=1and λl
iU+−ν+
lp
i=1. The matrix Hlneeds to
be a symmetric positive definite matrix. Therefore, the matrix |l|has to be chosen carefully.
In Appendix B.3, the matrices to construct the dissipation operator (2.96) for the compressible
Euler equations are given. There it can be also seen which average values are used to evaluate
the states U−,U+in the matrices.
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Journal of Scientific Computing (2020) 82 :69 Page 19 of 42 69
The dissipation operator (2.96) is used to modify the Cartesian numerical surface flux at
the element interfaces as follows
GES
l:= GEC
l−1
2Hl[[W]],l=1,2,3,(2.97)
where the Cartesian fluxes GEC
l,l=1,2,3, satisfy (2.65), (2.66), (2.67). The contravariant
surface numerical fluxes ˜
GEC
ˆnare computed by (2.77). We note that the dissipation operator
(2.96) is not used to modify the entropy conservative fluxes in the derivative projection
operator (2.64).
The numerical fluxes ˜
GES
ˆndo not provide an entropy conservative scheme, but the result
in Theorem 2.4 can be used to prove that the moving mesh DGSEM becomes entropy stable,
such that the discrete mathematical entropy is bounded at any time by its initial data, when
the numerical fluxes ˜
GES
ˆnare used at the element interfaces and it is assumed that the time
integration is exact and that properties like positivity preservation (of the water height, density
or pressure) are satisfied on the discrete level.
Corollary 2.7 Suppose the flux functions ↔
GEC in the derivative projection operator (2.64)are
computed by Cartesian fluxes GEC
l,l =1,2,3, with the properties (2.65),(2.66),(2.67),the
numerical surface fluxes ˜
G∗
ˆn=˜
GES
ˆnare computed by the Cartesian fluxes GES
l,l =1,2,3,
given by (2.97)and periodic boundary conditions are used. Then the semi-discrete moving
mesh DGSEM (2.71)satisfies the discrete entropy inequality
¯
S(τ)≤¯
S(0),∀τ∈[0,T].(2.98)
Furthermore, with proper dissipative boundary conditions, the method satisfies again the
inequality (2.98)for non-periodic problems.
Proof We proceed as in the proof of Theorem 2.4 and obtain the equation
∂
∂τ ¯
S(τ)=
Boundary
faces

∂E,N˜
ˆn−˜ν∗
ˆn−WT˜
GES
ˆndS
−
Interior
faces 
∂E,N[[ ˜
ˆn]]−{{˜νˆn}}[[]]−[[W]] T˜
GES
ˆndS.
(2.99)
Since the numerical surface fluxes ˜
GES
ˆnare computed by the Cartesian fluxes (2.97)andthe
fluxes GEC
l,l=1,2,3, satisfy (2.67), it follows
[[ ˜
ˆn]]−[[]]{{ ˜νˆn}} − [[W]] T˜
GES
ˆn
=
3

l=1ˆsnl[[l]]−[[]]{{νl}} − [[W]] TGEC
l+1
2[[W]]THl[[ W]]
=1
2
3

l=1ˆsnl[[W]] THl[[W]].
(2.100)
Since the matrices Hl,l=1,2,3, are symmetric positive definite and the outward normal
vectors of the curved elements are positive oriented, the Eq.(2.100) provides
−
Interior
faces 
∂E,N[[ ˜
ˆn]]−{{˜νˆn}}[[]]−[[W]] T˜
GES
ˆndS ≤0.(2.101)
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69 Page 20 of 42 Journal of Scientific Computing (2020) 82 :69
Hence, we obtain the inequality
∂
∂τ ¯
S(τ)≤
Boundary
faces

∂E,N˜
ˆn−˜ν∗
ˆn−WT˜
GES
ˆndS.(2.102)
The right hand side of the inequality vanishes, since the method is investigated with periodic
boundary conditions. Hence, we obtain the the desired inequality (2.98)byintegrating(2.102)
over the temporal interval [0,T].
2.8 Free Stream Preservation for the Moving Mesh DGSEM
In this section, we check the discretization of the geometric and metric terms in time. Since DG
methods with the forward Euler discretization are unstable [9,16], we investigate directly the
discretization by an explicit RK method with s≥2 stages and the characteristic coefficients
aσ s
,σ =1,{bσ}s
σ=1,{cσ}s
σ=1. It is worth to mention that a Courant–Friedrichs–Lewy (CFL)
restriction is necessary when an explicit s-stage RK method is used in the DG framework.
In order to present the RK discretization of the semi-discrete DGSEM (2.71), it is beneficial
to write the method in the equivalent nodal representation. This representation is for all
i,j,k=0,...,N,givenby
∂Jijk
∂τ =V(ν)ijk,(2.103a)
∂JijkUijk
∂τ =G(ν)ijk,Uijk,(2.103b)
where the right hand sides are given by
V(ν)ijk:= 
DN·
˜νijk +1
ωiωjωk
∂E,N
ijk˜ν∗
ˆn−˜νˆndS,(2.104)
G(ν)ijk,Uijk:= −
DN·↔
˜
GEC
ijk −1
ωiωjωk
∂E,N
ijk˜
G∗
ˆn−˜
GˆndS (2.105)
with the tensorial Lagrange polynomials ijkgiven by (2.47).
It should be noted that the solutions Jijk,i,j,k=0,...,N, of the ordinary differential
equations (ODEs) (2.103a) need to be positive. We note that the solutions Jijk,i,j,k=
0,...,Nof the ODEs (2.103a) are not used to compute the volume weighted contravariant
coordinate vectors Jal,l=1,2,3, in the right hand sides (2.104)and(2.105). These vectors
are computed from the mapping by the conservative curl form (2.52). Hence, the right hand
sides (2.104) are are independent of Jijk,i,j,k=0,...,N. Therefore, the solutions of the
ODEs (2.103a) are positive, if the grid velocity does not cause to much distortion in the mesh
which is ensured when the assumptions (A1)-(A3) are satisfied.
Next, the interval [0,T]is divided in time points tn. The step size of the time discretization
is t. The DGSEM solutions, the fluxes and the grid velocity field are approximated in the
time points tn, e.g. U(tn)≈Un. Then, the RK discretization of the semi-discrete DGSEM
is given by
for =1,...,s:
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Journal of Scientific Computing (2020) 82 :69 Page 21 of 42 69
J()
ijk =Jn
ijk +t
−1

