A Kinetic Energy Preserving DG Scheme based on Gauss …ortleb/DG_KEP_GaussLegend... · 2016. 12. 13. · energy preserving discontinuous Galerkin scheme with Gauss-Lobatto nodes

A Kinetic Energy Preserving DG Scheme based onGauss-Legendre Points1

Sigrun Ortleb

University of Kassel, Department of Mathematics and Natural Sciences, [email protected]

Abstract

In the context of numerical methods for conservation laws, not only the preser-vation of the primary conserved quantities can be of interest, but also the balanceof secondary ones such kinetic energy in case of the Euler equations of gas dynam-ics. In this work, we construct a kinetic energy preserving discontinuous Galerkinmethod on Gauss-Legendre nodes based on the framework of summation-by-partsoperators. For a Gauss-Legendre point distribution, boundary terms require spe-cial attention. In fact, stability problems will be demonstrated for a combinationof skew-symmetric and boundary terms that disagrees with exclusively interiornodal sets. We will theoretically investigate the required form of the correspond-ing boundary correction terms in the skew-symmetric formulation leading to aconservative and consistent scheme. In numerical experiments, we study the orderof convergence for smooth solutions, the kinetic energy balance and the behaviourof different variants of the scheme applied to an acoustic pressure wave and aviscous shock tube. Using Gauss-Legendre nodes results in a more accurate ap-proximation in our numerical experiments for viscous compressible flow. Moreover,for two-dimensional decaying homogeneous turbulence, kinetic energy preservationyields a better representation of the energy spectrum.

1 Introduction

When discretizing the Euler equations of gas dynamics, the straightforward way is tostart directly with the conservative divergence form of this system of hyperbolic equa-tions. Thereby, discrete conservation is almost automatically guaranteed and the Lax-Wendroff theorem then states that if the scheme is convergent, then it converges to aweak solution and hence yields correct shock speeds. Nonetheless, not only the preser-vation of primary conservative quantities but also the balance of secondary quantitiesas the entropy or the kinetic energy can be important for specific applications. Forexample in the context of the simulation of turbulent flows, an accurate simulation ofthe kinetic energy should be aimed at, see e.g. [20, 31, 26, 19, 10, 16]. For finite vol-ume methods, specifically designed numerical fluxes may guarantee either preservationof entropy or energy, see [15]. Similar construction principles to guarantee enhancedconservation properties can be found in the context of shallow water flows, for exampleenergy head conservation for simulations involving rapidly varied flow e.g. due to largegradients in bathymetry [25] or conservation of total energy in addition to mass andmomentum [30].

The specific enforcement of kinetic energy preservation is often based on the ob-servation that dissipative techniques used for shock capturing may compromise the

1The final publication is available at Springer via http://dx.doi.org/10.1007/s10915-016-0334-2

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calculation of viscous flows and should hence be reduced or avoided if possible. Now,a discrete scheme satisfying certain energy estimates may establish stability withoutadditional numerical dissipation - at least for smooth solutions - and should enable amore accurate viscous flow computation. Of course, this feature is also desirable forhigher order methods and is reflected by so-called summation-by-parts (SBP) finitedifference schemes applied to skew-symmetric forms of conservation laws. While theskew-symmetric form is used to enforce enhanced conservation of specific secondaryquantities, the SBP property mimics integration by parts and thus yields a discreteconservative operator enforcing conservation of the primary conserved quantities. Infact, as stated in [7], a sufficient prerequisite for the applicability of the Lax-Wendrofftheorem only is a discretely conservative difference operator. It is not necessary toderive this operator from the divergence form of the continuous equations. On the con-trary, carefully designed discrete operators applied to linearly split forms - e.g. diagonalnorm SBP-operators - are equivalent to telescoping operators fulfilling the assumptionsof the Lax-Wendroff theorem, see [7]. Moreover, SBP operators in combination withso-called simultaneous approximation terms (SAT) enable the construction of stronglystable schemes for quite general cases including linear advection and advection-diffusionequations, variable coefficient equations and Burgers’ equation, see e.g. [27].

At present, the theory and application of summation-by-parts(SBP) operators is anactive field of research for the numerical solution of time-dependent partial differentialequations. While originally, SBP operators have been introduced to finite differenceschemes [18] in order to construct high order conservative and time-stable methods[4, 21], recent results also focus on SBP operators within various popular classes ofnumerical schemes, e.g. finite volume schemes on unstructured dual grids [22], dis-continuous Galerkin(DG) schemes with Gauss-Lobatto nodes [8], the correction-via-reconstruction scheme which includes so-called spectral-difference methods, see [23], aswell as the direct construction of generalized nodal SBP operators in one space dimension[6] and even on simplex elements [12]. In fact, the generalizations in [6] show that 1DDG schemes on arbitrary nodal sets yield generalized SBP operators, since by construc-tion, their quadrature rule is sufficiently exact in order to mimic integration-by-parts atthe discrete level.

Closely linked to the framework of SBP finite difference schemes, in [9], a kineticenergy preserving discontinuous Galerkin scheme with Gauss-Lobatto nodes has beenconstructed based on a skew-symmetric formulation of the Euler equations. This con-struction rested upon the classical SBP property of the Gauss-Lobatto DG scheme.However, as stated above, the generalized SBP property is not restricted to Gauss-Lobatto nodes. Hence, the question is evident if a kinetic energy preserving DG schemecan also be based on the more accurate Gauss-Legendre nodes.

In this spirit, this gap is closed in this work and a kinetic energy preserving DGscheme using Gauss-Legendre nodes is constructed which is similar to the skew-symmetricDG scheme [9] designed for Gauss-Lobatto nodes.

Since quadrature rules based on the Gauss-Legendre nodes provide a higher degreeof exactness in comparison to an equal number of Gauss-Lobatto nodes, a lower errorof the corresponding DG approximation may be expected. In fact, higher accuracy ofthe DG scheme using Gauss-Legendre nodes will be experimentally shown for viscouscompressible flow. While Gauss-Legendre nodes lead to a diagonal mass matrix and

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hence a diagonal norm SBP operator as in the case of Gauss-Lobatto nodes, the interfaceoperator is not diagonal as in [9] as the Gauss-Legendre nodes do not include the intervalboundaries of a grid cell. Hence, the skew-symmetric DG scheme constructed in [9] cannot be used in combination with Gauss-Legendre nodes. In fact, in this work, we willshow stability issues that may arise when using a combination of skew-symmetric termswith inconsistent boundary treatment that disagrees with exclusively interior nodal sets.

This paper is organized as follows. After an introduction of the DG scheme in SBPframework in Sect. 2, including the precise setting of a generalized SBP property forDG schemes in 1D according to [6], the derivation of a skew-symmetric, kinetic energypreserving DG scheme on Gauss-Legendre nodes is presented in Sect. 3. Herein, wewill theoretically investigate the necessary form of inner correction terms and fluxeswhen additional skew-symmetric terms are present in the DG formulation. The correctskew-symmetric form of the DG scheme then has to be supplied with a kinetic energyconserving numerical flux for which different variants are taken into account. Accord-ingly, the numerical experiments in Sect. 4 study different variants of skew-symmetricDG schemes including skew-symmetric terms as in [9] which may not be used for Gauss-Legendre nodes. Thereby, the experimental order of convergence is studied for smoothsolutions using the method of manufactured solutions for the Euler equations of gasdynamics, while a direct study of the kinetic energy balance is carried out with the helpof reduced equations. Furthermore, the scheme is applied to an acoustic pressure waveas in [9] and a viscous shock tube similar to the set-up in [1]. Sect. 5 contains an exten-sion of the investigations made in one-dimensional space to the case of two-dimensionalcartesian grids. Here, numerical results are shown for a test case of two-dimensionalhomogeneous turbulence which is extensively studied in literature. The last Sect. 6contains conclusions to these investigations and an outlook on future work.

2 The DG scheme on Gauss nodes in SBP framework

For ease of presentation of the classical DG scheme and its SBP property, we considera scalar hyperbolic conservation law in one space dimension given by

∂

∂tu(x, t) +

∂

∂xf(u(x, t)) = 0, t > 0, x ∈ Ω = [a, b] ⊂ R. (1)

After a subdivision of the spatial domain Ω into sub-intervals, Ω =⋃i Ωi =

⋃i[xi, xi+1],

the DG scheme constructs an approximation uh of u which is piece-wise continuous, i.e.one each sub-interval Ωi we have

uh(x, t)|Ωi = uih(x, t) =N+1∑j=1

uij(t)Φij(x) (2)

using basis functions Φij , usually given by polynomial functions Φij ∈ PN ([xi, xi+1]).

The DG scheme in weak form on a certain sub-interval Ωi is obtained as usual byfirst multiplying the conservation law (1) with global test functions Φ̃ij constructed fromthe local ones by

Φ̃ij(x) =

{Φij(x) if x ∈ Ωi,

0 otherwise.3

Second, partial integration of the term containing the flux f is carried out and a nu-merical flux function is introduced to ensure coupling between the sub-intervals. Thisprocedure yields the DG scheme in weak form

d

dt

∫Ωi

uhΦikdx+ f

∗i−1,iΦ

ik(xi)− f∗i,i+1Φik(xi+1)−

∫Ωi

f(uh)dΦikdx

dx = 0, (3)

with f∗i−1,i = f∗ (ui−1h (xi), uih(xi)) denoting the values of a consistent numerical flux

function. The numerical flux f∗ hereby requires the left and right hand side limits,i.e. ui−1h (xi), u

ih(xi) as defined in (2), as input arguments. The last integral term on

the left hand side of equation (3) is generally not solved analytically but by numericalquadrature rules. Thus, for numerical integration we consider a set of quadrature nodes

ξν ∈ [−1, 1], ν = 1, . . . , N + 1,

transformed to the specific sub-interval under consideration. The corresponding weightswill be denoted by

ων , ν = 1, . . . , N + 1.

