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22nd International Congress of Mechanical Engineering (COBEM 2013)
November 3-7, 2013, Ribeirão Preto, SP, Brazil
Copyright c
2013 by ABCM
DIFFUSION-BASED LIMITERS FOR DISCONTINUOUS GALERKIN
METHODS - PART I: ONE-DIMENSIONAL EQUATIONS
Rodrigo Costa Moura
Rodrigo Castellari Affonso
André Fernando de Castro da Silva
Marcos Aurélio Ortega
Technological Institute of Aeronautics - ITA, São José dos Campos, Brazil
moura@ita.br, rodrigoaffonso@aluno.ita.br, andref@ita.br, ortega@ita.br
Abstract. Within the context of high-order accurate methods, the treatment of shock waves in the numerical solution of
non-linear hyperbolic equations is still a highly relevant field of research. In the last years, limiting techniques tradi-
tionally used with low-order schemes have been successfully adapted and extended for high-order methods. The same
can be said about artificial viscosity approaches; specifically, for the Discontinuous Galerkin (DG) method, recent works
based on such concept brought to the DG formulation the so called sub-cell shock resolution, namely, the capability of
representing shocks within a single mesh element. Typically, however, opting for artificial viscosity techniques in the DG
context requires the additional use of specific schemes to account for the artificial diffusion terms. Moreover, the introduc-
tion of an elliptic bias to originally hyperbolic equations severely restrains the (explicit) time-stepping stability envelope.
The choice for limiters does bypass such difficulties but, on the other hand, usually comes with drawbacks such as im-
plementation complexity, residue convergence problems and, normally, lack of sub-cell resolution. This study presents a
new type of shock capturing operator which practically does not suffer from any of the aforementioned disadvantages.
In summary, it works as a filter damping high-frequency modes, but is based on the concept of artificial viscosity, being
however applied as a limiter. In its final form, such diffusion-based limiter is implemented in an extremely simple way.
The operator is also relatively inexpensive and yet capable of providing results of surprising quality. Numerical tests
demonstrate fine sub-cell resolution as well as local accuracy scaling with h/p.
Keywords: Shock Treatment, Discontinuous Galerkin Method, High-order Schemes, CFD.
1. INTRODUCTION
Shock waves are well known physical structures that exist as a result of the compressibility of fluids, specially the air.
The importance of shock waves in engineering becomes evident in the aerospace context, where compressible flows in-
teract with vehicles traveling at supersonic speeds (Anderson, 2002). Therefore, minding CFD simulation and subsequent
design, the correct treatment of shock waves is of paramount importance when it comes to the numerical solution of fluid
flow equations.
Within the last two decades, the so called high-order methods (Ekaterinaris, 2005; Wang, 2007) are emerging in order
to properly address flow problems which particularly require highly accurate treatment, such as wave propagation for
aeroacoustics, vortex-dominated flows, turbulence simulation and also flows with complex shock interactions (Lê et al.,
2011). However, one of the issues difficulting a wider adoption of high-order methods amongst CFD practitioners is the
recognized need for robust and accurate shock capturing techniques suited for those methods (Vincent and Jameson, 2011;
Wang et al., 2013). The numerical treatment of flow discontinuities is addressed since several decades ago (von Neumann
and Richtmyer, 1950), but can still be considered a highly relevant field of research, specially in the context of high-order
schemes.
The present article focus on a modern but already popular high-order method known as Discontinuous Galerkin (DG)
(Cockburn et al., 2000; Hesthaven and Warburton, 2008). The reader is referred to the work of Moura (2012) for a
detailed introduction to the method. Concerning the treatment of shock waves, a variety of algorithms have been proposed
specifically for the DG formulation in the last years. For instance, different limiting techniques have been adapted and
extended for the DG scheme (Cockburn and Shu, 2001; Qiu and Shu, 2005; Krivodonova, 2007; Kuzmin, 2010). The same
can be said about artificial viscosity approaches (Persson and Peraire, 2006; Barter and Darmofal, 2007, 2010; Atkins and
Pampell, 2011); its worth mentioning that such viscous approaches brought to the DG formulation the so called sub-cell
shock resolution, namely, the distinctive capability of representing shocks within a single mesh element.
Working, however, with shock-capturing techniques based on artificial viscosity for the solution of hyperbolic equa-
tions (such as the Euler equations of gas dynamics) in a DG context requires the additional use of specific schemes to
account for the artificial diffusion terms (Arnold et al., 2002). Furthermore, the introduction of an elliptic bias to originally
hyperbolic equations severely restrains the time-stepping stability envelope (Klockner et al., 2011) when using explicit
time discretizations. The choice for limiters does bypass such difficulties but, on the other hand, usually comes with
Moura, R. C., Affonso, R. C., Silva, A. F. C. and Ortega, M. A.