σ=1
aσ V(ν)n+σ
ijk ,(2.106a)
U()
ijk =Un
ijk +t
J()
ijk
−1

σ=1
aσ G(ν)n+σ
ijk ,U(σ)
ijk−V(ν)n+σ
ijk Un
ij,(2.106b)
Jn+1
ijk =Jn
ijk +t
s

σ=1
bσV(ν)n+σ
ijk ,(2.106c)
Un+1
ijk =Un
ijk +t
J(n+1)
ijk
s

σ=1
bσG(ν)n+σ
ijk ,U(σ)
ijk−V(ν)n+σ
ijk Un
ijk,(2.106d)
where (ν)n+σ
ijk := νξ1
i,ξ2
j,ξ3
k,tn+cσtand ξ1
iN
i=0,ξ2
iN
i=0,ξ3
iN
i=0are sets of LGL
points. Next, we prove that the fully-discrete split form RK-DGSEM (2.106) satisfies the
free stream preservation property.
Theorem 2.8 Suppose the fully-discrete split form RK-DGSEM (2.106)is investigated with
periodic boundary conditions and the solution of the scheme is given by Un
ijk =C:=
c1,...,cpT∈Rpfor all elements eκ(tn),κ=1,...,K , and the numerical fluxes satisfy
(2.65). Then, the constant states cl,l =1,..., p, are preserved in each Runge–Kutta stage
(2.106b). In particular, the solution of the fully-discrete DGSEM method at time level tn+1
is Un+1
ijk =C.
Proof Let ∈{1,...,s}be an arbitrary fixed index. We are interested to investigate the
-th RK stage. Hence, without loss of generality, we can assume that U(σ) =Cfor all
σ=0,...,−1. Then, since the flux ↔
GEC satisfies (2.65), it follows

DN·↔
˜
GEC
ijk =2
N

m=0
Dim{{Ja1}}(i,m)jk ·↔
F(C)
+2
N

m=0
Djm{{Ja2}}i(j,m)k·↔
F(C)
+2
N

m=0
Dkm{{Ja3}}ij(k,m)·↔
F(C)−
DN·
˜νn+σ
ijk
C.
(2.107)
Furthermore, since the metric terms are computed by the conservative curl form (2.52), we
obtain
2
N

m=0Dim{{Ja1}}(i,m)jk +Djm{{Ja2}}i(j,m)k+Dkm {{Ja3}} ij(k,m)
=
N

m=0Dim Ja1mjk +Djm Ja2imk +Dkm Ja3ijm=0.
(2.108)
Here we used the split form Lemma from Gassner et al. [23, Lemma 1] in the first step and in
the second step we used the identity (2.60) for the discrete metric identities. Thus, it follows
that
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69 Page 22 of 42 Journal of Scientific Computing (2020) 82 :69

DN·↔
˜
GEC
ijk =−
DN·
˜νn+σ
ijk
C.(2.109)
Similar, since the flux ↔
G∗satisfies (2.65), follows
˜
G∗
ˆn−˜
Gˆn=
3

l=1ˆsnlFl(C)−{{νn+σ
l}}C−ˆsn·↔
F(C)−(ν)n+σC
=−ˆsn·{{(ν)n+σ}} − (ν)n+σC=−˜ν∗,n+σ
ˆn−˜νn+σ
ˆnC.
(2.110)
Thus, the Eqs. (2.109)and(2.110)give
G(ν)n+σ
ijk ,C=⎛
⎜
⎝
DN·
˜νn+σ
ijk +1
ωiωjωk
∂E,N
ijk˜ν∗,n+σ
ˆn−˜νn+σ
ˆndS⎞
⎟
⎠C
=V(ν)n+σ
ijk C.(2.111)
Hence, the solution of the RK stage (2.106b)isgivenby
U()
ijk =C+t
J()
ijk
−1

σ=1
aσ G(ν)n+σ
ijk ,C−V(ν)n+σ
ijk C=C.(2.112)
Since the parameter was arbitrary chosen, it follows U()
ijk for all =1,...,s. In particular,
it follows
Un+1
ijk =C+t
Jn+1
ijk
s