Hereby, we restrict the presentation to nodal sets containing exactly dimPN = N + 1points. In this case, we may as well use these nodes to construct an interpolation

fh ∈ PN of the nonlinear function f(uh). Exact integration of∫

Ωifh

dΦikdx dx is then

equivalent to numerical integration of the last integral in (3) for an arbitrary functionf if and only if the quadrature rule is exact for polynomials of degree 2N − 1.

To construct the DG scheme and highlight its SBP property, the test and basisfunctions for the expansion of uh and within the variational formulation (3) are chosenas the corresponding nodal Lagrange polynomials Φik = L

ik with L

ik(Λi(ξν)) = δνk, where

Λi denotes the transformation of the reference cell [−1, 1] to the specific sub-interval Ωi,i.e.

Λi(ξ) = ξxi+1 − xi

2+xi + xi+1

2.

Hence, considering the Lagrange polynomials Lk : [−1, 1] → R corresponding to thequadrature nodes on the reference interval [−1, 1], the basis functions with respect to aspecific cell are given by Lik = Lk ◦Λ

−1i . For a specific cell Ωi, the weak form of the DG

scheme using quadrature rules and Lagrange basis functions is then given by

d

dt

∫Ωi

uhLikdx+ f

∗i−1,iL

ik(xi)− f∗i,i+1Lik(xi+1)−

∫Ωi

fhdLikdx

dx = 0, (4)

where fh is defined by the expansion

fh(x, t)|Ωi =N+1∑j=1

f(uij)Lij(x), (5)

with point-wise values uij = uh (Λi(ξj), t). Transforming to the reference cell [−1, 1], we

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obtain the corresponding weak form

∆xi2

d

dt

∫ 1−1uh (Λi(ξ), t)Lk(ξ) dξ + f

∗i−1,iLk(−1)− f∗i,i+1Lk(1)

−∫ 1−1fh (Λi(ξ), t)L

′k(ξ) dξ = 0,

(6)

where ∆xi = xi+1 − xi. Equation (6) rewrites in a simpler form using matrix-vectornotation. For this purpose, we define the matrices M and S by their entries

Mjk =

∫ 1−1LjLkdξ = Mkj , (7)

Sjk =

∫ 1−1LjL

′kdξ, (8)

as well as the solution vector ui and vector of flux values f i given by

ui = (ui1, . . . , uiN+1)

T with uij = uh (Λi(ξj), t) ,

f i = (f i1, . . . , fiN+1)

T with f ij = f(uij).

Furthermore, we use the abbreviations f∗,i(1) = f∗i,i+1 and f∗,i(−1) = f∗i−1,i and collect

the basis functions in the vector valued function

L(ξ) = (L1(ξ), . . . , LN+1(ξ))T . (9)

Using the expansions of uh and fh into the Lagrange basis functions as in (2) and (5),the above variational form (6) is then equivalent to the matrix-vector form

∆xi2

Mdui

dt− ST f i = −[f∗,iL]1−1.

The DG scheme in strong from is obtained by a second partial integration of (4) resultingin the variational formulation∫

Ωi

∂uh∂t

Likdx+

∫Ωi

∂fh∂x

Likdx

=[f∗i−1,i − f ih(xi)]Lik(xi)− [f∗i,i+1 − f ih(xi+1)]Lik(xi+1).(10)

The above equation is the equivalent to the matrix-vector formulation

∆xi2

Mdui

dt+ Sf i = [(f ih − f∗,i)L]1−1,

where f ih(ξ, t) =∑N+1j=1 f(u

ij)Lj(ξ) is the transformation of fh(x, t)|Ωi to the reference

cell, i.e. f ih(ξ, t) = fh(Λi(ξ), t). For pairwise distinct nodes, the Lagrange polynomialsrepresent a set of linearly independent functions. Hence, the matrix M , given by the

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entries (7) is symmetric and positive definite and thus invertible. The relation to SBPschemes can now be observed by multiplying with the inverse of M to obtain

∆xi2

dui

dt+Df i = M−1[(f ih − f∗,i)L]1−1. (11)

where the entries of D = M−1S are given by

Djk = L′k(ξj), (12)

due to the interpolation property of the Lagrange polynomials. In fact, defining thematrix D by the entries (12) leads to

(MD)ij =∑k

MikDkj =

∫ 1−1Li(ξ)

∑k

Lk(ξ)L′j(ξk)dξ =

∫ 1−1Li(ξ)L

′j(ξ)dξ = Sij .

In the following section 3, we will consider the DG scheme in cell-wise fashion. Hence,in the subsequent presentation, we will drop the index i in equation (11) referring tothe specific cell under consideration and write

∆x

2

du

dt+Df = M−1[(fh − f∗)L]1−1. (13)

However, when referring to contributions of interface fluxes, corresponding interfaceindices (i− 1, i) and (i, i+ 1) will still be used as in (6) and (10).

According to [6], Definition 2, a scheme of the form (13) is an SBP scheme, or else,the matrix D is an SBP operator, if the subsequent conditions are fulfilled.

1. The matrix D is an approximation to ∂∂ξ with Dξj = jξj−1 for all 0 ≤ j ≤ q,

where q denotes the degree of the approximation to the first derivative and ξj =

(ξj1, . . . , ξjN+1)

T .

2. The matrix M is symmetric and positive definite.

3. Integration by parts is mimicked by MD + DTM = B = BT , where B is an

interface and boundary operator with the property (ξl)TBξm = [ξl+m]1−1 for all0 ≤ l,m ≤ r, where r ≥ q denotes the degree of the SAT terms used for impositionof boundary and interface conditions.

For the DG scheme (13) written for a spatial variable ξ on the reference cell [−1, 1]we now have the following SBP property. We note, that so far only a quadrature ruleof degree 2N − 1 has been assumed.

Lemma 1. The DG scheme (13) is an SBP scheme with the SBP operator D = M−1S

given in (12). Hereby, D approximates ∂∂ξ to degree q = N and the degree of B is

r = q = N as well. Furthermore, given a function g(ξ) with point-wise values g, theinterface and boundary operator B acts on g as

Bg = [ghL]1−1,

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where gh =∑N+1j=1 gjLj(ξ) denotes the polynomial interpolation of the point-wise values

g.

Proof. The first task is to prove that Dξj = jξj−1 holds for all 0 ≤ j ≤ N . On this,we consider a polynomial function ξj , j ≤ N , expanded in Lagrange polynomials asξj =

∑N+1k=1 ξ

jkLk(ξ). Since the derivative of ξ

j is given by the function jξj−1, we have

jξj−1 =∑k ξ

jkL′k(ξ), or else

jξj−1 =

N+1∑k=1

ξjkL′k(ξ), (14)

in vector notation, collecting the point-wise values of ξj−1. Now, an application of thematrix D to a vector results in a suitable linear combination of the columns of D, i.e.

Dξj =

N+1∑k=1

ξjkL′k(ξ) = jξ

j−1,

where the last equality is due to (14). Secondly, the matrix M obviously is symmetricand positive definite by construction, as stated before. Lastly, we consider the boundaryoperator B. The entries of MD +DTM are given by

(MD +DTM)jk = Sjk + Skj =

∫ 1−1

(LjL′k + L

′jLk)dξ = [LjLk]

1−1 = Bjk.

Therefore, we obtain the required assertion

(ξl)TBξm =

N+1∑j=1

N+1∑k=1

ξljBjkξmk =

N+1∑j=1

ξljLj(ξ) ·N+1∑k=1

ξmk Lk(ξ)

1−1

= [ξlξm]1−1,

for all 0 ≤ l,m ≤ N . Finally, denoting the columns of B by Bk, we have the action ofB represented by

Bg =

N+1∑k=1

Bkgk =

[L(ξ)

N+1∑k=1

gkLk(ξ)

]1−1

= [ghL]1−1,

as stated.

At this point, we remark that if the chosen quadrature rule exactly integrates poly-nomials of degree 2N , e.g. for Gauss-Legendre nodes, we have Mjk = δjkωj due to exactintegration of the integrals in (7). Hence, M is diagonal in this case and D is called adiagonal-norm SBP operator. Non-diagonal matrices M lead to so-called non-diagonalnorm SBP operators. However, in the case of Gauss-Lobatto nodes, mass lumping againresults in a diagonal mass matrix with Mjk = δjkωj and a diagonal norm SBP operatoras shown in [9].

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Choosing the classical Gauss-Legendre nodes which do not contain the interval endpoints ξ = −1 and ξ = 1 yields exact integration of polynomials up to degree 2N+1. Dueto the improved accuracy of the resulting DG scheme, these points might be preferredto the Gauss-Lobatto variant with mass lumping. Higher efficiency of Gauss nodesespecially for a non-linear example based on the two-dimensional Euler equations isnumerically demonstrated in [17]. However, as also stated in [17], in addition to thelower cost based on the fact that boundary interpolation is not required, the DG schemeon Gauss-Lobatto nodes also allows larger time steps in case of explicit time integration.Time steps may be taken roughly twice as large in comparison to Gauss nodes as shownin [11]. Further subtleties arise as Gauss integration may increase robustness for non-linear problems and underresolved simulations, see e.g. [11, 2]. Hence, the questionof efficiency will depend on the specific application including accuracy requirements.In addition, the situation of reduced exactness for Gauss-Lobatto integration may bedifferent if the exact full matrix M for Gauss-Lobatto nodes is used, as defined byexact integration in (7). In this case, a technique in [28] lowers the cost of applyingthe inverse of M to an arbitrary vector. Matrix-vector multiplication then results inan O(N) operation, i.e. it becomes as expensive as mass lumping. However, enforcinga balance of certain secondary quantities might necessitate a diagonal mass matrix aswell as including the interval boundaries into the nodal set.

In the case of the kinetic energy, we will show in the following Sect. 3 that theinclusion of boundary nodes is not required. Hence, a kinetic energy preserving DGscheme can be constructed on the more accurate Gauss-Legendre nodes as well.