Diffusion-based Limiters for Discontinuous Galerkin Methods - Part I: One-dimensional Equations
drawbacks such as implementation complexity, residue convergence problems and, normally, lack of sub-cell resolution
(Vincent and Jameson, 2011; Wang et al., 2013).
This article presents a new type of shock-capturing operator which practically does not suffer from any of the afore-
mentioned problems. In resume, it works as a filter damping high-frequency modes, but is based on the concept of
artificial viscosity, being however applied as a limiter. In its final form, such diffusion-based limiter is implemented in
an extremely simple way. The operator is also relatively inexpensive and yet capable of providing results of surprising
quality. Numerical tests demonstrate fine sub-cell resolution as well as local accuracy scaling with h/p (where his the
local mesh size and pis the degree of the polynomial expansion representing the solution), i.e. the same local accuracy
obtained with modern artificial viscosity approaches (Persson and Peraire, 2006; Barter and Darmofal, 2010).
The present study is organized in the following way. Section 2 introduces the reader to the DG discretization for
convection-diffusion problems in one dimension. It also discuss the idea behind shock-capturing artificial viscosity ap-
proaches and how they affect the (explicit) time-marching stability envelope. In section 3 the diffusion-based limiter
is presented, both in terms of theory and implementation. Numerical tests are addressed in section 4, where the (in-
viscid) Burgers’ equation is used to evaluate the operator local order of accuracy. Moreover, a specific form of the
one-dimensional Euler equations is solved to present the reader some of the fine results one can expect for “practical”
problems. At last, concluding remarks are given in section 5 along with future research possibilities.
2. THE DISCONTINUOUS GALERKIN FORMULATION
2.1 Discretizing the model equation
A generalized convection-diffusion (scalar) equation is here adopted as model equation for the DG formulation. Being
Ωthe one-dimensional domain of interest and given suited initial and boundary conditions, one have
∂u
∂t +∂f
∂x =∂fv
∂x in Ω, (1)
where f=f(u)and fv=fv(u, ∂u/∂ x)are, in that order, the inviscid and viscous flux functions. Traditionally, to solve
Eq. (1), the second-order PDE must be split into a system of two first-order PDEs, namely,
∂u
∂t +∂
∂x (f(u)−fv(u, g)) = 0 , (2)
g−∂u
∂x = 0 , (3)
by which the gradient function g(x)is introduced. At this point, one must proceed with the hp discretizations.
By hdiscretization, one means partitioning the domain Ωinto non-overlapping elements Ωesuch that Ω = SeΩe.
The pdiscretization consists of representing the (approximated) numerical solution u(x, t)within each element through
the sum of local basis functions φi, normally polynomials of degree equal or less than P, weighted by coefficients cito
be adjusted. It must be empathized that while the coefficients ciare time-dependent, the modal functions φiare not, being
however pre-defined in the standard domain Ωst = [−1,1]. In mathematical terms,
u(x, t)|Ωe∼
=
P
X
i=0
c(e)
i(t)φi(ξ), (4)
where the variables ξ∈Ωst and x∈Ωeare related by linear mappings given by
x(ξ) = 1−ξ
2x
e+1 + ξ
2x⊕
e,ξ∈Ωst = [−1,1] , (5)
ξ(x) = −1+2 x−x
e
x⊕
e−x
e
,x∈Ωe= [x
e, x⊕
e], (6)
in which x
eand x⊕
eare the left and right coordinates of element Ωe, respectively.
Here, the Legendre polynomials are chosen to generate the basis functions, since an orthogonal space of functions
can be spanned by this polynomial family. The Legendre polynomial of degree kcan be obtained as a particular case of
Rodrigue’s formula (Askey, 2005), being given by
Pk(ξ) = (−1)k
2kk!
dk
dξk(1 −ξ2)k,ξ∈Ωst . (7)
22nd International Congress of Mechanical Engineering (COBEM 2013)
November 3-7, 2013, Ribeirão Preto, SP, Brazil
In a numerical context, however, such polynomials (and its derivatives) are evaluated recursively through the following
expressions, already simplified from the textbook of Karniadakis and Sherwin (2005):
(k+ 1) Pk+1 (ξ) = (2k+ 1) ξPk(ξ)−kPk−1(ξ), (8)
(1 −ξ2)d
dξ Pk(ξ) = −k ξ Pk(ξ) + kPk−1(ξ), (9)
where the initial values needed for the recursive process can be calculated analytically from Eq. (7).