σ=1
bσG(ν)n+σ
ijk ,C−V(ν)n+σ
ijk C=C.(2.113)
This completes the proof of Theorem 2.8.
3 Numerical Results
The numerical computations in this section are performed with the open source code FLEXI1
and the three-dimensional high-order meshes for the simulations are generated with the open
source tool HOPR.2
We present tests on three dimensional moving hexahedral curved meshes for the com-
pressible Euler equations. Based on these tests we will evaluate the theoretical findings of
the previous sections. The three dimensional compressible Euler equations are given by
∂u
∂t+
∇·↔
f=0.(3.1)
1www.flexi-project.org.
2www.hopr-project.org.
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Journal of Scientific Computing (2020) 82 :69 Page 23 of 42 69
The state vector and the components of the block vector flux, ↔
f,aregivenby
u=⎡
⎢
⎢
⎢
⎢
⎣
ρ
ρu1
ρu2
ρu3
E
⎤
⎥
⎥
⎥
⎥
⎦
,f1=⎡
⎢
⎢
⎢
⎢
⎣
ρu1
ρu2
1+p
ρu1u2
ρu1u3
(E+p)u1
⎤
⎥
⎥
⎥
⎥
⎦
,f2=⎡
⎢
⎢
⎢
⎢
⎣
ρu2
ρu1u2
ρu2
2+p
ρu2u3
(E+p)u2
⎤
⎥
⎥
⎥
⎥
⎦
,f3=⎡
⎢
⎢
⎢
⎢
⎣
ρu3
ρu1u3
ρu2u3
ρu2
3+p
(E+p)u3
⎤
⎥
⎥
⎥
⎥
⎦
,(3.2)
where the conservative variables are the density ρ, the momentum ρu=[ρu1,ρu2,ρu3]T
and the total energy E. In order to close the system, we assume an ideal gas such that the
pressure is defined as
p=(γ −1)E−ρ
2|u|2,(3.3)
where γis the adiabatic exponent. We choose γ=1.4 in the following experiments. The
system (3.1) is investigated in the domain =[xmin,xmax]3. At initial time t=0the
domain is divided in Knon-overlapping and conforming cartesian hexahedral elements
eκ(0),κ=1,...,K. For each element eκ(0),κ=1,...,K, the temporal distribution of a
grid point
xκ(0)=xκ
1(0),xκ
2(0),xκ
3(0)T∈eκ(0)(3.4)
is given by
xκ(t)=xκ(0)+0.05 Lsin (2πt)sin 2π
Lxκ
1(0)sin 2π
Lxκ
2(0)sin 2π
Lxκ
3(0),
(3.5)
where L:= xmax −xmin.InFig.2, we show a slice through a three dimensional mesh
with K=163elements at initial time and at its maximal distortion. The mesh velocity is
calculated by exact differentiation of Eq. (3.5). Note that the formula (3.5) is a common
way to construct a deforming domain. For instance, similar formulas were used for the DG
methods in [37,64] and the collocation method in [66]. Furthermore, the five stage fourth order
low-storage two-register explicit RK method (RK4(3)5[2R+]) from Kennedy, Carpenter and
Lewis [34, Section 3.4.] is used for the time-integration in the numerical experiments. The
CFL restriction is computed as in [12]
t
min
1≤κ≤K|hκ(tn)|≤CCFL
(2N+1)λmax
,(3.6)
where hκ(tn)is the minimum element size of eκ(tn),CCFL ∈(0,1]and λmax is the largest
advective wave speed at the current time level traveling in either the x1,x2,x3-direction.
3.1 Experimental Convergence Rates
In this section, we verify the high-order approximation of the moving DGSEM (2.71). For
this purpose, we investigate the domain =[−1,1]3and apply the method of manufactured
solutions. Thus, we assume a solution of the form
123
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69 Page 24 of 42 Journal of Scientific Computing (2020) 82 :69
Fig. 2 A slice through a three dimensional mesh with K=163elements at initial time (left) and at its maximal
distortion (right)
U(x,t)=⎡
⎢
⎢
⎢
⎢
⎣
ρ(x,t)
ρu1(x,t)
ρu2(x,t)
ρu3(x,t)
E(x,t)
⎤
⎥
⎥
⎥
⎥
⎦=⎡
⎢
⎢
⎢
⎢
⎣
2+0.1sin(π(x1+x2+x3−2·0.3t))
2+0.1sin(π(x1+x2+x3−2·0.3t))
2+0.1sin(π(x1+x2+x3−2·0.3t))
2+0.1sin(π(x1+x2+x3−2·0.3t))
[2+0.1sin(π(x1+x2+x3−2·0.3t))]2
⎤
⎥
⎥
⎥
⎥
⎦
.(3.7)
We plug solution (3.7) into the Euler system and compute the residual using a computer
algebra system. This term is used as a source term in our convergence tests. We note that this
term is handled and discretized as a solution independent part in the numerical computation.
We run the convergence test with periodic boundary conditions. Furthermore, the moving
mesh DGSEM (2.71) is applied with the flux function in Appendix B.1 as volume and surface
flux. In addition, the surface flux is stabilized by the dissipation operator in Appendix B.3.
Besides using the grid point distribution given in (3.5), we also compute static reference
solutions, by setting the grid velocity to zero. In this case, the moving mesh DGSEM (2.71)
degenerates to the split form DGSEM for static meshes [23,24].
In Table 1, we list the experimental order of convergence (EOC) and L2errors for the
conservative variables that we obtain for polynomials with odd degree N=3onastatic
mesh (top) and on a moving mesh (bottom). To calculate the L2norm, we interpolate the
polynomial solution to a higher degree (at least twice the degree of the solution) and perform
integration on that higher degree. The convergence rates on the moving mesh are not as
good as on a static mesh, which can be justified by the high distortion in the mesh from the
grid point distribution formula (3.5). However, with an increasing number of elements the
same convergence rates as on a static mesh are almost reached. Moreover, the experimental
order of convergence (EOC) and L2errors for the conservative variables that we obtain for
polynomials with even degree N=4 are listed in the Table 2. We observe a similar behavior
as for the odd degree N=3. This indicate the high-order approximation properties of the
moving mesh DGSEM.
3.2 Entropy Analysis Validation
The three dimensional Euler equations (3.1) are equipped with the entropy/entropy flux pairs
s=− ρς
γ−1,fs
l=−ρςul
γ−1,l=1,2,3,(3.8)
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Journal of Scientific Computing (2020) 82 :69 Page 25 of 42 69
Table 1 Experimental order of convergence (EOC) and L2errors at time T=5 for the Euler manufactured solution test (3.7)
KL
2(ρ)EOC(ρ) L2(ρu1)EOC(ρu1)L2(ρu2)EOC(ρu2)L2(ρu3)EOC(ρu3)L2(E)EOC(E)
232.84E−02 – 2.74E−02 – 2.74E−02 – 2.74E−02 – 5.47E−02 –
435.54e-03 2.36 5.43E−03 2.34 5.43E−03 2.34 5.43E−03 2.34 1.03E−03 2.40
834.35E−05 6.99 4.28E−05 6.99 4.28E−05 6.99 4.28E−05 6.99 1.06E−04 6.61
1632.10E−06 4.37 2.07E−06 4.37 2.08E−06 4.37 2.07E−06 4.37 5.33E−06 4.31
3231.26E−07 4.06 1.24E−07 4.06 1.24E−07 4.06 1.24E−07 4.06 3.19E−07 4.06
6437.82E−09 4.01 7.67E−09 4.01 7.67E−09 4.01 7.67E−09 4.01 1.97E−08 4.01
234.16E−02 – 3.73E−02 – 3.73E−02 – 3.73E−02 – 5.61E−02 –
433.77E−03 3.46 3.52E−03 3.41 3.52E−03 3.41 3.52E−03 3.41 6.06E−03 3.21
831.99E−04 4.25 1.75E−04 4.33 1.75E−04 4.33 1.75E−04 4.33 3.24E−04 4.23
1635.37E−06 5.21 4.91E−06 5.16 4.91E−06 5.16 4.91E−06 5.16 1.20E−05 4.75
3232.18E−07 4.62 2.07E−07 4.57 2.07E−07 4.57 2.07E−07 4.57 5.83E−07 4.36
6431.45E−08 3.92 1.34E−08 3.95 1.34E−08 3.95 1.34E−08 3.95 3.95E−08 3.88
The moving mesh DGSEM is used with N=3 on a static mesh (top) and on a moving mesh (bottom) with the grid point distribution (3.5)
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69 Page 26 of 42 Journal of Scientific Computing (2020) 82 :69
Table 2 Experimental order of convergence (EOC) and L2errors at time T=5 for the Euler manufactured solution test (3.7)
KL
2(ρ)EOC(ρ) L2(ρu1)EOC(ρu1)L2(ρu2)EOC(ρu2)L2(ρu3)EOC(ρu3)L2(E)EOC(E)
236.99E−03 – 6.64E−03 – 6.64E−03 – 6.64E−03 – 1.16E−02 –
434.02E−04 4.12 3.97E−04 4.06 3.97E−04 4.06 3.97E−04 4.06 7.96E−04 3.87
834.50E−06 6.48 4.50E−06 6.47 4.50E−06 6.47 4.50E−06 6.47 1.16E−05 6.10
1631.37E−07 5.04 1.38E−07 5.02 1.38E−07 5.02 1.38E−07 5.02 3.66E−07 4.98
3234.33E−09 4.98 4.40E−09 4.97 4.40E−09 4.97 4.40E−09 4.97 1.16E−08 4.97
6431.36E−10 4.99 1.38E−10 4.99 1.38E−10 4.99 1.38E−10 4.99 3.66E−10 4.99
231.02E−02 – 9.06E−03 – 9.06E−03 – 9.06E−03 – 1.45E−02 –
434.53E−04 4.50 4.13E−04 4.46 4.13E−04 4.46 4.13E−04 4.46 7.18E−04 4.33
831.10E−05 5.37 1.02E−05 5.35 1.02E−05 5.35 1.02E−05 5.35 1.86E−05 5.27
1631.91E−07 5.85 1.72E−07 5.88 1.72E−07 5.88 1.72E−07 5.88 3.81E−07 5.61
3237.28E−09 4.71 6.33E−09 4.77 6.33E−09 4.77 6.33E−09 4.77 1.38E−08 4.78
6432.79E−10 4.71 2.38E−10 4.74 2.38E−10 4.74 2.38E−10 4.74 5.40E−10 4.68
The moving mesh DGSEM is used with N=4 on a static mesh (top) and on a moving mesh (bottom) with the grid point distribution (3.5)
123
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Journal of Scientific Computing (2020) 82 :69 Page 27 of 42 69
where ς=log pρ−γ. We are interested in the behavior of the discrete entropy conservation
error
S(T)=¯
S(T)−¯
S(0),(3.9)
where ¯
S(·)is computed by (2.78). Therefore, we investigate the inviscid Taylor-Green vortex
(TGV) test case [56] in the domain =[0,2π]3. The inviscid TGV can be a challenging test
case regarding the robustness of a numerical scheme, partly because the dynamics produce
arbitrarily small scales. The flow field is thus by design under-resolved, which makes it a
suitable test case to investigate the entropy conservation properties of the scheme. The TGV
evolves from the initial data
ρ=1,
u=[sin (x1)cos (x2)cos (x3),−cos (x1)sin (x2)cos (x3),0]T,
p=p0+1
16 (cos (2x1)+cos (2x2)) (cos (2x3)+2).
(3.10)
To render the simulation close to incompressible, the Mach number M0=1
√γp0is set to 0.1