3 Kinetic energy preservation based on the Eulerequations in skew-symmetric form

In this section, a kinetic energy preserving DG scheme on arbitrary nodal sets withpairwise distinct nodes will be constructed. The corresponding scheme is still conser-vative with respect to mass, momentum and energy. In order to construct this scheme,a skew-symmetric formulation of the Euler equations of gas dynamics is used but thediscretization is then related to the divergence form of the Euler equations given by

∂

∂tρ+

∂

∂x(ρv) = 0, (15)

∂

∂t(ρv) +

∂

∂x(ρv2 + p) = 0, (16)

∂

∂t(ρE) +

∂

∂x((ρE + p)v) = 0, (17)

for the density ρ, the velocity v, the specific total energy E and the pressure p. Thissystem is closed by the equation of state for ideal gases p = (γ − 1)ρ(E − v2/2), withconstant adiabatic coefficient γ.

Furthermore, a specific skew-symmetric form has been given by Morinishi in [19].

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The resulting system of PDEs, which is also discretized in [9], is

∂

∂tρ+

∂

∂x(ρv) = 0, (18)

1

2

[∂

∂t(ρv) + ρ

∂

∂tv

]+

1

2

[∂

∂x(ρv2) + ρv

∂

∂xv

]+

∂

∂xp = 0, (19)

∂

∂t(ρe) +

∂

∂x(ρve+ vp)− v ∂

∂xp = 0, (20)

where e denotes the specific inner energy and the equation of state can be rewritten as(γ−1)ρe = p. The following derivation will clarify that the first two equations, (18) and(19), are responsible for the conservation of mass and momentum as well as the correctbalance of kinetic energy. The only requirement concerning the third equation (20) isconservation of total energy, which is already fulfilled by the standard DG discretizationof the energy equation (17) in divergence form. Therefore, we will mainly consider thefollowing alternative skew-symmetric form of the Euler equations,

∂

∂tρ+

∂

∂x(ρv) = 0, (21)

1

2

[∂

∂t(ρv) + ρ

∂

∂tv

]+

1

2

[∂

∂x(ρv2) + ρv

∂

∂xv

]+

∂

∂xp = 0, (22)

∂

∂t(ρE) +

∂

∂x(ρvE + vp) = 0. (23)

The quantities in equations (18)–(23) will be evaluated at the quadrature nodes, hence

we consider the vectors ρ = (ρ1, . . . , ρN+1)T, v = (v1, . . . , vN+1)

T, e = (e1, . . . , eN+1)

T, p =

(p1, . . . , pN+1)T

and E = (E1, . . . , EN+1)T

of nodal values. We will first directly dis-cretize the above continuous formulations. As a second step, the discretization will bereformulated using the classical conservative variables at the quadrature nodes,

u1 = (u1,1, . . . , u1,N+1)T

with u1,ν = ρν ,

u2 = (u2,1, . . . , u2,N+1)T

with u2,ν = ρνvν ,

u3 = (u3,1, . . . , u3,N+1)T

with u3,ν = ρνeν + ρνv2ν/2,

in order to analyze the properties of the DG scheme applied to these skew-symmetricforms.

Once the point-wise values of the conservative variables are known, the point-wise

values of v, p, e can be calculated as vν =u2,νu1,ν

, eν =(u3,νu1,ν− 12

u22,νu21,ν

)and pν = (γ −

1)(u3,ν − 12

u22,νu1,ν

)for ν = 1, . . . , N + 1. Furthermore, we consider the flux vector of the

compressible Euler equations in conservative form and denote vector of point-wise fluxvalues by

f1

= (f1,1, . . . , f1,N+1)T

with f1,ν = ρνvν = u2,ν ,

f2

= (f2,1, . . . , f2,N+1)T

with f2,ν = ρνv2ν + pν = vνf1,ν + pν ,

f3

= (f3,1, . . . , f3,N+1)T

with f3,ν = ρνvνeν + ρνv3ν/2 + vνpν = vν(u3,ν + pν).

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In order to use matrix-vector notation in the following derivation of the kinetic energypreserving DG scheme, we need the following notation for certain diagonal matrices. Fora given vector w ∈ RN+1 of point-wise quantities, we denote by w the diagonal matrix

w = diag[w1, . . . , wN+1],

obtained by injecting the entries of w into the diagonal.In the following paragraphs, each of the equations within the skew-symmetric for-

mulations will be discretized in an analogous manner to equation (13) which basicallymeans that the partial derivative ∂∂x will be substituted by the matrix D. The precisechoice of numerical flux functions needed within the discretization will not be specifieduntil the kinetic energy balance is considered. For the continuity equation in divergenceform given in (18) or (21) we then obtain the standard DG discretization.

The discrete continuity equation The discrete continuity equation is given by adirect discretization of (18) (or (21), respectively) via the DG scheme, i.e.

∆x

2

d

dtu1 +Df1 = M

−1[(f1,h − f∗1 )L]1−1. (24)

Thus, conservation of mass is automatically satisfied. Conservation of momentumand total energy as well as the kinetic energy balance are subject of the following Sections3.1, 3.2 and 3.3. The DG discretization of equations (18) (or (22), respectively) and (19)will yield certain additional terms. These terms have to be chosen in a way to guaranteeconservativity and consistency as stated more precisely in the following Lemma whichis proven in the Appendix Sect. 7.1. Due to the references to interface fluxes neededboth in the statement of Lemma 2 and in the proof, the cell index i is not neglectedthis time.

Lemma 2. For a scalar conservation law, ∂∂tu +∂∂xf = 0, we consider a cell-wise

discretization of the form

∆xi2

d

dtui +Df i +

[−Dαiβi + αiDβi + βiDαi

]= M−1

([(f ih − f∗,i)L]1−1 + [αi(βih − β∗,i)L]1−1

)+M−1([kihL]

1−1 + k

∗,+i L(−1)− k

∗,−i L(1)),

(25)

with arbitrary nodal values αi, βi, additional inner correction terms kih and numerical

fluxes k∗,±i which have to be specified. In (25), the functions f∗ and β∗ denote locally

Lipschitz continuous numerical flux functions consistent to f and β, respectively. Underthese premises, the following assertions hold.

1. The scheme (25) is conservative, if and only if

kih(−1)− k∗,+i = −(αβ)

ih(−1) + αih(−1)β∗i−1,i + C∗i−1,i (26)

and

kih(1)− k∗,−i = −(αβ)

ih(1) + α

ih(1)β

∗i,i+1 + C

∗i,i+1, (27)

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for interface-dependent functions C∗i−1,i, C∗i,i+1 which may also depend on the val-

ues of α, β, u in the corresponding left and right cells.

2. If we assume that

• kih only depends on a combination of the interior values αi, βi,

• k∗,+i only depends on β∗i−1,i and αih(−1),

• k∗,−i only depends on β∗i,i+1 and αih(1),

then C∗i−1,i = C∗i−1,i(β

∗i−1,i) and C

∗i,i+1 = C

∗i,i+1(β

∗i,i+1) in (26), (27), i.e. C

∗ onlydepends on the values of the given flux function β∗.

3. The scheme (25) with correction terms set to

kih = −(αβ)ih,k∗,+i = −α

ih(−1)β∗i−1,i + C∗i−1,i,

k∗,−i = −αih(1)β

∗i,i+1 + C

∗i,i+1,

reduces to a consistent finite volume scheme for N = 0 if and only if C∗i−1,i =C∗i,i+1 = 0.

Remark. The assumptions in assertion 2 of Lemma 2 basically have the purposeof keeping the computational cost of (25) at a minimum by reducing the level of datadependence. For instance, no additional numerical flux function needs to be evaluated.

3.1 The discrete momentum equation

In the following derivations the skew-symmetric formulation of the momentum equa-tion will be discretized by the DG scheme using Gauss-Legendre nodes. The availablefreedom in choosing the numerical flux function will be used to obtain a schemes whichconserves momentum and additionally fulfills a kinetic energy balance with respect tothe cell means of kinetic energy. This part follows closely the Gauss-Lobatto case in [9].

A direct discretization of the momentum equation, neglecting the cell index i, yields

∆x

2

1

2

[d

dtu2 + u1

d

dtv

]+

1

2

[Dρv2 + ρ vDv

]+Dp = M−1[(g2,h − g∗2)L]1−1, (28)

with terms g2,h and g∗2 which are not yet specified. We will choose these terms in a

manner to guarantee conservation of momentum. Therefore, we multiply (24) from leftby 12v and add the resulting equation to (28). Due to continuity in time, this yields

∆x

2

d

dtu2 +D(ρv

2 + p) +1

2

[−Dρv2 + ρ vDv + vDf

1

]= M−1

([(g2,h − g∗2)L]1−1 +

1

2v[(f1,h − f∗1 )L]1−1

).

(29)

11

Setting α = 12v, β = ρv = f1 and β∗ = f∗1 , due to Lemma 2, we obtain a consistent

and conservative scheme if we set

g2,h = f2,h + k2,h = f2,h − (αβ)h =(

1

2ρv2 + p

)h

(30)

and

g∗2(−1) = f∗2 −1

2v+f∗1 , g

∗2(1) = f

∗2 −

1

2v−f∗1 ,

where v+ = vh(−1), v− = vh(1) denote the one-sided limits of vh at cell interfaces. Theformulation (29) can then finally be rewritten as

∆x

2

d

dtu2 +Df2 + s2 = M

−1[(f2,h − f∗2 )L]1−1 +M−1sbc2 , (31)

with the volume terms

s2 =1

2

[−Dρv2 + ρ vDv + vDρv

]and the boundary correction

sbc2 =1

2

(v[(f1,h − f∗1 )L]1−1 − [(ρv2)h − v±f∗1 )L]1−1

).