The Legendre polynomials are said to be orthogonal in Ωst in the sense that
ZΩst
Pi(ξ)Pj(ξ)dξ =δij
i+ 1/2, (10)
where δij is the Kronecker delta, being one if i=jand zero otherwise. Therefore, it is handy to normalize them in order
to use an orthonormal set of basis functions φi, namely,
φi(ξ) = pi+ 1/2Pi(ξ), (11)
such that
ZΩst
φi(ξ)φj(ξ)dξ =δij . (12)
Now, returning to Eqs. (2) and (3), one must follow the (standard) Galerkin method so as to make the projection of
both PDEs to vanish within the approximating element-wise polynomial space of functions. This condition must hold
locally for each element Ωe. Working first with Eq. (2), the procedure consists of enforcing
ZΩe
φj∂u
∂t +∂
∂x (f−fv)dx = 0 for j= 0, ..., P . (13)
Such set of equations can be translated into corresponding expressions for the coefficients c(e)
i, once
ZΩe
φj
∂u
∂t dx =ZΩe
φj P
X
i=0
dci
dt φi!dx =
P
X
i=0 dci
dt ZΩe
φjφidx=Je
dcj
dt , (14)
since, by Eq. (12),
ZΩe
φiφjdx =JeZΩst
φiφjdξ =Jeδij , (15)
where Jeis the transformation Jacobian associated with Ωe, which, through Eq. (5), is given by
Je=dx
dξ =x⊕
e−x
e
2. (16)
The remaining terms related to Eq. (13) can be worked out by noticing that
φj
∂
∂x (f−fv) = ∂
∂x [(f−fv)φj]−(f−fv)∂φj
∂x . (17)
Communication between adjacent elements is now introduced in the formulation by making
ZΩe
φj
∂
∂x (f−fv)dx ∼
=h(˜
f−˜
fv)φjiΩ⊕
e
Ω
e
−ZΩe
(f−fv)∂φj
∂x dx , (18)
where ˜
f=˜
f(uL, uR)and ˜
fv=˜
fv(uL, gL, uR, gR)are numerical fluxes (to be defined) based on the solution values
at the left side (L) and at the right side (R) of the boundaries Ω
eand Ω⊕
e, since the global solution is allowed to be
discontinuous and therefore double-valued at interfaces.
Using Eqs. (14) and (18) in Eq. (13) yields
Je
dci
dt =ZΩe
(f−fv)∂φi
∂x dx −h(˜
f−˜
fv)φiiΩ⊕
e
Ω
e
for i= 0, ..., P . (19)
Moura, R. C., Affonso, R. C., Silva, A. F. C. and Ortega, M. A.
Diffusion-based Limiters for Discontinuous Galerkin Methods - Part I: One-dimensional Equations
Working now with Eq. (3), the same procedure must be applied. In order to do that, the gradient function g(x)can be
written within each element Ωeas
g(x, t)|Ωe∼
=
P
X
i=0
γ(e)
i(t)φi(ξ), (20)
where the coefficients γirely upon the solution u(x, t), being obtained through the set of equations
ZΩe
φjg−∂u
∂x dx = 0 for j= 0, ..., P . (21)
Such expressions can be worked out by noticing that
φj
∂u
∂x =∂
∂x (uφj)−u∂φj
∂x , (22)
which can be used to yield
Jeγi= [˜u φi]Ω⊕
e
Ω
e
−ZΩe
u∂φi
∂x dx for i= 0, ..., P , (23)
where ˜u= ˜u(uL, uR)is a numerical average (to be defined) which further increases communication across boundaries
and is fundamental to the stability of the formulation.
In summary, at each time instant, the numerical solution u(x, t)is used to obtain g(x, t)via Eqs. (20) and (23), so
that the flux functions f(u)and fv(u, g)can be evaluated and used in Eq. (19), yielding dci/dt by which the numerical
solution can be calculated at the next time step through the application of time discretization techniques. To close the
formulation, the numerical fluxes ˜
fand ˜
fv, as well as the numerical average ˜u, must be discussed and defined.
A variety of formulas for the numerical fluxes are available in the literature (Karniadakis and Sherwin, 2005). For the
sake of simplicity, two well-known techniques are here adopted. For the inviscid numerical flux ˜
f, the Lax-Friedrichs
formula (Rider and Lowrie, 2002) is employed:
˜
f(uL, uR) = 1
2(f(uL) + f(uR)) −|˜
λ|
2(uR−uL), (24)
where, as usual, uLand uRare the solution values respectively taken from the left and from the right side of the interface
in which the flux is being evaluated. For instance, at the interface x⊕
e=x
e+1 shared by elements Ωe= [x
e, x⊕
e]and
Ωe+1 = [x
e+1, x⊕
e+1], they are given by
uL=u(e)(x⊕
e) =
P
X
i=0
c(e)
iφi(+1) =
P
X
i=0
c(e)
iφ⊕
i, (25)
uR=u(e+1)(x
e+1) =
P
X
i=0
c(e+1)
iφi(−1) =
P
X
i=0
c(e+1)
iφ
i. (26)
The value of ˜
λcorresponds to the eigenvalue related to the inviscid flux function (λ=∂f /∂u) calculated in some mean
value of the solution at the considered interface. For the scalar case, the simple average is a valid choice:
˜
λ=λ((uL+uR)/2) . (27)
For the viscous terms ˜
fvand ˜u, the BR2 scheme (Bassi et al., 1997; Brezzi et al., 2000) is here adopted, in which
˜u=1
2(uL+uR), (28)
˜
fv(uL, gL, uR, gR) = 1
2(fv(uL, g?