by adjusting the pressure correspondingly. We run the simulation with K=163elements
and periodic boundary conditions. The final time is chosen to be T=13. Furthermore,
we apply the flux function in Appendix B.1 to compute the derivative projection operator
(2.64). To calculate the discrete integral entropy, the SBP mass matrices are used directly.
In Fig. 3we present a log-log plot of the entropy conservation error for N=3,4. We
note that the flux in Appendix B.1 is used as surface flux without a dissipation term in
these computations, rendering the semi-discrete discretization fully entropy conserving. As
expected, we observe the reduction of the remaining entropy conservation error according to
the order of the RK method for decreasing CFL numbers. In Fig. 4the temporal evolution of
the entropy conservation errors S(T)is given. The CFL number is set to CCFL =0.125 and
polynomial degrees N=3andN=4 are used. We observe that the entropy conservation
error S(T)is constant in time (dashed line) when the flux in Appendix B.1 is used without
a dissipation term. This indicates the entropy conservation in the TVG test case. On the other
hand the entropy conservation error S(T)is decreasing in time (solid line) when the surface
flux is stabilized by the dissipation term in Appendix B.3. Thus, the moving mesh DGSEM
is an entropy stable scheme in this test case. These observations agree with the results in
Theorem 2.4 and Corollary 2.6.
3.3 Robustness Test
As has been stated in Sect. 3.2 and noted in literature [50,62], the inviscid TGV is a notoriously
challenging test case for the stability of the numerical scheme. While for lower polynomial
degrees calculations may be possible, high-order simulations are known to crash even if
aliasing-reducing methods like polynomial dealiasing are used [50]. Thus, we use the TGV
test case (3.10) to demonstrate the increased robustness of the entropy stable moving mesh
DGSEM. To do so, we run the simulation up to T=13 using a polynomial degree of N=7
on three different meshes employing K1=143,K2=193and K3=263elements. These
cases correspond to the most restrictive simulations from [50]. Again, the point distribution
given in (3.5) is used. We use the flux function in Appendix B.1 as volume and surface flux
and stabilize the surface flux by the dissipation operator in Appendix B.3.
123
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69 Page 28 of 42 Journal of Scientific Computing (2020) 82 :69
Fig. 3 Log-log plot of the entropy conservation errors S(T)for the Euler equations with initial data (3.10).
The errors are given at time T=13 for polynomials with degree N=3 (solid line) and N=4 (dotted line)
on a curved moving mesh with K=163elements
Fig. 4 Temporal evolution of the entropy conservation errors S(T)for the Euler equations with initial data
(3.10). The flux in Appendix B.1 is used as surface flux without dissipation (solid line) and with the dissipation
term in Appendix B.3 (dashed line)
Using the entropy stable moving DGSEM, we were able to run all simulations until
final time. This shows that the consistent dissipation operators in combination with the
entropy conservative volume fluxes can lead to superior stability properties. We note that
simulations without dissipative surface fluxes crash before reaching the final time for higher-
order simulations (N≥3). This highlights the role that entropy conservation plays in the
stabilization of challenging numerical problems.
3.4 Free Stream Preservation Validation
We consider the Euler equations (3.1) on the domain =[0,2π]3with the initial data
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Journal of Scientific Computing (2020) 82 :69 Page 29 of 42 69
Table 3 Free stream preservation test for N=3 (top) and N=4 (bottom)
CCFL L∞(ρ)L∞(ρu1)L∞(ρu2)L∞(ρu3)L∞(E)
0.95 2.47E−14 1.40E−12 4.46E−12 4.48E−12 1.33E−12
0.5 2.47E−14 1.40E−12 4.46E−12 4.48E−12 1.33E−12
0.25 2.70E−14 1.40E−12 4.46E−12 4.48E−12 1.36E−12
0.125 3.10E−14 1.40E−12 4.46E−12 4.48E−12 1.43E−12
0.0625 3.78E−14 1.40E−12 4.46E−12 4.48E−12 1.56E−12
0.95 2.07E−14 1.24E−12 5.28E−12 5.21E−12 1.12E−12
0.5 2.49E−14 1.24E−12 5.28E−12 5.21E−12 1.30E−12
0.25 2.81E−14 1.24E−12 5.28E−12 5.21E−12 1.34E−12
0.125 3.32E−14 1.24E−12 5.28E−12 5.21E−12 1.40E−12
0.0625 4.24E−14 1.24E−12 5.28E−12 5.21E−12 1.59E−12
The L∞errors measure the difference between the initial data (3.11) and the numerical solution at time T=20
for different constants CCFL
U(x,t)=⎡
⎢
⎢
⎢
⎢
⎣
ρ(x,t)
ρu1(x,t)
ρu2(x,t)
ρu3(x,t)
E(x,t)
⎤
⎥
⎥
⎥
⎥
⎦=⎡
⎢
⎢
⎢
⎢
⎣
1
0.3
0
0
17
⎤
⎥
⎥
⎥
⎥
⎦
.(3.11)
The entropy stable DGSEM is applied with the flux function in Appendix B.1 as volume and
surface flux as well as the dissipation operator in Appendix B.3 to stabilize the surface flux.
We apply K=163elements, the formula (3.5) to describe the displacement of the mesh
points and periodic boundary conditions are used in the simulation. Furthermore, the final
time is set to T=20. In Table 3,wepresenttheL
∞errors between the initial data (3.11)
and the numerical solution at time T=20 for polynomials of degree N=3 (top), N=4
(bottom) and different CFL numbers CCFL. The errors are computed by super sampling the
polynomial solution, at least doubling the amount of nodes per direction from the underlying
solution. We observe that the errors are close to zero and vary slightly for the different CFL
numbers. These results indicate the compliance of the free stream preservation property.
4 Conclusions
In this work a moving mesh DGSEM to solve non-linear conservation laws has been con-
structed and analyzed. The semi-discrete method is provably entropy stable and the free
stream preservation property is satisfied for each explicit s-stage Runge–Kutta method.
The moving mesh DGSEM has been presented for three dimensional conservation laws.
The derivatives in space are approximated with high-order derivative matrices which are
SBP operators. Furthermore, the split form DG framework [23,24] has been used to avoid
aliasing in the discretization of the volume integrals. In addition, two-point flux functions
with the generalized entropy condition (2.67) are used in the split form DG framework. These
modules in the spatial discretization are the basis to prove that the moving mesh DGSEM
is an entropy stable scheme, when periodic boundary conditions are used. Non-periodic
boundary conditions require the construction of suitable dissipative boundary conditions to
enforce the entropy stability. The discrete entropy analysis requires the assumptions that the
123
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69 Page 30 of 42 Journal of Scientific Computing (2020) 82 :69
time derivatives can be evaluated exactly and that properties like positivity preservation (of
the water height, density or pressure) are satisfied on the discrete level. Operations like the
integration-by-parts formula are mimicked by the SPB operators on the discrete level.
The three dimensional Euler equations have been considered to verify the proven properties
of the moving mesh DGSEM in our numerical experiments. We presented convergence tests
for smooth test problems to verify that the split form DG framework provides also on a
moving mesh a high-order accurate approximation. Furthermore, the numerical robustness
tests in the Sect. 3.3 emphasize the relevance of the entropy stable DGSEM, since the method
was able to run the challenging inviscid TGV test case until final time.
Acknowledgements Open Access funding provided by Projekt DEAL. Gero Schnücke and Gregor Gassner
are supported by the European Research Council (ERC) under the European Union’s Eights Framework
Program Horizon 2020 with the research project Extreme, ERC Grant Agreement No. 714487. The authors
gratefully acknowledge the support and the computing time on “Hazel Hen” provided by the HLRS through
the project “hpcdg”.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,
and indicate if changes were made. The images or other third party material in this article are included in the
article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is
not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
A Proof for Lemma 2.1
In this section, we prove Lemma 2.1. For the sake of simplicity we present merely the proof
in two dimensions. The three dimensional proof can be done by the same argumentation.
A way to compute the two dimensional bijective isoparametric transfinite mapping is given
in the book of Kopriva [36, Chapter 6, equation (6.18)]. The mapping is for all ξ1,ξ2T∈E
and τ∈[0,T]given by
eκ(t)x(t)=χξ1,ξ2,τ=1
21−ξ1IN
4ξ2,τ+1+ξ1IN
2ξ2,τ
+1
21−ξ2IN
1ξ1,τ+1+ξ2IN
3ξ1,τ
−1
41+ξ1"1−ξ2IN
1(1,τ)+1+ξ2IN
3(1,τ)#
−1
41−ξ1"1−ξ2IN
1(−1,τ)+1+ξ2IN
3(−1,τ)#.
(A.1)
It is worth to mention that the mapping χξ1,ξ2,τmatches with the boundary faces in the
interpolation points. The location of the curved faces 
1ξ1,τ,
2ξ2,τ,
3ξ1,τand