We may as well derive a weak formulation of this skew-symmetric discretization.Using the SBP property MD = B −DTM with Bf = [fhL]1−1 as well as the definitionof g2,h and f2,h = (ρv

2 + p)h, equation (29) results in

∆x

2

d

dtMu2 = D

TM

(1

2ρv2 + p

)− 1

2

[ρ vMDv − vDTMf

1

]−B

(1

2ρv2 + p

)− 1

2vBf

1+

([(g2,h − g∗2)L]1−1 +

1

2v[(f1,h − f∗1 )L]1−1

),

which is equivalent to the weak formulation

∆x

2

d

dtMu2 = D

TM

(1

2ρv2 + p

)− 1

2

[ρ vMDv − vDTMf

1

]− [g∗2L]1−1 −

1

2v[f∗1L]

1−1

= ST g2− 1

2

[ρ vSv − vST f

1

]− [f∗2L]1−1 +

1

2

[(v±I − v

)f∗1L

]1−1 .

(32)

In fact, for efficiency reasons, this weak formulation should be implemented instead ofthe strong form as it does not require boundary interpolation of the flux values f1 andg2.

3.2 The discrete kinetic energy balance

Before considering the energy equation, i.e. the third equation of the system of Eulerequations, we will derive the kinetic energy balance of the scheme. This will lead toadditional constraints on the numerical flux function f∗.

12

First, we multiply the discrete momentum equation (28) with v and obtain

∆x

2

1

2

[vd

dtu2 + u2

d

dtv

]+

1

2

[vD ρv2 + ρ v2Dv

]+ vDp = M−1v[(g2,h − g∗2)L]1−1,

where we have used the fact that M is diagonal.Time derivatives can be recast to

1

2

[vd

dtu2 + u2

d

dtv

]=

d

dt

(1

2ρv2)

hence, assuming time continuity, the kinetic energy balance is given by

∆x

2

d

dtekin =

∆x

2

d

dt

(1

2ρv2)

=− 12

[vD ρv2 + ρ v2Dv

]− vDp+M−1v[(g2,h − g∗2)L]1−1.

(33)

The balance of the cell means of kinetic energy is then obtained by multiplying theabove equation by 1TM . That is, the total amount of kinetic energy within the cell issubject to the following rate of change

∆x

2

d

dt1TMekin = −

1

2

[vTMDρv2 + (ρv2)TMDv

]− vTMDp+ vT [(g2,h − g∗2)L]1−1

= −12

[vTB ρv2 − (Dv)TM ρv2 + (ρv2)TMDv

]−vTBp+ (Dv)TMp+ vT [(g2,h − g∗2)L]1−1

= −vTB(

1

2ρv2 + p

)+ (Dv)TMp+ vT [(g2,h − g∗2)L]1−1,

where the SBP property MD = B −DTM as well as the equality

−(Dv)TM ρv2 + (ρv2)TMDv = 0

have been used. Furthermore, since vTBg2

= vT [g2,hL]1−1 by Lemma 1, we obtain

d

dt1TMekin =− vTBg2 + (Dv)

TMp+ vT [(g2,h − g∗2)L]1−1

=(Dv)TMp− vT [g∗2L]1−1.(34)

The volume term (Dv)TMp represents a change of kinetic energy due to a pres-sure variation which is physically correct. Hence, only the contribution containingthe auxiliary flux g∗2 remains to be dealt with. For a correct balance of kinetic en-ergy, only the terms contained in g∗2 which are related to transport have to vanish.In this regard, we split g∗,±2 = f

∗2 − 12v

±f∗1 into a transport and a pressure term,

g∗,±2 = f̃∗2 + p

∗− 12v±f∗1 = g̃

∗,±2 + p

∗ as in [9]. For the contribution of interface fluxes atthe interface (i− 1, i) to the cell means of kinetic energy as in (34) we then demand

(vi)T g̃∗,−2 L(−1) = (vi−1)T g̃∗,+2 L(1).

13

This yields the condition

0 =

(f̃∗2 −

1

2v+f∗1

)(vi)TL(−1)−

(f̃∗2 −

1

2v−f∗1

)(vi−1)TL(1)

=

(f̃∗2 −

1

2v+f∗1

)v+ −

(f̃∗2 −

1

2v−f∗1

)v−

= f̃∗2 (v+ − v−) + 1

2((v−)2 − (v+)2)f∗1 ,

thus, the numerical flux f̃∗2 corresponding to the transport part ρv2 of f2 is required to

fulfill

f̃∗2 =v+ + v−

2f∗1 . (35)

With this property of the numerical flux function, we have

g∗,±2 = f̃∗2 + p

∗ − 12vf∗1 =

v+ + v−

2f∗1 + p

∗ − 12v±f∗1 =

1

2v∓f∗1 + p

∗ (36)

and the condition

f∗2 = f̃∗2 + p

∗ =v+ + v−

2f∗1 + p

∗. (37)

There are many possibilities to construct numerical flux functions satisfying (37). Inthis work we will consider the following numerical flux functions.

• The so-called KEP flux f∗A = f∗KEP by Jameson, e.g. given in [15],

f∗A,1 = ρ̄v̄,

f∗A,2 = ρ̄v̄2 + p̄,

f∗A,3 = ρ̄v̄H̄, H = E +p

ρ,

(38)

where, given left and right values q±, the quantity q̄ denotes the arithmetic averageq̄ = 12 (q

+ + q−).

• The kinetic energy preserving and entropy consistent numerical flux given in [5],which will be denoted KEP-EC in this work,

f∗B,1 = ρ̂v̄,

f∗B,2 =ρ̄

2β̄+ v̄f∗B,1,

f∗B,3 =1

2(γ − 1)β̂− v

2

2f∗B,1 + v̄f

∗B,2,

(39)

14

where β = ρ2p and, given left and right values q±, the quantity q̂ denotes the

logarithmic average

q̂ =q− − q+

ln(q−)− ln(q+). (40)

For ql ≈ qr the numerically stable approximation to (40) given in [14] will be used.

• A modified version of the classical van Leer flux f∗V L given in [29] specificallydesigned to fulfill the KEP property (37). This flux f∗C is denoted by KEP-VLand given by

f∗C,1 = f∗V L,1,

f∗C,2 = v̄f∗V L,1 + p̄,

f∗C,3 = f∗V L,3.

(41)

3.3 The discrete energy equation

In the previous sections we have constructed a skew-symmetric DG discretization of thecontinuity and momentum equations which is i) mass and momentum conserving andii) fulfills a balance of kinetic energy. Moreover, a direct discretization of the divergenceform (17) accomplishes the task of total energy conservation. The resulting conservativescheme then exhibits all the desired properties and will be numerically tested in Sect. 4.

In addition, as in [9], a conservative discretization may be obtained for the alternativeskew symmetric form (20). Again, the derivations for the DG scheme on Gauss-Legendrenodes leads to modifications of the skew-symmetric terms with respect to the interfacecontributions. First, the discretization of equation (20) takes on a more general formgiven by

∆x

2

d

dt(ρe) +D(ρve+ vp)− vDp = M−1[(G3,h −G∗3)L]1−1, (42)

where G3,h and G∗3 now denote matrix valued quantities which have to be specified.

Using the discrete kinetic energy balance (33) in the form

−vDp = ∆x2

d

dt

(1

2ρv2)

+1

2

[vD ρv2 + ρ v2Dv

]−M−1v[(g2,h − g∗2)L]1−1,

the corresponding term −vDp in (42) may be substituted.Hence, for the total energy u3 =

(ρe+ 12ρv

2)

we have

∆x

2

d

dtu3 +D(ρve+ vp) +

1

2

[vD ρv2 + ρ v2Dv

]= M−1[(G3,h −G∗3)L+ v(g2,h − g∗2)L]1−1,

15

which can be rearranged to

∆x

2

d

dtu3 +Df3 +

1

2

[−Dρv3 + vD ρv2 + ρ v2Dv

]= M−1[(G3,h −G∗3)L+ v(g2,h − g∗2)L]1−1

= M−1[(G3,h −G∗3)L+ v(ph − (p∗ +

1

2(f̃∗2 − v±f∗1 )))L+

1

2v((ρv2)h − f̃∗2 )L

]1−1,

(43)

using the representations of g2,h and g∗2 as in (30) and (36), respectively.

Also in this case, Lemma 2 can be applied again to (43) using α = v, β = 12ρv2 and

β∗ = 12 f̃∗2 . A conservative scheme is then obtained by setting

G3,h + vph = (f3,h − (αβ)h)I = (ρve+ vp)hI,

where I denotes the identity matrix, and

G∗,±3 + v

(p∗ +

1

2(f̃∗2 − v±f∗1 )

)= (f∗3 − α±β∗)I =

(f∗3 −

1

2v±f̃∗2

)I,

This yields

G3,h + vph +1

2v(ρv2)h

= f3,hI −1

2

(ρv3)hI +

1

2v(ρv2)h

and

G∗,±3 + v

(p∗ +

1

2(f̃∗2 − v±f∗1 ) +

1

2f̃∗2

)=

(f∗3 −

1

2v±f̃∗2

)I +

1

2vf̃∗2 .

Hence, the formulation (43) results in

∆x

2

d

dtu3 +Df3 + s3 = M

−1[(f3,h − f∗3 )L]1−1 +M−1sbc3 , (44)

with the volume terms

s3 =1

2

[−Dρv3 + ρ v2Dv + vDρv2

]and the boundary correction

sbc3 =1

2

(v[((ρv2)h − f̃∗2 )L]1−1 − [(ρv3)h − v±f̃∗2 )L]1−1

),

with f̃∗2 as in (35).

3.4 Summary of skew-symmetric DG discretizations

For convenience, we summarize the alternative discretizations of the Euler equations atthis point. The resulting discrete continuity, momentum and energy equation have thegeneral form

∆x

2

d

dtui +Df i + si = M

−1[(fi,h − f∗i )L]1−1 +M−1sbci , i = 1, 2, 3.

16

With respect to the investigations in this work, four alternative choices for the volumeterms si and boundary corrections s

bci may be considered, i.e. the standard DG scheme

which is conservative regarding mass, momentum and energy, two skew-symmetric formswhich additionally preserve the kinetic energy balance as well as a naive application ofthe kinetic energy preserving DG scheme in [9] to Gauss-Legendre nodes. That is, wehave the following specifications.