L) + fv(uR, g?
R)) , (29)
where g?
Land g?
Rare modified gradient functions at the left and right sides of the interface, which for the basis functions
here adopted, are given by
g?
L=∂u
∂x L
+η
2JL
(uR−uL)
P
X
i=0
(φ⊕
i)2, (30)
22nd International Congress of Mechanical Engineering (COBEM 2013)
November 3-7, 2013, Ribeirão Preto, SP, Brazil
g?
R=∂u
∂x R
+η
2JR
(uR−uL)
P
X
i=0
(φ
i)2, (31)
in which ηis a stabilization parameter that can be fixed as η= 3 for the one-dimensional case, JLand JR, recall Eq.
(16), are the Jacobians of the elements sharing the considered interface from the left and right sides, and the values of
[∂u/∂x]Land [∂u/∂x]Rare obtained directly by the expressions
∂u
∂x L
="∂
∂x
P
X
i=0
ciφi#L
=1
JL"P
X
i=0
ci
∂φi
∂ξ #L
=1
JL
P
X
i=0
cL
i∂ξφ⊕
i, (32)
∂u
∂x R
="∂
∂x
P
X
i=0
ciφi#R
=1
JR"P
X
i=0
ci
∂φi
∂ξ #R
=1
JR
P
X
i=0
cR
i∂ξφ
i. (33)
2.2 Treating shocks
The sub-cell shock resolution capability of the DG formulation was first recognized through the work of Persson and
Peraire (2006), where a shock capturing operator based on the concept of artificial viscosity was proposed. The idea of
the model is to solve the equations of interest with the inclusion of a diffusion term whose magnitude becomes non-zero
in the vicinity of discontinuities. For instance, in the case of purely hyperbolic problems, a modified PDE must be used,
similar to Eq. (1), where its right-hand side term would represent the artificial diffusion, such that
fv=(u)∂u
∂x , (34)
where (u)is the viscosity magnitude.
Without proper treatment, Gibbs-like oscillations are known to appear in the presence of shocks, degrading the solution
accuracy and possibly causing the simulation to diverge. In the model here considered, an element-wise constant viscosity
is used, where the magnitude of is triggered by the activation of an oscillation sensor, or, more specifically, by a sensor
that estimates the solution’s lack of resolution. When using orthonormal basis functions, the sensor in question is given,
within each element, by
σ=c2
P
c2
0+... +c2
P
. (35)
After calculating σ, the viscosity is introduced gradually by means of a smooth switching, namely,
=
0if s < so−κ,
1
2oh1 + sin π(s−so)
2κiif so−κ≤s≤so+κ,
oif s > so+κ,
(36)
where s= log10 σ,so=−(A+Blog10 P)and o=C h/P , being hthe size of the considered element. Typical values
of the model parameters are A∼
=4,B∼
=4,C∼
=1/2and κ∼
=1/2, but they are strongly dependent on the particular
problems being simulated and may adjusted to furnish optimal results. Also, regarding numerical implementation, it
is advisable to add a small constant (∼
=10−10) to the value of σwhen calculating s= log10 σin order to avoid the
divergence of the logarithmic function in the limit of a zero argument (away from discontinuities).
When using viscosity-based shock capturing approaches, discontinuities are turned into continuous gradient layers of
length proportional to the viscosity magnitude (von Neumann and Richtmyer, 1950). When artificial diffusion is applied
in a DG context, the intention is to spread discontinuities just to the extent where the approximating space is able to
resolve shock transitions well, without spurious oscillations. In general, the resolution provided by polynomial functions
in a hp discretization scales with δ∼h/p. In other words, when using a viscosity of magnitude O(h/p), a gradient layer
of thickness δcan be adequately represented by a polynomial space of degree pin a mesh of local size h.
The nominal order of accuracy of the DG method for smooth solutions is P+ 1/2(Johnson and Pitkaranta, 1986;
Richter, 1992). Such error estimates were obtained for simple equations (scalar, linear) and rely upon somewhat restrictive
hypothesis. Fortunately, however, order P+ 1 is often verified even for the complete Navier-Stokes equations in three
dimensions, see for instance the cases analyzed in the work of Nogueira et al. (2010). On the other hand, in the vicinity
of shocks, the accuracy is typically reduced to first order. When using the artificial viscosity model here considered, the
error level observed near shocks is compatible with the magnitude of the artificial term introduced on the original PDE,
namely, O(h/p), which is normally much better than "plain" first order, O(h), but still far from the usual order of accuracy
expected at smooth regions, O(hp+1).