4ξ2,τis sketched in Fig. 5.
In the following, e1(t)and e2(t)are two neighboring elements which share the same
boundary face. Without loss of generality the elements share the face 
1
3=
2
1as it is
illustrated in Fig. 6. Then, for the elements el(t),l=1,2, the grid velocity field is given by
νlξ1,ξ2,τ=1
21−ξ1IN∂
∂τ 
l
4ξ2,τ+1+ξ1INd
dt 
l
2ξ2,t
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Journal of Scientific Computing (2020) 82 :69 Page 31 of 42 69
Fig. 5 Left the reference element E=[−1,1]2and on the right a general quadrilateral element ek(t)with
the curved faces 
1ξ1,τ,
2ξ2,τ,
3ξ1,τand 
4ξ2,τ. The mapping χξ1,ξ2,τconnects
Eand ek(t)
Fig. 6 Two elements e1(t)and
e2(t)of a conforming mesh
sharing the same curved
boundary (dotted curve)
+1
21−ξ2IN∂
∂τ 
l
1ξ1,τ+1+ξ2IN∂
∂τ 
l
3ξ1,τ
−1
41+ξ11−ξ2IN∂
∂τ 
l
1(1,τ)
+1+ξ2IN∂
∂τ 
l
3(1,τ)
−1
41−ξ11−ξ2IN∂
∂τ 
l
1(−1,τ)
+1+ξ2IN∂
∂τ 
l
3(−1,τ),(A.2)
since it holds the identity
∂
∂τ IN
l
i=IN∂
∂τ 
l
i,l=1,2,and i=1,2,3,4.(A.3)
Furthermore, since for l=1,2, and i=1,2,3,4, the faces 
l
i(·,·,t)are continuously
differentiable in the time interval [0,T], it holds
IN∂
∂τ 
1
4(1,τ)=IN∂
∂τ 
1
3(−1,τ),IN∂
∂τ 
1
2(1,τ)=IN∂
∂τ 
1
3(1,τ),
(A.4)
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69 Page 32 of 42 Journal of Scientific Computing (2020) 82 :69
IN∂
∂τ 
2
4(−1,τ)=IN∂
∂τ 
2
1(−1,τ),IN∂
∂τ 
2
2(−1,τ)=IN∂
∂τ 
2
1(1,τ),
(A.5)
∂
∂τ 
1
3ζj,τ=d
dt 
2
1ζj,τ,j=0,...,N,(A.6)
where ζjN
j=0are interpolation points.
For the element e1(t)the points along the interface with e2(t)are mapped in the set
{(ξ,1):ξ∈[−1,1]}. Hence, the grid velocity becomes
ν1(ξ,1,τ)=IN∂
∂τ 
1
3(ξ,τ),∀ξ∈[−1,1](A.7)
by (A.4). On the opposite, for the element e2(t)the points along the interface with e1(t)are
mapped in the set {(ξ,−1):ξ∈[−1,1]}and we obtain
ν2(ξ,−1,τ)=IN∂
∂τ 
2
1(ξ,τ),∀ξ∈[−1,1],(A.8)
by (A.5). Thus, we obtain ν1(·,1,τ)=ν2(·,−1,τ)by (A.6). This proves that the grid
velocity is continuous in the interface points of the two neighboring elements.
B Entropy Stable Moving Mesh Euler Fluxes
We present entropy stable Cartesian fluxes GEC
l,l=1,2,3, for the compressible Euler
equations (3.1) equipped with the entropy/entropy flux pairs (3.8). Then the entropy variables
are given by
w=γ−ς
γ−1−β|u|2,2βu1,2βu2,2βu3,−2βT
,with β:= ρ
2p(B.1)
and the entropy functionals are given by
φ=wTu−s=ρ, ψl=wTfl−fs
l=ρul,l=1,2,3.(B.2)
B.1 Entropy Conservative Euler Flux Based on the Flux in [8]
In the following the logarithmic mean {{·}}log will be used. For two positive states a−and a+,
the logarithmic mean is defined by
{{a}}log := *[[ a]]
[[log(a)]] ,if a−= a+,
a−,if a−=a+.(B.3)
A numerically stable procedure to compute the logarithmic mean (B.3) is provided by Ismail
and Roe [33, Appendix B]. Friedrich et al. [21, Theorem 3] constructed the following state
function
U#=⎡
⎢
⎢
⎢
⎢
⎢
⎣
{{ρ}}log
{{ρ}}log {{u1}}
{{ρ}}log {{u2}}
{{ρ}}log {{u3}}
{{ρ}}log
2(γ−1){{β}}log +1
2{{ρ}}log |u|2
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,(B.4)
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Journal of Scientific Computing (2020) 82 :69 Page 33 of 42 69
where
|u|2=2{{ u1}}2+{{u2}}2+{{u3}}2−{{ u2
1}} + {{u2
2}} + {{u2
3}}.(B.5)
The state function (B.4) is consistent, symmetric and it holds
[[w]]TU#=[[ρ]] = [[φ]] .(B.6)
In [40, Eg. (3.4) p. 1974] LeFloch et al. gave a general formula to compute entropy con-
servative numerical state flux-functions. Furthermore, Chandrashekar constructed in [8]a
kinetic energy preserving and entropy conservative (KEPEC) numerical flux function for the
compressible Euler equations. In the x1-direction Chandrashekar’s KEPEC flux is given by
FEC_CH
1=⎡
⎢
⎢
⎢
⎢
⎢
⎣
{{ρ}}log {{u1}}
{{ρ}}log {{u1}}2+{{ρ}}
2{{β}}
{{ρ}}log {{u1}}{{u2}}
{{ρ}}log {{u1}}{{u3}}
{{ρ}}log {{u1}} 1
2(γ−1){{β}}log +1
2|u|2+{{ ρ}} {{ u1}}
2{{β}}
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.(B.7)
The flux (B.7) is consistent with f1and symmetric. In particular, Chandrashekar proved that
[[w]]TFEC_CH
1=[[ρu1]] = [[ψ1]] .(B.8)
The state function (B.4) and the flux function (B.7) are used to construct the flux
GEC_CH
1=FEC_CH
1−{{ν1}} U#=⎡
⎢
⎢
⎢
⎢
⎢
⎣
{{ρ}}log {{u1−ν1}}
{{ρ}}log {{u1−ν1}}{{u1}} + {{ρ}}
2{{β}}
{{ρ}}log {{u1−ν1}}{{u2}}
{{ρ}}log {{u1−ν1}}{{u3}}
{{ρ}}log {{u1−ν1}} 1
2(γ−1){{β}}log +1
2|u|2+{{ ρ}} {{ u1}}