1. The standard DG discretization is given by

si = sbci = 0, i = 1, 2, 3. (45)

2. The skew-symmetric DG discretization based on Morinishi’s skew-symmetric form,

s1 = 0,

s2 =1

2


],

s3 =1

2


],

sbc1 = 0,

sbc2 =1

2

(v[(f1,h − f∗1 )L]1−1 − [(ρv2)h − v±f∗1 )L]1−1

),

sbc3 =1

2

(v[((ρv2)h − f̃∗2 )L]1−1 − [(ρv3)h − v±f̃∗2 )L]1−1

)with f̃∗2 as in (35).

3. The skew-symmetric DG discretization based on the first two equations of Morin-ishi’s skew-symmetric form, (18), (19), as well as the energy equation in divergenceform (17). This alternative formulation is given by

s1 = s3 = 0,

s2 =1

2


],

sbc1 = sbc3 = 0,

sbc2 =1

2

(v[(f1,h − f∗1 )L]1−1 − [(ρv2)h − v±f∗1 )L]1−1

).

(46)

4. An additional formulation is obtained, when the DG discretization on Gauss-Lobatto nodes in [9] is naively transferred to the Gauss-Legendre case. Then wehave

s1 = 0,

s2 =1

2


],

s3 =1

2


],

sbci = 0, i = 1, 2, 3.

(47)

17

However, in this case, the boundary treatment is inconsistent to the skew-symmetricterms if Gauss nodes are used. This will result in stability problems as shown inSect. 4.

These alternative discretizations are completed by the choice of a numerical fluxfunction f∗. If the kinetic energy balance is to be preserved, one of the numerical fluxesgiven in Sect. 3.2 will be chosen.

4 Numerical experiments

In this section, numerical experiments will be performed in order to compare the DGdiscretizations with or without skew-symmetric terms, i.e. (45), (46) and (47), using thekinetic energy preserving numerical fluxes in Sect. 3.2 in addition to the classical vanLeer flux [29]. For time discretization, the classical fourth order Runge-Kutta schemeis used. Furthermore, in all test cases, we set the adiabatic coefficient to γ = 1.4.

4.1 Experimental order of convergence

The first test case numerically investigates the order of convergence of the kinetic energypreserving scheme based on classical Gauss-Legendre nodes. As in [9], a manufacturedsolution is used to test the order of the scheme. For this purpose, the Euler equations(15),(16), (17) are augmented by a source term Q(x, t). More precisely, we consider theexact solution

Ums(x, t) =

u(x, t)u(x, t)u2(x, t)

, u(x, t) = 2 + 0.1 sin(2π(x− t)),of the balance law

∂

∂tU +

∂

∂xF (U) = Q(x, t), U =

ρρvρE

, F (U) = ρvρv2 + p

v(ρE + p)

,where

Q(x, t) =

0p(x, t)p(x, t)

, p(x, t) = 0.28π cos(2π(x− t)) + 0.008π sin(4π(x− t)).Thus, the source term is specifically designed to enforce Ums(x, t) as the exact solu-

tion of the continuous system of equations. The initial conditions on the computationaldomain Ω = [0, 1] are given by U(x, 0) = Ums(x, 0) and periodic boundary conditionsare chosen. Tables 1, 2 and 3 list the L2 errors and corresponding experimental order ofconvergence obtained by the DG scheme with skew symmetric terms (46) for polynomialdegrees N = 2, 3 and N = 4, respectively, using the different numerical flux functionsgiven in Sect. 3.2 as well as the classical van Leer flux for reference. All computations

18

Kvan Leer KEP (38) KEP-EC (39) KEP-VL (41)

L2 error EOC L2 error EOC L2 error EOC L2 error EOC10 7.23e-05 - 8.66e-05 - 8.68e-05 - 1.57e-03 -20 5.95e-06 3.60 4.34e-06 4.32 4.33e-06 4.32 1.15e-04 3.7740 6.64e-07 3.16 4.26e-07 3.35 4.26e-07 3.35 8.33e-06 3.7880 8.18e-08 3.02 5.26e-08 3.02 5.26e-08 3.02 8.35e-07 3.32

Table 1: L2 errors and experimental order of convergence (EOC) of the skew-symmetricGauss-Legendre DG scheme, case N = 2, using different numerical fluxes.




have been carried out until tend = 10 with time steps small enough in order to make tem-poral errors negligible. Similar to the results in [9], the scheme using central numericalfluxes which disregard upwind information show an order reduction for odd polynomialdegrees, i.e. for N = 3 the order is reduced to EOC = 3 instead of EOC = N + 1in the even cases N = 2, 4. Hence, this observation made in [9] can be attested alsoin the case of Gauss-Legendre nodes. In a direct comparison of the numerical fluxesused in the DG scheme for a constant polynomial degree, the KEP-VL flux (41) yieldsthe largest errors while the original van Leer flux performs best in this setting. Hence,preserving the kinetic energy is no guarantee for better accuracy. Rather, kinetic energypreservation is a property which mimics a qualitative behaviour of the exact solution.In fact, preservation of kinetic energy itself by different variants of the DG scheme hasto be studied more carefully. This is the purpose of the next test case in Sect. 4.2.




19

4.2 Conservation of mean kinetic energy

In the following, we consider a special set-up to study the conservation of kinetic energy.Measuring the KEP property of a scheme is not straightforward as kinetic energy isgenerally not conserved in the exact solution but in balance with the term vDp, seeequation (33). In this second test we therefore neglect the pressure term in the Eulerequations, i.e. we consider the case of constant pressure. As a result, the energy equationis automatically fulfilled and can be dropped from the system of equations. Of course,to remain with a consistent formulation, the pressure is also neglected in the numericalflux function. Under these assumptions as specified above, the Euler equations reduceto the system

∂

∂tρ+

∂

∂x(ρv) = 0,

∂

∂t(ρv) +

∂

∂x(ρv2) = 0.

The components of the corresponding numerical flux function are given by

f∗1 = (ρv)∗ = ρ̄v̄, f∗2 = (ρv

2)∗ = ρ̄v̄2 = v̄f∗1 ,

which is precisely the reduction of the KEP flux (38) to the reduced system with fluxesf1 = ρv, f2 = ρv

2. Initial conditions are given by the initial density and velocitydistributions

ρ(x, 0) = 2, v(x, 0) = cos(2πx),

on the computational domain Ω = [0, 1] with periodic boundary conditions.First, Figure 1 depicts the time evolution of the mean kinetic energy ēkin(t), given

by

ēkin(t) =∑i

∆xi2

1TMekin(t) =∑i

∫ xi+1xi

(1

2ρv2)h

(t) dx,

for different variants of the skew symmetric terms in Sect. 3.4 in the case N = 1.Here, we compare the DG scheme with the correctly derived skew-symmetric termsand boundary treatment (46), denoted by ’DG skew1’, the standard DG scheme (45)without skew-symmetric terms as well as the skew symmetric terms with inconsistentboundary treatment as in (47), denoted by ’DG skew2’. We may observe that onlythe Gauss-Legendre ’DG skew1’ scheme preserves the mean kinetic energy while thestandard DG scheme dissipates this quantity. Skew symmetric terms with inconsistentboundary treatment for ’DG skew2’ lead to an non-physical increase of kinetic energy.This increase of kinetic energy leads to oscillations of the DG solution, visible in Figure2 showing the distribution of kinetic energy for the different variants of the DG schemeat output time tend = 0.12. In the density distribution depicted Figure 3 this effectis not present but it can be observed in terms of less pronounced oscillations for thevelocity distribution shown in Fig. 4.

4.3 Non-linear acoustic pressure wave

In [9], Gassner proposes a test case which is sensitive to dissipation and dispersionerrors to study the performance of the kinetic energy preserving DG scheme based on

20

0 0.02 0.04 0.06 0.08 0.1 0.120.4996

0.4998

0.5

0.5002

0.5004

0.5006

0.5008

0.501

0.5012

t

Kin

etic E

nerg

y

DG skew1

DG standard

DG skew2

Figure 1: DG scheme forN = 1 and 100cells using different variants of skew-symmetric terms. Time evolution of ki-netic energy.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

x

Kin

etic E

nerg

y

DG skew1

DG standard

DG skew2

Figure 2: Distribution of kinetic energyat time tend = 0.12.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

2

3

4

5

6

7

8

x

Density

DG skew1

DG standard

DG skew2

Figure 3: Distribution of density attime tend = 0.12.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

Velo

city

DG skew1

DG standard

DG skew2

Figure 4: Distribution of velocity attime tend = 0.12.

21

Gauss-Lobatto nodes on course grids and for a low polynomial degree of N = 1. Incomparison to the standard DG scheme disretizing the conservative Euler equations,the approximation by the skew-symmetric scheme using the KEP flux appeared to becloser to the reference solution with respect to the pressure wave and the kinetic energydistribution.