Moura, R. C., Affonso, R. C., Silva, A. F. C. and Ortega, M. A.
Diffusion-based Limiters for Discontinuous Galerkin Methods - Part I: One-dimensional Equations
2.3 Time step reduction
In the simulation of hyperbolic equations, the employment of explicit time discretizations is not only simple but also
efficient. When dealing with steady-state problems, the use of local time-stepping techniques is advisable in order to
improve the residue convergence rate to the desired solution. Within the DG context, the CFL-based formula (Toulorge
and Desmet, 2011) for the time step is given by
∆t=hCFL
(2P+ 1)|λ|, (37)
in which his the local mesh size, λis the average wave speed within the considered element and CFL is the prescribed
Courant-Friedrichs-Lewy number, typically of order one.
For simulations where artificial viscosity models are employed, the time-step stability limits are greatly reduced (for
explicit time-marching). The adaptation of the local time-step formula consists of dividing the expression in Eq. (37) by
a viscous factor (Klockner et al., 2011) which can be written as
vf=s1 + 2P+ 1
h
|λ|2
, (38)
where is the local viscosity magnitude. It is important to highlight that vfcan be much greater than one, specially for
higher-order simulations. This is why the choice for artificial viscosity approaches can be significantly worse than using
limiting techniques if one wants to avoid implicit time discretizations.
3. THE DIFFUSION-BASED LIMITER
3.1 Solving the diffusion PDE
The idea behind the limiting technique here proposed consists in applying the diffusion equation as a filtering operator
at troubled elements. The oscillating local expansion is taken as initial condition in the solution of the classical diffusion
PDE, with suitable boundary conditions prescribed at the edges of the troubled element. During the diffusion process
the operator works as a filter damping high-frequency modes of the expansion. This idea was borrowed from the field of
image processing, where a similar technique is used to smooth out noisy figure-related data, see for instance the work of
Wu et al. (2008).
The diffusion equation, also known as the heat equation, is given by
∂u
∂t =∂2u
∂x2, (39)
where the element-wise artificial viscosity is here taken as diffusivity constant. In order to apply the DG formulation,
the approach described in section (2.1) is now reproduced, yielding
∂u
∂t =∂fv
∂x ,fv= g , (40)
g=∂u
∂x , (41)
such that, in a troubled element Ω, one has, according to Eqs. (19) and (23),
Jdci
dt =−ZΩ
g∂φi
∂x dx +h˜
fvφii⊕
, (42)
Jγj=−ZΩ
u∂φj
∂x dx + [˜u φj]⊕
, (43)
whose integrations can be moved to Ωst, so that, recalling Eqs. (4) and (20),
Jdci
dt =h˜
fvφii⊕
−X
j
γjµji , (44)
Jγj= [˜u φj]⊕
−X
k
ckµkj , (45)
22nd International Congress of Mechanical Engineering (COBEM 2013)
November 3-7, 2013, Ribeirão Preto, SP, Brazil
where the summations from here on always varies from 0to P, and µis defined as
µab =ZΩst
φa
∂φb
∂ξ dξ . (46)
Using Eq. (45) into Eq. (44) gives
J2dci
dt =Jh˜
fvφii⊕
−X
j
µji ([˜u φj]⊕
−X
k
ckµkj ), (47)
but noting that
X
j
µji [˜u φj]⊕
= ˜u⊕X
j
µji φ⊕
j−˜uX
j
µji φ
j=
˜uX
j
µji φj
⊕
, (48)
X
j
µji X
k
ckµkj !=X
j,k
ckµkj µji =X
k
ck
X
j
µkj µji
, (49)
turns Eq. (47) into
J2dci
dt =Jh˜
fvφii⊕
−
˜uX
j
µji φj
⊕
+X
k
ck
X
j
µkj µji
, (50)
which can be seen as an ODE system, where the first two terms on the right-hand side contain the boundary conditions
for the diffusion PDE and the third term is a linear combination of the coefficients ci.
3.2 On the local boundary conditions
An important characteristic of a limiter is the conservativity property, which in the context of high-order methods
means that the average value of the numerical solution within a troubled element must remain unchanged during the
limiting process. When dealing with a modal DG formulation, it is sufficient to keep untouched the first coefficient of the
local expansion, since
ZΩ
u dx =1
φ0ZΩ
φ0X
i
ciφidx =Jc0
φ0
. (51)
Now, noting from Eq. (46) that µj0= 0 ∀j, Eq. (50) yields
Jdc0
dt =h˜
fvφ0i⊕
=˜
f⊕
vφ⊕
0−˜
f
vφ
0=˜
f⊕
v−˜
f
vφ0, (52)
so that exact conservativity would require ˜
f⊕
v=˜
f
v= 0. This, however, is not an option because of stability issues.