2{{β}}
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
(B.9)
The flux (B.9) is consistent with g1=f1−ν1u, symmetric and it follows
[[w]]TGEC_CH
1=[[w]] TFEC_CH
1−{{ν1}} [[w]]TU#=[[ψ1]] − {{ν1}} [[φ]] (B.10)
by (B.6)and(B.8). In the same way, the x2-direction and the x3-direction of Chandrashekar’s
KEPEC flux can be used to construct
GEC_CH
2=⎡
⎢
⎢
⎢
⎢
⎢
⎣
{{ρ}}log {{u2−ν2}}
{{ρ}}log {{u1}}{{u2−ν2}}
{{ρ}}log {{u2}}{{u2−ν2}} + {{ρ}}
2{{β}}
{{ρ}}log {{u2−ν2}}{{u3}}
{{ρ}}log {{u2−ν2}} 1
2(γ−1){{β}}log +1
2|u|2+{{ ρ}} {{ u2}}
2{{β}}
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(B.11)
and
GEC_CH
3=⎡
⎢
⎢
⎢
⎢
⎢
⎣
{{ρ}}log {{u3−ν3}}
{{ρ}}log {{u1}}{{u3−ν3}}
{{ρ}}log {{u2}}{{u3−ν3}}
{{ρ}}log {{u3}}{{u3−ν3}} + {{ρ}}
2{{β}}
{{ρ}}log {{u3−ν3}} 1
2(γ−1){{β}}log +1
2|u|2+{{ ρ}} {{u3}}
2{{β}}
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.(B.12)
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69 Page 34 of 42 Journal of Scientific Computing (2020) 82 :69
The fluxes (B.11), (B.12) are consistent with g2=f2−ν2u,g3=f3−ν3u, symmetric and
satisfy
[[w]]TGEC_CH
2=[[w]] TFEC_CH
2−{{ν2}} [[w]]TU#=[[ψ2]]−{{ν2}}[[φ]] ,(B.13)
[[w]]TGEC_CH
3=[[w]] TFEC_CH
3−{{ν3}} [[w]]TU#=[[ψ3]]−{{ν3}}[[φ]] .(B.14)
B.2 Entropy Conservative Euler Flux Based on the Flux in [55]
Ranocha constructed in his PHD thesis [55] another KEPEC numerical flux for the com-
pressible Euler equations. We proceed as in the Appendix B.1 and use the state (B.4)and
Ranocha’s flux to construct the following two-point flux functions
GEC_R
1=⎡
⎢
⎢
⎢
⎢
⎢
⎣
{{ρ}}log {{u1−ν1}}
{{ρ}}log {{u1−ν1}}{{u1}} + {{ p}}
{{ρ}}log {{u1−ν1}}{{u2}}
{{ρ}}log {{u1−ν1}}{{u3}}
{{ρ}}log {{u1−ν1}} 1
2(γ−1){{β}}log +1
2|u|2+2{{ p}} {{u1}} − {{ pu1}}
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,(B.15a)
GEC_R
2=⎡
⎢
⎢
⎢
⎢
⎢
⎣
{{ρ}}log {{u2−ν2}}
{{ρ}}log {{u1}}{{u2−ν2}}
{{ρ}}log {{u2}}{{u2−ν2}} + {{ p}}
{{ρ}}log {{u2−ν2}}{{u3}}
{{ρ}}log {{u2−ν2}} 1
2(γ−1){{β}}log +1
2|u|2+2{{ p}} {{u2}} − {{ pu2}}
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,(B.15b)
and
GEC_R
3=⎡
⎢
⎢
⎢
⎢
⎢
⎣
{{ρ}}log {{u3−ν3}}
{{ρ}}log {{u1}}{{u3−ν3}}
{{ρ}}log {{u2}}{{u3−ν3}}
{{ρ}}log {{u3}}{{u3−ν3}} + {{ p}}
{{ρ}}log {{u3−ν3}} 1
2(γ−1){{β}}log +1
2|u|2+2{{ p}} {{u3}} − {{ pu3}}
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
(B.15c)
The flux functions (B.15) are consistent with g1,g2,g3, symmetric and satisfy
[[w]]TGEC_R
l=[[ψl]]−{{νl}}[[φ]],l=1,2,3.(B.16)
B.3 Matrix Dissipation Term for the Euler Flux
We will use the Euler fluxes from the previous Appendices B.1 and B.2 with the dissipation
operators form Winters et al. [63]. In the following, the matrices to construct the entropy
stable dissipation operators (2.96) are listed. The average components of the dissipation term
in the x1-direction are given by
R
1=⎡
⎢
⎢
⎢
⎢
⎣
11001
{{u1}} − ¯c{{u1}} 00{{u1}} + ¯c
{{u2}} {{u2}} 10 {{ u2}}
{{u3}} {{u3}} 01 {{ u3}}
¯
h−{{u1}} ¯c1
2|u|2{{ u2}} {{u3}} ¯
h+{{u1}} ¯c
⎤
⎥
⎥
⎥
⎥
⎦
,(B.17)
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Journal of Scientific Computing (2020) 82 :69 Page 35 of 42 69
T
1=diag ⎛
⎝+{{ρ}}log
2γ,+(γ−1)
γ{{ρ}}log ,+{{ρ}}
2{{β}} ,+{{ρ}}
2{{β}} ,+{{ρ}} log
2γ⎞
⎠,(B.18)
1=diag (|{{u1−ν1}} − ¯c|,|{{u1−ν1}} |,|{{u1−ν1}}|,|{{ u1−ν1}}|,|{{u1−ν1}} + ¯c|),
(B.19)
where
¯c:= +γ{{ρ}}
2{{ρ}}log {{β}} ,¯
h:= γ
2(γ−1){{β}}log +1
2|u|2.(B.20)
In the x2-direction the components are given by
R
2=⎡
⎢
⎢
⎢
⎢
⎣
10101
{{u1}} 1{{u1}} 0{{ u1}}
{{u2}} − ¯c0{{u2}} 0{{ u2}} + ¯c
{{u3}} 0{{u3}} 1{{ u3}}
¯
h−{{u2}} ¯c{{u1}} 1
2|u|2{{ u3}} ¯
h+{{u2}} ¯c
⎤
⎥
⎥
⎥
⎥
⎦
,(B.21)
T
2=diag ⎛
⎝+{{ρ}}log
2γ,+{{ρ}}
2{{β}} ,+(γ−1)
γ{{ρ}}log ,+{{ρ}}
2{{β}} ,+{{ρ}} log
2γ⎞
⎠,(B.22)
2=diag (|{{u2−ν2}} − ¯c|,|{{u2−ν2}} |,|{{u2−ν2}}|,|{{ u2−ν2}}|,|{{u2−ν2}} + ¯c|),
(B.23)
andinthex3-direction the components are given by
R
3=⎡
⎢
⎢
⎢
⎢
⎣
1001 1
{{u1}} 10{{u1}} {{u1}}
{{u2}} 01{{ u2}} {{u2}}
{{u3}} − ¯c00{{u3}} {{u3}} + ¯c
¯
h−{{u3}} ¯c{{u1}} {{u2}} 1
2|u|2¯
h+{{u3}} ¯c
⎤
⎥
⎥
⎥
⎥
⎦
,(B.24)
T
3=diag ⎛
⎝+{{ρ}}log
2γ,+{{ρ}}
2{{β}} ,+{{ρ}}
2{{β}} ,+(γ−1)
γ{{ρ}}log ,+{{ρ}}log
2γ⎞
⎠,(B.25)
3=diag (|{{u3−ν3}} − ¯c|,|{{u3−ν3}} |,|{{u3−ν3}}|,|{{ u3−ν3}}|,|{{u3−ν3}} + ¯c|).
(B.26)
C Proofs of Entropy Conservation for Advection Terms
In this section, we apply the following identities which result from the properties of the SBP
operator Q
N