The same test case is used in this work to investigate the performance of the DG-KEP scheme based on the classical Gauss points. Therefore, the Euler equations areaugmented by viscous terms of the compressible Navier-Stokes equations. The equationsto be solved are given by

∂

∂tU +

∂

∂xF (U) =

∂

∂xF visc(U,Ux),

where F again denotes the inviscid convective fluxes contained in the Euler equationsand F visc(U,Ux) contains the viscous fluxes given by

F visc(U,Ux) =

0µ 43vxµ 43vvx + kTx

.Herein, the viscosity coefficient µ = µ(T ) possibly depends on the temperature T =pρR =

pγρ(γ−1)cp =

γcpe, where R denotes the gas constant of the ideal gas law and cp is

the specific heat at constant pressure. The head conduction coefficient is furthermoregiven by k =

cpµPr , with the Prandtl number Pr. As in [9], the dependence of the viscosity

µ on temperature is neglected to simplify the equation and its discretization.We then note that the viscous terms can be re-written as F visc(U,Ux) = A(U)Ux

using the diffusion matrix

A(U) =µ

ρ

0 0 0− 43v 43 0−(

43v

2 + γPr (e− v2)) (

43 −

γPr

)v γPr

.Now, to specify the set-up of the numerical experiment as in [9], the initial conditions

for the acoustic pressure wave are given by the initial density, velocity and pressuredistribution

ρ(x, 0) = 1, v(x, 0) = 1, p(x, 0) = 1 + 0.1 sin(2πx)

on the computational domain Ω = [0, 1] with periodic boundary conditions. The vis-cosity coefficient is set to µ = 0.002 and the Prandtl number is Pr = 0.72. The viscousterms are discretized by the BR2 approach developed by Bassi and Rebay, see [3]. Inorder to study long time integration, the numerical computations are then carried outuntil the final time tend = 20 is reached. For this test case computed by the Gauss-Legendre DG scheme, the results showed no difference in accuracy for the DG schemewith or without skew-symmetric terms - even in case of the inconsistent boundary treat-ment. However, we may study the effect of a higher order quadrature rule when usingGauss-Legendre nodes instead of the Gauss-Lobatto variant considered in [9]. Thus,Figure 5 reports the output of the DG scheme for N = 1 in case of Gauss-Legendre as

22

well as Gauss-Lobatto nodes using the KEP-VL flux (41) and correct combinations ofskew-symmetric and boundary terms, i.e. terms (46) for the Gauss-Legendre case andterms (47) for the Gauss-Lobatto case.

In order to account for the differences in arithmetic operations in case of Gauss-Legendre or Gauss-Lobatto nodes, respectively, the DG scheme on Gauss nodes uses40 cells while for the Gauss-Lobatto variant 80 cells are taken. Thus, the stabilityconstraint for explicit time integration is roughly the same, see [11]. The additionalcost introduced by the finer grid for the Gauss-Lobatto nodes as well as the additionalboundary interpolation and boundary correction terms for Gauss-Legendre nodes canthen be quantified and compared. More precisely, the skew-symmetric DG scheme (32)on Gauss nodes needs to interpolate the conservative variables as well as velocity tothe two boundaries of grid cells, which results in 16(N + 1) arithmetic operations percoarse grid cell, and to evaluate the additional surface correction, i.e. the last termon the right hand side of (32), second line, for which we have 9(N + 1) arithmeticoperations per coarse grid cell. On the other hand, the Gauss-Lobatto variant needstwice as many grid cells, hence twice as many flux evaluations and multiplications by ST ,

resulting in 7(N+1)+6(N+1)2 arithmetic operations per coarse grid cell. Furthermore,double the amount of skew-symmetric term evaluations is necessary, i.e. additional4((N + 1)2 +N + 1) arithmetic operations in the weak formulation and twice as manyevaluations of numerical fluxes, resulting in 18 additional arithmetic operations percoarse grid interface for the KEP flux. For N = 1 and periodic boundary conditions inone space dimension, i.e. an equal number of grid cells and interfaces, this results in 50additional arithmetic operations per coarse grid cell for the DG-KEP scheme on Gaussnodes in comparison to the Gauss-Lobatto variant on the coarser grid and 80 additionalarithmetic operations for the Gauss-Lobatto variant on the finer grid in comparison tothe coarser. Thus, the Gauss-Lobatto set-up is designed to be more expensive for thisproblem. However, in a comparison with a reference solution obtained by the standardDG scheme for a polynomial degree N = 3 and 500 cells, this Gauss-Lobatto variantclearly is not as accurate as the Gauss-Legendre variant on the coarser grid as shown inFig. 5 where the DG solution with Gauss-Legendre nodes almost cannot be distinguishedfrom the reference solution. For this test case, it has hence payed of to consider Gauss-Legendre nodes, though kinetic energy preservation seems to be less critical for thistest case. Again one should remark that preserving a qualitative behaviour does notguarantee better accuracy in general.

4.4 Viscous Sod shock tube

In order to investigate the behaviour of KEP schemes near their limits of applicability,i.e. exact solutions with shocks, Allaneau and Jameson studied their performance forviscous shock test cases e.g. in [1]. For finite volume approximations on coarse meshes,a comparison of numerical fluxes gave better, oscillation-free, results in case of diffusivenumerical fluxes. However, the KEP flux (38) lead to stable computations whereasthe regular central scheme blew up. In [1], high order DG schemes on coarse mesheswhere considered as well, where again the KEP flux performed better than a regularcentral scheme in damping oscillations due to odd/even decoupling. With these formerinvestigations in mind, it should be interesting to study DG schemes incorporating

23

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.99

0.995

1

1.005

1.01

x

Pre

ssure

DG Gauss−Legendre

DG Gauss−Lobatto

Reference Solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.485

0.49

0.495

0.5

0.505

0.51

0.515

0.52

xK

inetic E

nerg

y

DG Gauss−Legendre

DG Gauss−Lobatto

Reference Solution

Figure 5: DG scheme using skew-symmetric terms for N = 1, Gauss-Legendre (40cells) vs. Gauss-Lobatto nodes (80 cells) using kinetic energy preserving flux f∗C . Left:pressure. Right: kinetic energy.

different skew-symmetric terms in addition to a kinetic energy preserving numericalflux. Hence, a corresponding test case of a viscous Sod shock tube is also considered inthis work. The system of equations to be solved is again the Navier-Stokes equations asin the previous test case using a constant viscosity coefficient µ = 0.0001. The initialconditions on the computational domain Ω = [0, 1] are given by

U(x, 0) =

{Uleft for, x < 0.5,Uright, otherwise,

with left and right states Uleft and Uright given by ρleftvleftpleft

= 10

1

, ρrightvright

pright

= 0.1250

0.1

.Both of the computational boundaries are supplemented by inflow boundary conditionsand the computations are carried out until the final time tend = 0.15.

Figure 6 shows the DG solution for N = 1 on a coarse grid of 100 cells using thedifferent skew-symmetric terms (46) and (47) as well as the kinetic energy preservingKEP-VL flux (41). We clearly see oscillations at the shock position produced by theDG scheme using skew symmetric terms with inconsistent boundary treatment. No suchinstability phenomenon is present in the case of the usual van Leer flux, see Figure 7,whereas Figure 8 again shows larger oscillations for the inconsistent boundary treatmentwhen choosing the KEP flux (38). For the DG scheme with N = 3 and 100 cells, theresolution is high enough and the precise choice of numerical flux function has less effectas shown in Figure 9.

24

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

x

Pre

ssure

DG skew1

DG skew2

0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

Pre

ssure

DG skew1

DG skew2

Figure 6: DG scheme for N = 1 and 100 cells using different skew-symmetric terms andkinetic energy preserving flux f∗C . Right: close-up at instability region.

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

Pre

ssure

DG skew1

DG skew2

Figure 7: DG scheme for N = 1 and100 cells using different skew-symmetricterms and van Leer flux f∗V L.

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

Pre

ssure

DG skew1

DG skew2

Figure 8: DG scheme for N = 1 and100 cells using different skew-symmetricterms and kinetic energy preservingKEP flux f∗A.

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

x

Pre

ssure

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

x

Pre

ssure

Figure 9: DG scheme for N = 3 and 100 cells using correct skew-symmetric terms andKEP fluxes f∗A (left) vs. f

∗C (right).

5 Extension to two-dimensional cartesian grids

The skew-symmetric, kinetic energy preserving DG scheme on Gauss-Legendre nodeseasily extends to two-dimensional cartesian grids using tensor-product basis functionsLi(ξ)Lj(η) on the reference elementK = [−1, 1]2. In this case, the DG scheme in matrix-vector formulation can be constructed based on the 1D formulation by using Kroneckerproducts. Applied to a skew-symmetric formulation of the Euler equations in two spacedimensions, this tensor-product DG formulation allows for a kinetic energy preservingDG scheme in 2D with similar additional terms as in the 1D case. In particular, thenecessary boundary correction terms for the interior node distributions are a directextension of the one-dimensional case. In this section, this 2D extension is derived andthe advantage of using a kinetic energy preserving scheme is demonstrated in the contextof two-dimensional turbulent flow.

5.1 The DG scheme with tensor-product SBP operators

For given pointwise data values gi,j ≈ g(ξi, ηj) at the two-dimensional Gauss nodesξi, ηj , i, j = 1, . . . , N + 1, the vector g collects these values in the form

gν(i,j) = g(i−1)(N+1)+j = g(i,j),

i.e. in lexicographical order. Denoting by M1D

the mass matrix of the 1D DG schemeand by I

1Dthe corresponding identity matrix whilst keeping the definition of D and S

as in Sect. 2 the DG scheme in two space dimensions then uses the mass, stiffness anddifferentiation matrices

M = M1D⊗M

1D,

Sξ = S ⊗M1D, Sη = M1D ⊗ Sη,

Dξ = D ⊗ I1D = M−1Sξ, Dη = I1D ⊗D = M

−1Sη,

26

as well as the boundary operators

Bξ = B ⊗M1D, Bη = M1D ⊗B.

Furthermore, the properties of the Kronecker product directly yield the SBP prop-erties Sξ + Sξ

T = Bξ and Sη + SηT = Bη.

The action of the boundary operators on pointwise values is related to the discreteboundary integral as follows. Let ωk denote the Gaussian weights as given in Section2. Furthermore, we will introduce the index e enumerating the edges of the referencesquare K in countner-clockwise manner starting with e = 1 referring to the lower edge.The corresponding normal vectors are denoted by ne = (neξ, n

eη), i.e. n

1ξ = 0, n

1η = −1

and the nodes (ξek, ηek) denote the Gaussian quadrature nodes on edge e. The following

Lemma then transfers the actions of Bξ and Bη to a numerical quadrature on ∂K. The

proof of this Lemma is given in the Appendix Sect. 7.2.