More specifically, the ODE in Eq. (50) can be verified to be unstable when such condition is adopted. Despite of this,
conservativity can still be pursued in a weaker sense.
In the following analysis, the letters Rand Lwill be used to denote properties outside the troubled element, referring
to properties of virtual elements at the right (R) and left (L) sides of the troubled element. For now, the central element
should be regarded as disconnected from the global mesh and from its real neighboring elements. In what follows, the
(local) boundary conditions used for the solution of Eq. (50) within a troubled element will be discussed.
Using Eqs. (29) to (33), one can write
˜
f⊕
v=1
2(
JX
i
ci∂ξφ⊕
i+R∂u
∂x R
+η
2 uR−X
i
ciφ⊕
i!X
i
Jφ⊕
i2+R
JRφ
i2), (53)
˜
f
v=1
2(L∂u
∂x L
+
JX
i
ci∂ξφ
i+η
2 X
i
ciφ
i−uL!X
iL
JLφ⊕
i2+
Jφ
i2). (54)
Since the whole expressions of ˜
f⊕
vor ˜
f
vcannot be made zero, a first idea in that direction would be to prescribe values
for [∂u/∂x]Rand [∂u/∂x]Lsuch that
JX
i
ci∂ξφ⊕
i+R∂u
∂x R
=L∂u
∂x L
+
JX
i
ci∂ξφ
i= 0 , (55)
Moura, R. C., Affonso, R. C., Silva, A. F. C. and Ortega, M. A.
Diffusion-based Limiters for Discontinuous Galerkin Methods - Part I: One-dimensional Equations
by which the numerical viscous fluxes would become
˜
f⊕
v=η
4 uR−X
i
ciφ⊕
i!X
i
Jφ⊕
i2+R
JRφ
i2, (56)
˜
f
v=η
4 X
i
ciφ
i−uL!X
iL
JLφ⊕
i2+
Jφ
i2. (57)
At this point, one must note that if uRand uLwere calculated as a composition of the solution values at the interfaces
of the real mesh, i.e. taking into account the global solution, than the property jumps (the first expressions in parenthesis
above) would be of magnitude O(h/p), since that is the error level of the global solution in the vicinity of shocks, as
discussed in section (2.2). Following this approach, the error introduced by the limiter in the solution average value,
obtained by the variation ∆c0, recall Eq. (52), would also be of magnitude O(h/p). For the sake of consistency, this
would also guarantee that ∆c0→0through hor prefinement. Therefore, the limiting technique here proposed can be
considered conservative within the order of accuracy of the formulation near shocks.
Whatever might be the definitions of uRand uL, the values of ˜uneeded in Eq. (50) must be prescribed coherently:
˜u⊕=1
2 X
i
ciφ⊕
i+uR!, (58)
˜u=1
2 uL+X
i
ciφ
i!. (59)
Different formulas for uRand uLwere tested, but the best results were obtained with a “crossed averaging”, namely,
uR=1
|S|+|SR| |S|X
i
cR
iφ
i+|SR|X
i
ciφ⊕
i!, (60)
uL=1
|SL|+|S| |SL|X
i
ciφ
i+|S|X
i
cL
iφ⊕
i!, (61)
where ci,cR
iand cL
iare the coefficients of the solution expansion in the troubled element and in its (real) neighbors, the
same being true for S,SRand SL. Here, Sstands for an estimate of the solution slope within the considered elements,
given by the second expansion coefficients in a modal DG context, i.e. S=c1,SR=cR
1and SL=cL
1. The idea is
that the local boundary conditions should be obtained mainly from outside the troubled element, so as to avoid the use of
“noisy data”.
3.3 Solving the diffusion ODE
Using Eqs. (56) to (59) into Eq. (50), one can arrive at
2J2
dci
dt =X
j
µji uLφ
j−uRφ⊕
j+η ζ
2uLφ
i+uRφ⊕
i+
+X
k
ck
X
j
µji (µkj −µj k)−η ζ
2φ
iφ
k+φ⊕
iφ⊕
k
, (62)
where the variable ζwas defined by the approximations
ζ=X
khφ⊕
k2+φ
k2i∼
=X
kφ⊕
k2+J
JR
R
φ
k2∼
=X
kJ
JL
L
φ⊕
k2+φ
k2, (63)
which can be done by referring again to the concept of virtual adjacent elements, so that JL,JR,Land Rmay be defined
suitably.