i,j=0
Qij[[a]](i,j){{b}} (i,j)=
N

i,j=0
Qijaibj−(aNbN−a0b0),(C.1)
N

i,j=0
Qij[[a]](i,j){{b}} (i,j){{c}}(i,j)=
N

i,j=0
2Qijai{{b}}(i,j){{c}} (i,j)−(aNbNcN−a0b0c0),
(C.2)
123
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69 Page 36 of 42 Journal of Scientific Computing (2020) 82 :69
where {a}N
i=0,{b}N
i=0and {c}N
i=0are generic nodal values. These identities can be proven in
a similar way as the discrete split forms in Lemma 1 in [23]. Thus, we skip a proof in this
paper.
C.1 Proof for Eq. (2.84)
The flux ↔
GEC satisfies the symmetry property (2.66) and the SBP property (2.59) provides
2ωiDim =2Qim =Qim −Qmi +Bim,i,m=0,...,N.(C.3)
Thus, we obtain

DN·↔
˜
GEC,W N
=
N

j,k=0
ωjωk
N

i,m,k=0
Qim[[W]] T
(i,m)j↔
GEC νijk,νmjk,Uijk,Umjk·{{Ja1}}(i,m)jk
+
N

j,k=0
ωjωk
N

i,m=0
BimWT
ijk ↔
GEC νijk,νmjk,Uijk,Umjk·{{Ja1}}(i,m)jk
+
N

i,k=0
ωiωk
N

j,m=0
Qjm[[W]] T
i(j,m),k↔
GEC νijk,νimk,Uijk,Uimk·{{Ja2}}i(j,m)k
+
N

i,k=0
ωiωk
N

j,m=0
BjmWT
ijk ↔
GEC νijk,νimk,Uijk,Uimk·{{Ja2}}i(j,m)k
+
N

i,j=0
ωiωj
N

j,m=0
Qkm[[W]]T
ij(k,m)↔
GEC νijk,νijm,Uijk,Uijm·{{Ja3}}ij(k,m)
+
N

i,j=0
ωiωj
N

j,m=0
BkmWT
ijk ↔
GEC νijk,νijm,Uijk,Uijm·{{Ja3}}ij(k,m)
(C.4)
by the same calculation as in [21, Appendix C.1., Equations (C.4) and (C.5)] or [24, Appendix
B.3., Equation (B.31)]. To evaluate the fluxes ↔
GEC at the element interfaces, we apply the
consistence condition (2.65), such that e.g.
↔
GEC νNjk,νNjk,UNjk,UNjk=↔
FUNjk−{{ν}}(N,N)jkUNjk =↔
GNjk,
j,k=0,...,N.(C.5)
This provides the identity
N

j,k=0
ωjωk
N

i,m=0
BimWT
ijk ↔
GEC νijk,νmjk,Uijk,Umjk·{{Ja1}}(i,m)jk
+
N

i,k=0
ωiωk
N

j,m=0
BjmWT
ijk ↔
GEC νijk,νimk,Uijk,Uimk·{{Ja2}}i(j,m)k
123
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Journal of Scientific Computing (2020) 82 :69 Page 37 of 42 69
+
N

i,j=0
ωiωj
N

k,m=0
BkmWT
ijk ↔
GEC νijk,νijm,Uijk,Uijm·{{Ja3}}ij(k,m)
=
N

j,k=0
ωjωkWT
Njk↔
GNjk ·Ja1Njk−WT
0jk ↔
G0jk ·Ja10jk
+
N

i,k=0
ωiωkWT
iNk ↔
GiNk ·Ja2iNk−WT
i0k↔
Gi0k·Ja2i0k
+
N

i,j=0
ωiωjWT
ijN ↔
GijN ·Ja3ijN−WT
ij0↔
Gij0·Ja3ij0
=
∂E,N
WT↔
˜
G·ˆndS.(C.6)
Next, we investigate the first sum on the right hand side in the Eq.(C.4). Since the fluxes
GEC
l,l=1,2,3, satisfy the entropy condition (2.67), follows
N