Lemma 3. Let gξ, gη ∈ R(N+1)2 denote arbitrary sets of pointwise data values. For thesum of boundary terms Bξg

ξ +Bηgη we then have

Bξgξ +Bηg

η =

4∑e=1

N+1∑k=1

ωk

(neξg

ξh(ξ

ek, η

ek) + n

eηgηh(ξ

ek, η

ek))L(ξek, η

ek),

where L(ξek, ηek) = L(ξ

ek)⊗ L(ηek) and g

ξh and g

ηh denote the polynomial interpolations

gξh(ξ, η) =

N+1∑i=1

N+1∑j=1

gξν(i,j)Li(ξ)Lj(η), gηh(ξ, η) =

N+1∑i=1

N+1∑j=1

gην(i,j)Li(ξ)Lj(η).

Remark. To reduce formalism in the later definition of the DG scheme the shortnotation 〈τ〉∂K =

∑4e=1

∑N+1k=1 ωkτ(ξ

ek, η

ek) will be used to denote surface terms of the

above form, i.e. we have

Bξgξ +Bηg

η =〈(nξg

ξh + nηg

ηh

)L(ξ, η)

〉∂K

.

With these notations, the extension of the standard DG scheme in weak form to anequation ∂∂tu +

∂∂xf

x + ∂∂yfy = 0 in two space dimensions on a cartesian grid is given

by the cell-wise formulation

∆x∆y

4M

du

dt− SξT fξ − SηT fη = −〈f∗L(ξ, η))〉∂K ,

where fξ and fη approximate the grid values of the fluxes fx and fy, respectively,on the specific grid cell with length scales ∆x,∆y. The corresponding strong form isobtained by multiplication with M−1and application of Lemma 3, i.e.

∆x∆y

4

du

dt+Dξf

ξ +Dηfη = M−1

〈(nξf

ξh + nηf

ηh − f

∗)L(ξ, η)

〉∂K

. (48)

27

5.2 The KEP construction in two space dimensions

For the construction of a kinetic energy preserving DG scheme on two-dimensionalcartesian grids, we consider the skew-symmetric form of the Euler equations in twospace dimensions given by

∂

∂tρ = − ∂

∂x(ρv1)−

∂

∂y(ρv2),

1

2

[∂

∂t(ρv1) + ρ

∂v1∂t

]= −1

2

[∂

∂x(ρv21) + ρv1

∂v1∂x

]− 1

2

[∂

∂y(ρv1v2) + ρv2

∂v1∂y

]− ∂p∂x,

1

2

[∂

∂t(ρv2) + ρ

∂v2∂t

]= −1

2

[∂

∂x(ρv1v2) + ρv1

∂v2∂x

]− 1

2

[∂

∂y(ρv22) + ρv2

∂v2∂y

]− ∂p∂y,

∂

∂t(ρE) = − ∂

∂x(ρv1E + v1p)−

∂

∂y(ρv2E + v2p).

The scheme (48) applied to the continuity equation is given by

∆x∆y

4

d

dtu1 +Dξf

ξ

1+Dηf

η

1= M−1

〈(nξf

ξ1,h + nηf

η1,h − f

∗1

)L(ξ, η)

〉∂K

In the same manner, we may discretize the energy equation as it is given in divergenceform. For the skew-symmetric forms of the momentum equations, discretization of ∂∂xby Dξ and

∂∂y by Dη as well as interface terms corresponding to the right-hand side of

(48) yield

∆x∆y

4

1

2

[du2dt

+ u1

dv1dt

]+

1

2

[Dξ ρv

21 + u2Dξv1 +Dη u3v1 + u3Dηv1

]+Dξp

= M−1〈

(nξgξ2,h + nηg

y2,h − g

∗2)L〉∂K

,

(49)

and

∆x∆y

4

1

2

[du3dt

+ u1

dv2dt

]+

1

2

[Dξ u2v2 + u2Dξv2 +Dη ρv

22 + u3Dηv2

]+Dηp

= M−1〈

(nξgξ3,h + nηg

η3,h − g

∗3)L〉∂K

,

(50)

with terms g2,h, g∗2 , g3,h, g

∗3 to be specified. As in the 1D case, multiplication of the

semi-discrete continuity equation from left by 12v1, adding the result to (49) and usingcontinuity in time, we have

∆x∆y

4

du2dt

+Dξfξ

2+Dηf

η

2+ sξ2 + s

η2

= M−1(〈

(nξgξ2,h + nηg

η2,h − g

∗2)L〉∂K

+1

2v

1

〈(nξf

ξ1,h + nηf

η1,h − f

∗1 )L

〉∂K

),

with skew-symmetric terms sξ2, sη2 given by

sξ2 =1

2

[−Dξ u2v1 + u2Dξv1 + v1Dξu2

],

sη2 =1

2

[−Dη u3v1 + u3Dηv1 + v1Dηu3

].

28

Using the SBP properties of Dξ and Dη a corresponding weak formulation may be

obtained. Analogous to the 1D case, we multiply by the diagonal matrix M to obtain

∆x∆y

4Mdu2dt

+ (Bξ − SξT )(fξ

2− 1

2u

2v1

)+ (Bη − SηT )

(fη

2− 1

2u

3v1

)+

1

2

[u

2Sξv1 + v1(Bξ − Sξ

T )u2 + u3Sηv1 + v1(B − SηT )u3

]=

(〈(nξg

ξ2,h + nηg

η2,h − g

∗2)L〉∂K

+1

2v

1

〈(nξf

ξ1,h + nηf

η1,h − f

∗1 )L

〉∂K

).

(51)

Choosing gξ2,h =(

12ρv

21 + p

)h

and gη2,h =(

12ρv1v2

)h

in accordance with the 1D case andusing Lemma 3 to cancel out boundary terms yields

∆x∆y

4Mdu2dt

= SξT

(fξ

2− 1

2u

2v1

)+ Sη

T

(fη

2− 1

2u

3v1

)− 1

2

[u

2Sξv1 − v1Sξ

Tu2 + u3Sηv1 − v1SηTu3

]−(〈g∗2L〉∂K −

1

2v

1〈f∗1L〉∂K

).

(52)

Multiplying (52) from left with 1T cancels the volume terms and we obtain

∆x∆y

41TM

du2dt

= −(〈g∗21

TL〉∂K− 1

2vT1 〈f∗1L〉∂K

)= −

〈(g∗2 +

1

2v1,hf

∗1

)L

〉∂K

.

Hence consistency demands g∗2 = f∗2 − v1,hf∗1 in accordance with the one-dimensional

case. Analogous derivations for the y-direction of the momentum equations yield thecorresponding weak formulation

∆x∆y

4Mdu3dt

= SξT (fξ

3− 1

2u

2v2) + Sη

T (fη3− 1

2u

3v2)

− 12

[u

2Sξv2 − v2Sξ

Tu2 + u3Sηv2 − v2SηTu3

]−〈(

f∗3 I +1

2

(v

2− v2,hI

)f∗1

)L

〉∂K

.

(53)

Furthermore, the same derivations as in the 1D case yield a semi-discrete balanceequation for the kinetic energy ekin =

12ρ(v21 + v

22

). Similar to equation (34) we arrive

at

d

dt1TMekin =

((Dξv1

)T+(Dηv2

)T)Mp− vT1 〈g∗2L〉∂K − v

T2 〈g∗3L〉∂K .

Hence, for a correct kinetic energy balance the left and right hand sided transportterms within the surface fluxes have to cancel out as in the 1D case. Decomposingg∗,±2 = g̃

∗,±2 +nξp

∗ and g∗,±3 = g̃∗,±3 +nηp

∗ with g̃∗,±k = f̃∗k − 12v

±k−1f

∗1 a condition of the

formv−1 g̃

∗,−2 + v

−2 g̃∗,−3 = v

+1 g̃∗,+2 + v

+2 g̃∗,+3

29

needs to be fulfilled. Thus, a suitable choice for kinetic energy preservation similar tothe 1D case is

f̃∗k = v̄k−1f∗1 , k = 2, 3.

In particular, this holds for the KEP flux using rotational invariance of the Euler equa-tions. This numerical flux is given by

f∗1f∗2f∗3f∗4

=

f∗,1D1 (u−, u+, n)

nξf∗,1D2 (u

−, u+, n)− nηf∗,1D3 (u−, u+, n)nηf

∗,1D2 (u

−, u+, n) + nξf∗,1D3 (u

−, u+, n)

f∗,1D4 (u−, u+, n)

,where f∗,1D denotes the 1D KEP flux in normal direction

f∗1D,1 = ρ̄v̄n,

f∗1D,2 = ρ̄v̄2n + p̄,

f∗1D,3 = ρ̄v̄nv̄t,

f∗1D,4 = ρ̄v̄nH̄,

using the normal and tangential velocity vn = nξv1 + nηv2 and vt = nξv2 − nηv1.

5.3 Numerical simulation of 2D homogeneous turbulence

Two-dimensional homogeneous turbulence is an energy-decaying system which is exten-sively used to study accuracy and efficiency of numerical methods, e.g. in [13, 24, 32].Our purpose is to demonstrate the improved resolution of the proposed kinetic energypreserving skew-symmetric DG scheme for this test case compared to the standard DGscheme. The computational domain is the square [0, 2π]2 supplied with periodic bound-ary conditions. The initial energy spectrum is given in Fourier space by

E(k) =as2

1

kp

(k

kp

)2s+1exp

[−(s+

1

2

)(k

kp

)2],

where k =√k2x + k

2y. The initial energy spectrum attains its maximum at the wavenum-

ber kp. As in the references, the parameters are set to kp = 12, as =(2s+1)s+1

2ss! , s = 3.From this initial energy spectrum an initial velocity distribution is obtained using trans-fer procedures desribed in [13, 24] where a random phase is introduced into the vorticityfield. The initial velocity distribution in physical space is then given by the inverseFourier transform. For the compressible flow computations the initial density is set toρ0 = 1 while the pressure is computed setting the initial Mach number to Ma = 0.1.The viscosity coefficient µ may be varied to study the quality of the numerical solutionsfor different Reynolds numbers. We then compute the numerical solution to this testproblem both with the standard DG scheme and the kinetic energy preserving skew-symmetric DG scheme based on the Gauss nodes until time T = 10. The resultingenergy spectrum E(k) at this final time is then computed from the velocity distribution

30

10 20 40 60 8010010

−14

10−10

10−6

10−2

k

E(k

)

DG standard 40 cells

DG skew 40 cells

Reference t=10

Initial energy spectrum

10 20 40 60 8010010

−14

10−10

10−6

10−2

k

E(k

)


DG skew 80 cells

Reference t=10


Figure 10: Comparison of DG scheme and DG-KEP on Gauss nodes for N = 1 andRe = 100. Energy spectrum at time T = 10. Left: 40 cells. Right: 80 cells.