The ODE system given in Eq. (62) is now written in vector form, namely,
2J2
dc
dt =Ac +b, (64)
22nd International Congress of Mechanical Engineering (COBEM 2013)
November 3-7, 2013, Ribeirão Preto, SP, Brazil
where cis the coefficients vector, Ais the system matrix and bis a vector containing the edge conditions uLand uR.
Both Aand bare constants for the local ODE, being given by
(A)m,n =X
j
µjm (µnj −µj n)−η ζ
2φ
mφ
n+φ⊕
mφ⊕
n, (65)
(b)m=
η ζ
2φ
m+X
j
µjm φ
j
uL+
η ζ
2φ⊕
m−X
j
µjm φ⊕
j
uR. (66)
Since matrix Acan be verified to be diagonalizable, an analytical solution for the considered ODE is available. Let
D={...vn...}be a matrix whose columns are the (right) eigenvectors of A, and λnthe respective eigenvalues, such that
Avn=λnvn. Then, the solution c(t)can be shown to be
c(t) = DEcD−1c0+DEbD−1b, (67)
where c0=c(t= 0), while Ecand Ebare diagonal time-dependent matrices, given by
(Ec)n,n = exp λn
t
2J2, (68)
(Eb)n,n =1
λn
exp λn
t
2J2−1
λn
. (69)
3.4 Implementation details
The diffusion-based limiter acts through the operator given in Eq. (67). The limiting should be performed between
each step the time-marching process, but only at troubled elements. The oscillating solution is introduced as c0and the
limited solution is obtained as c(t), where the duration of the diffusion process must be consistent with the time step used
for the time discretization of the original equation, i.e. t= ∆t.
It is important to remind that, for purely hyperbolic problems, the local time step ∆tcan be calculated directly by Eq.
(37). Since the artificial diffusion PDE is not accounted by the DG discretization, but solved analytically, there is no need
to use the viscous factor given in Eq. (38). For the same reason, one does not have to worry about employing specific
techniques, such as the BR2 scheme, for the discretization of artificial viscosity terms.
One can note by Eq. (65) that the matrix Ais not mesh-dependent, the same being true for its eigenvalues and
eigenvectors. Therefore, when applying Eq. (67), one needs only to adjust the diagonal matrices Ecand Eb(with the
local values of J,and ∆t), and to evaluate the vector busing the formulas for uLand uRgiven in Eqs. (60) and (61).
For each element, the viscosity magnitude can be obtained by the smooth switching given in Eq. (36).
4. NUMERICAL RESULTS
4.1 Accessing the local accuracy
In order to address the accuracy of the solution (with limiting) in the vicinity of shocks, the inviscid Burgers’ equation,
given by
∂u
∂t +∂f
∂x = 0 ,f=u2
2, (70)
is here solved for x∈(−∞,+∞)with the initial conditions
u=
1if x < −1,
−xif −1≤x≤1,
−1if x > 1.
(71)
The exact solution reaches a steady state for t= 1, in which u=−x/|x|everywhere except at x= 0, where the shock
takes place.
For the numerical solution, two spatial discretizations are considered. First, the domain is divided into an odd number
of elements to simulate a shock forming in the middle of an element. After that, the domain is divided into an even number
of elements to test the case in which the shock forms at the interface between adjacent elements. Full residue convergence
was achieved for all cases addressed. The error between exact and numerical solutions is measured in the L1norm.
Moura, R. C., Affonso, R. C., Silva, A. F. C. and Ortega, M. A.
Diffusion-based Limiters for Discontinuous Galerkin Methods - Part I: One-dimensional Equations
Figure 1. Error convergence for hrefinement with different values of P. Slopes equal to 1were verified for all cases.
Figure 2. Error convergence for Prefinement. Asymptotic slopes equal to −1were verified.
Figure 3. Steady-state shock profiles obtained with different values of P. At the left side, the transition takes place in the
middle of an element; at the right side, the shock forms at the interface between adjacent elements.
22nd International Congress of Mechanical Engineering (COBEM 2013)
November 3-7, 2013, Ribeirão Preto, SP, Brazil
The results obtained for hrefinement are given in Fig. 1. Different values of Pwere tested. For all the cases, the error
is shown to decay with h1, but the actual error level was verified to decrease with increasing P. Such results hold for odd
and even number of elements, as can be seen at the left and right sides of Fig. 1, respectively.
The effect of Prefinement was then analyzed, and the results are displayed in Fig. 2. The error behavior is again
considered for an odd number of elements (such that 3 elements of the same size rely within [−1,1]) as well as for an
even number (such that 4 elements of the same size rely within [−1,1]). For both cases, the error is shown to decay
asymptotically with p−1.