j,k=0
ωjωk
N

i,m=0
Qim[[W]] T
(i,m)jk ↔
GEC νijk,νmjk,Uijk,Umjk·{{Ja1}}(i,m)jk
=
N

j,k=0
ωjωk
N

i,m=0
Qim[[W]] T
(i,m)jkGEC
1νijk,νmjk,Uijk,Umjk{{Ja1
1}}(i,m)jk
+
N

j,k=0
ωjωk
N

i,m=0
Qim[[W]] T
(i,m)jkGEC
2νijk,νmjk,Uijk,Umjk{{Ja1
2}}(i,m)jk
+
N

j,k=0
ωjωk
N

i,m=0
Qim[[W]] T
(i,m)jkGEC
3νijk,νmjk,Uijk,Umjk{{Ja1
3}}(i,m)jk
=
N

j,k=0
ωjωk
N

i,m=0
Qim [[1]](i,m)jk −{{ν1}}(i,m)jk[[ ]](i,m)jk{{ Ja1
1}}(i,m)jk
+
N

j,k=0
ωjωk
N

i,m=0
Qim [[2]](i,m)jk −{{ν2}}(i,m)jk[[ ]](i,m)jk{{ Ja1
2}}(i,m)jk
+
N

j,k=0
ωjωk
N

i,m=0
Qim [[3]](i,m)jk −{{ν3}}(i,m)jk[[ ]](i,m)jk{{ Ja1
3}}(i,m)jk.
(C.7)
For l=1,2,3, the SPB properties (C.1)and(C.2) provide
N

j,k=0
ωjωk
N

i,m=0
Qim [[l]](i,m)jk −{{νl}}(i,m)jk[[]] (i,m)jk{{Ja1
l}}(i,m)jk
=−
N

j,k=0
ωjωk(l)Njk −(νl)NjkNjkJa1
lNjk
123
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69 Page 38 of 42 Journal of Scientific Computing (2020) 82 :69
−(l)0jk −(νl)0jk 0jkJa1
l0jk
+
N

i,j,k=0
ωiωjωk(l)ijk
N

m=0
Dim Ja1
lmjk
−
N

i,j,k=0
ωiωjωkijk
N

m=0
2Dim{{νl}}(i,m)jk{{ Ja1
l}}(i,m)jk.(C.8)
Hence, we obtain the identity
N

j,k=0
ωjωk
N

i,m=0
Qim[[W]] T
(i,m)jk ↔
GEC νijk,νmjk,Uijk,Umjk·{{Ja1}}(i,m)jk
=−
N

j,k=0
ωjωk
Njk −(ν)Njk Njk·Ja1Njk −
0jk
−(ν)0jk 0jkJa10jk
+
N

i,j,k=0
ωiωjωk
ijk ·N

m=0
Dim Ja1mjk
−
N

i,j,k=0
ωiωjωkijk
N

m=0
2Dim{{ ν}}(i,m)jk ·{{Ja1}}(i,m)jk.
(C.9)
By the same computation, the third sum on the right hand side in the Eq. (C.4) becomes
N

i,k=0
ωiωk
N

j,m=0
Qjm[[W]] T
i(j,m)k↔
GEC νijk,νimk,Uijk,Uimk·{{Ja2}}i(j,m)k
=−
N

i,k=0
ωiωk
iNk −(ν)iNk iNk·Ja2iNk −
i0k
−(ν)i0ki0kJa2i0k
+
N

i,j,k=0
ωiωjωk
ijk ·N

m=0
Djm Ja2imk
−
N

i,j,k=0
ωiωjωkijk
N

m=0
2Djm{{ ν}}i(j,m)k·{{Ja2}}i(j,m)k
(C.10)
and the sum next-to-last on the right hand side in the Eq. (C.4) becomes
N

i,j=0
ωiωj
N

k,m=0
Qkm[[W]]T
ij(k,m)↔
GEC νijk,νijm,Uijk,Uijm·{{Ja3}}ij(k,m)
=−
N

i,j=0
ωiωj
ijN −(ν)ijN ijN·Ja3ijN −
ij0
−(ν)ij0ij0Ja3ij0
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Journal of Scientific Computing (2020) 82 :69 Page 39 of 42 69
+
N

i,j,k=0
ωiωjωk
ijk ·N

m=0
Dkm Ja3ijm
−
N

i,j,k=0
ωiωjωkijk
N

m=0
2Dkm{{ ν}}ij(k,m)·{{Ja3}}ij(k,m).(C.11)
The definition of the derivative projection operator (2.62)intheEq.(2.71a) provides
−
N

i,j,k=0
ωiωjωkijk
N

m=0
2Dim{{ ν}}(i,m)jk ·{{Ja1}}(i,m)jk
−
N

i,j,k=0
ωiωjωkijk
N

m=0
2Djm{{ ν}}i(j,m)k·{{Ja2}}i(j,m)k
−
N

i,j,k=0
ωiωjωkijk
N

m=0
2Dkm{{ ν}}ij(k,m)·{{Ja3}}ij(k,m)
=−
N

i,j,k=0
ωiωjωkijk 
DN·
˜νijk=−
DN·
˜ν, N.
(C.12)
Next, we plug the Eqs. (C.6), (C.9), (C.10), (C.11)intheEq.(C.4) and apply the identity
(C.12). This results in the identity

DN·↔
˜
GEC,W N=
∂E,NWT↔
˜
G·ˆn−
˜
−
˜ν·ˆndS −
DN·
˜ν, N
+
N

i,j,k=0
ωiωjωk
ijk ·N

m=0Dim Ja1mjk +Djm Ja2imk
+Dkm Ja3ijm.
(C.13)
The definition of the contravariant vector flux functions (2.23) and contravariant block vector
flux functions (2.28) provide the equality
WT↔
˜
G·ˆn−
˜
−
˜ν·ˆn=
3

l,r=1
Jar
lWTGl−9l+νlˆnr
=
3

l,r=1
Jar
lFs
l−νlSˆnr
=ˆsn·
Fs−νS=˜
Fs
ˆn−˜νˆnS.
(C.14)
Therefore, the Eq. (C.13) simplifies to

DN·↔
˜
GEC,W N=
∂E,N˜
Fs
ˆn−˜νˆnSdS −
DN·
˜ν, N,(C.15)
since the discrete volume weighted contravariant vectors Jaα,α=1,2,3, are computed by
the conservative curl form (2.52) and the discrete metric identities (2.60)aresatisfiedinthe
LGL points.
123
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69 Page 40 of 42 Journal of Scientific Computing (2020) 82 :69
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Affiliations
Gero Schnücke1·Nico Krais2·Thomas Bolemann2·Gregor J. Gassner3
Nico Krais
krais@iag.uni-stuttgart.de
Thomas Bolemann
bolemann@iag.uni-stuttgart.de
Gregor J. Gassner
ggassner@math.uni-koeln.de
1Mathematical Institute, University of Cologne, Cologne, Germany
2Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart, Stuttgart, Germany
3Mathematical Institute, Center for Data and Simulation Science (CDS), University of Cologne,
Cologne, Germany
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