10 20 40 60

10−8

10−6

10−4

10−2

k

E(k

)


DG skew 80 cells

Reference t=10


10 20 40 60

10−8

10−6

10−4

10−2

k

E(k

)


DG skew 160 cells

Reference t=10


Figure 11: Comparison of DG scheme and DG-KEP on Gauss nodes for N = 1 andRe = 600. Energy spectrum at time T = 10. Left: 80 cells. Right: 160 cells.

by the same procedures as in [24]. Figure 10 depicts the comparison of the standardDG scheme on Gauss nodes for N = 1 and its kinetic energy preserving variant KEP-DG in terms of their energy spectrum for a lower Reynolds number of Re = 100. Areference solution is obtained by the standard DG scheme of 5th order on 80 grid cells.A more accurate representation of the energy spectrum is obvious for the second orderKEP-DG scheme, both on the very coarse grid with 40 cells and on the finer one of 80cells. Figure 11 shows the corresponding comparison of the second order standard DGscheme vs. the KEP-DG scheme on Gauss points for Re = 600. Now, the reference so-lution is obtained by the standard 5th order DG scheme on 160 grid cells, although thisnumerical solution is indistinguishable from the one on 80 grid cells in this case. Alsofor the higher Reynolds number, the energy spectrum is represented more accurately bythe KEP-DG scheme.

31

6 Conclusion and outlook

In this work, a kinetic energy preserving DG scheme on Gauss-Legendre nodes has beenconstructed. With respect to the experimental order of convergence, this new schemeperforms just as the skew-symmetric DG scheme with Gauss-Lobatto nodes in [9]. For aviscous Sod shock tube, the results are in accordance with the kinetic energy preservingschemes in [1], though the latter work does not include skew-symmetric volume terms.The test case based on an acoustic pressure wave in Sect. 4.3 demonstrates higher ac-curacy for Gauss-Legendre nodes while skew-symmetric terms have less effect on theaccuracy of the computed distributions of pressure and kinetic energy. The experimentscarried out in Sect. 4.2 and 4.4 show that it is important to consider interface contribu-tions in case of exclusively interior nodes as the direct use of the skew-symmetric terms(47) based on Gauss-Lobatto nodes does not preserve the mean kinetic energy and mayeven lead to additional stability problems. The simulations of decaying homogeneousturbulence suggest that also in the case of Gauss-Legendre nodes, the property of kineticenergy preservation leads to enhanced accuracy of the energy spectrum. Future workwill concentrate on the construction of kinetic energy preserving schemes on generalnode distributions including schemes on triangular grids.

7 Appendix

In the following, the proofs of Lemma 2 in Sect. 3 and Lemma 3 in Sect. 5.1 are given.

7.1 Proof of Lemma 2

For the assertion of conservativity, we multiply the scheme (25) from left by 1TM . Thus,the rate of change of the cell mean within a cell [xi, xi+1] multiplied by the cell size ∆xis given by

1TM∆xi

2

d

dtui = −1TMDf i − 1TM

[−Dαiβi + αiDβi + βiDαi

]+1T [(f ih + α

iβih + kih)L]

1−1 − 1T [f∗,iL]1−1 − 1T [αiβ∗,iL]1−1

+1T(k∗,+i L(−1)− k

∗,−i L(1)

)= −1T (B −DTM)f i

−1T[−(B −DTM)αiβi + αi(B −DTM)βi + βiMDαi

]+1T [(f ih + αβ

ih + k

ih)L]

1−1 − 1T [f∗,iL]1−1 − 1T [αiβ∗,iL]1−1

+1T(k∗,+i L(−1)− k

∗,−i L(1)

),

using the SBP property MD = B −DTM .Furthermore, the interpolation of boundary values and the corresponding boundary

terms are encoded in the matrix B. Thus, due to Lemma 1 we can replace the corre-

sponding terms by Bf i = [f ihL]1−1, Bα

iβi = [(αβ)ihL]1−1 and (α

i)TBβi = 1Tαi[βihL]1−1.

32

Since this cancels out the terms containing f ih and αiβih, it holds that

1TM∆xi

2

d

dtui = (D1)TMf i −

[(D1)TMαiβi − (αi)TDTMβi + (βi)TMDαi

]+ 1T [

((αβ)ih + k

ih

)L]1−1

− 1T [f∗,iL]1−1 − 1T [αiβ∗,iL]1−1 + 1T(k∗,+i L(−1)− k

∗,−i L(1)

).

Now, discrete differentiation yields D1 = 0 and obviously, −αTDTMβ + βTMDα = 0holds. This reduces the rate of change of mass contained in the specific cell to

1TM∆xi

2

d

dtui = 1T [

((αβ)ih + k

ih

)L]1−1 − 1T [f∗,iL]1−1 − 1T [αiβ∗,iL]1−1

+ 1T(k∗,+i L(−1)− k

∗,−i L(1)

)For conservativity, the remaining terms may only contain interface contributions.

Hence, the balance of flux contributions at cell interfaces has to be investigated. Weconsider the interface with index (i− 1, i) between two cells [xi−1, xi] and [xi, xi+1]. Asthe interpolation property yields 1TL(ξ) = 1, we obtain the following fluxes over cellinterfaces. Denoting by C∗,i−1i−1,i the corresponding flux from left to right based on thevalues in the left cell, neglecting the term f∗i−1,i, we have

C∗,i−1i−1,i = −(αβ)i−1h (1)− k

i−1h (1) + α

i−1h (1)β

∗i−1,i + k

∗,−i−1,

whereas the analogous quantity C∗,ii−1,i based on the right cell is given by

C∗,ii−1,i = −(αβ)ih(−1)− kih(−1) + αih(−1)β∗i−1,i + k

∗,+i .

Hence, conservativity precisely requires

C∗,i−1i−1,i = C∗,ii−1,i =: C

∗i−1,i,

yielding the first assertion.As in the second assertion, we now assume that kih only depends on a combination

of the interior values αi, βi and that k∗,+i only depends on β∗i−1,i and α

ih(−1). If we

then modify any of the input values other than αi, βi, β∗i−1,i, the value of C∗,ii−1,i does

not change and thus C∗i−1,i remains constant as well. If we furthermore assume that

k∗,−i only depends on β∗i,i+1 and α

ih(1), modifying α

i or βi does not change C∗,i−1i−1,i andhence does not change C∗i−1,i. Therefore, C

∗i−1,i may only depend on β

∗i−1,i which proves

assertion 2.Now, consistency in the finite volume sense refers to a consistent numerical flux.

With kih and k∗,±i set as in the third assertion, the scheme (25) for N = 0 reduces to

∆xid

dtui = [f∗,i]1−1 − [αiβ∗,i]1−1 +

(k∗,+i − k

∗,−i

)= [f∗,i]1−1 −

(C∗i,i+1 − C∗i−i,i

).

Since f∗ is consistent to f with f∗i−1,i(u, u) = f(u) and β∗ is consistent to β, this above

finite volume scheme is consistent if and only if

f∗i−1,i(u, u) + C∗i−1,i(β

∗i−1,i(u, u)) = f(u) + C

∗i−1,i(β(u)) = f(u),

for any interface index (i − 1, i) and any value of the conserved variable u. Hence, wehave the requirement Ci−1,i(β(u)) = 0, which yields the last assertion.

33

7.2 Proof of Lemma 3

The columns of the matrices Bξ = B ⊗M1D and Bη = M1D ⊗ B can be related tothe Lagrange polynomials as follows. Let ν(i, j) = (i − 1)(N + 1) + j. From the proofof Lemma 1 in Sect. 2, we recall that B has the entries Bjk = [LjLk]

1−1 and M1D is

diagonal with entries ωj . Therefore, the ν-th columns of the above matrices Bξ and

Bη are given by Bξν(i,j)

= [L(ξ)Li(ξ)]1−1 ⊗ ωj�j and Bη ν(i,j) = ωi�i ⊗ [L(ξ)Lj(ξ)]

1−1,

respectively, where �j denotes the j-th unit vector.Therefore, we obtain

Bξgξ =

N+1∑i=1

N+1∑j=1

Bξν(i,j)

gξν(i,j) =

N+1∑i=1

N+1∑j=1

([L(ξ)Li(ξ)]

1−1 ⊗ ωj�j

)gξν(i,j)

=

N+1∑j=1

ωj

(N+1∑i=1

Li(ξ)gξν(i,j)

)L(ξ)⊗ �j

1−1

=

N+1∑j=1

ωjgξh(ξ, ηj)L(ξ)⊗ L(ηj)

1−1

.

Since for the normal vectors and Gauss nodes on edges e = 2 and e = 4 we haven2ξ = 1, n

4ξ = −1 and ξ2j = 1, ξ4j = −1, j = 1, . . . , N + 1, it holds that

Bξgξ =

∑e=2,4

N+1∑j=1

neξωjgξh(ξ

ej , η

ej )L(ξ

ej )⊗ L(ηej )

With analogous arguments we obtain

Bηgη =

N+1∑i=1

ωi�i ⊗ L(η)N+1∑j=1

Lj(η)gην(i,j)

1−1

=∑e=1,3

N+1∑i=1

neηωigηh(ξ

ei , η

ei )L(ξ

ei )⊗ L(ηei ).

Summing up Bξgξ + Bηg

η and considering that n1ξ = n2η = n

3ξ = n

4η = 0 we obtain

the assertion.

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