In summary, the order of accuracy of the numerical solution after limiting is verified to be of magnitude O(h/p)in the
vicinity of shocks. The transition profiles obtained at x= 0 are shown in Fig. 3 for different values of P. As one can see,
fine shock layers are in fact given by the limiting technique here proposed. Solution continuity at interfaces can be more
strongly enforced with a higher value of η. For the simulations here considered, the value η= 3 was used.
4.2 The case of systems
Having fluid flow problems in mind, a modification of the one-dimensional Euler equations in is now considered. The
modification consists in the inclusion of a source term to account for flows with area variations, such as those inside ducts
or nozzles (Toro, 1999). In conservation form, this model equation can be written as
∂U
∂t +∂F
∂x =S, (72)
where the vector of conserved variables U, the flux vector Fand the source term Sare given by
U={ρ, ρu, e}T, (73)
F=ρu, ρu2+p, (e+p)uT, (74)
S=−d(ln A)/dx ρu, ρu2,(e+p)uT, (75)
where {·}Tindicates the transpose of a vector, ρstands for density, uis the velocity, e=ρ(ei+u2/2) is the total energy
per unit volume, and eiis the specific internal energy. The static pressure pis obtained using the equation of state for a
perfect gas, namely, p=ρ(γ−1)ei, where γis the fluid ratio of specific heats, which assumes the value γ= 7/5for the
air. Moreover, the cross-sectional area of the nozzle is denoted by A=A(x).
When applying the diffusion-based limiter to systems, each equation should be treated separately, as if they were
independent scalar equations. For instance, in order to evaluate the vector bin Eq. (67), the values of uLand uRmust
be obtained for each conserved variable of U. The slope factors used in Eqs. (60) and (61) can however be based on an
unique characteristic property. Here, the coefficients of the density expansion are used as slope factors.
For the detection of troubled elements, the sensor σdiscussed in section 2.2 was applied. In the case of systems,
one need to choose a scalar variable upon which the sensor must rely. Here, the element-wise Mach number expansion
was used, where the Mach number is given by M=|u|/a, being a=pγ p/ρ the speed of sound. As noted in the
work of Huerta et al. (2012), this choice grants better results than those obtained when density or entropy expansions are
employed.
The considered problem was solved for x∈[0,1], being the area function prescribed as A= 0.05 + 1.4(x−0.5)2.
Suited boundary conditions were adjusted so that the ratio between the pressure peat the exit of the nozzle and the
stagnation pressure p0at the inflow boundary was fixed as pe/p0= 7/10. The domain was discretized into 100 elements
of equal size and the solutions obtained with P= 1 and P= 5 are respectively shown in Figs. 4 and 5. Both figures
display the distributions of Mach number and static pressure inside the nozzle, along with the residue convergence history
and the limiting-related normalized artificial viscosity /0, recall Eq. (36).
5. CONCLUDING REMARKS
This article presented a new type of shock capturing operator for the Discontinuous Galerkin (DG) method. The
DG formulation for convection-diffusion problems was discussed and then a diffusion-based limiter was introduced in a
detailed manner. The limiter here proposed works as a filter damping high-frequency modes, but is based on the concept
of artificial viscosity. In terms of implementation, the limiting technique is simple, relatively inexpensive, and yet capable
of providing results of surprising quality.
Numerical tests demonstrated fine sub-cell resolution as well as local accuracy scaling with h/p (where his the
local mesh size and pis the degree of the polynomial expansion representing the solution). It was also verified that the
limiting procedure does not affect the (explicit) time-marching stability envelope nor hinders residue convergence. Such
characteristics are specially desired when simulating purely hyperbolic problems with explicit time-stepping.
Moura, R. C., Affonso, R. C., Silva, A. F. C. and Ortega, M. A.
Diffusion-based Limiters for Discontinuous Galerkin Methods - Part I: One-dimensional Equations
Figure 4. Numerical solution obtained with P= 1 for the compressible flow inside a nozzle. The mesh consists of 100
elements of equal size. As one can see, full convergence can be achieved with the diffusion-based limiter.
Figure 5. Numerical solution obtained with P= 5 for the compressible flow inside a nozzle. The mesh consists of 100
elements of equal size. As one can see, full convergence can be achieved with the diffusion-based limiter.
22nd International Congress of Mechanical Engineering (COBEM 2013)
November 3-7, 2013, Ribeirão Preto, SP, Brazil
Among future research possibilities, one can mention the development of better suited, parameter-free, oscillation
sensors for the detection of troubled elements, the analysis of different averaging formulas for the limiting-related local
boundary conditions and, of course, the extension of the diffusion based-limiter for fluid flow equations in two and three
dimensions.
6. ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support for this research provided by FAPESP (Sao Paulo Research Founda-
tion), under the Research Grant No. 2012/16973-5.
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8. RESPONSIBILITY NOTICE
The authors are the only responsible for the printed material included in this paper.
