NASA Contractor Report 178021

lCASE REPORT NO. 85-51

NASA-CR-178021

19860009594

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leASE

THE NUMERICAL VISCOSITY OF ENTROPY STABLE SCHEMES

FOR SYSTEMS OF CONSERVATION LAWS. I.

Eitan Tadmor

,".

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Contract No. NAS1-17070

November 1985

INSTITUTE FOR CO~WUTER APPLICATIONS IN SCIENCE AND ENGINEERING

NASA Langley Research Center, Hampton, Virginia 23665

Operated by the Universities Space Research Association

NI\SI\

National Aeronautics and

Space Administration

Langley Research center

Hampton.Virginia 23665

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EOUATIONS/*VISCOSITY

THE NUMERICALVISCOSITY OF ENTROPYSTABLE SCHEMES

FOR SYSTEMSOF CONSERVATIONLAWS. I.

Eitan Tadmor

School of Mathematical Sciences, Tel-Aviv University

and

Institute for Computer Applications in Science and Engineering

ABSTRACT

Discrete approximations to hyperbolic systems of conservation laws are

studied. We quantify the amount of numerical viscosity present in such

schemes, and relate it to their entropy stability by means of comparison. To

this end, conservative schemes which are also entropy conservative are

constructed. These entropy conservative schemes enjoy second-order accuracy;

moreover, they admit a particular interpretation within the finite-element

framework, and hence can be formulated on various mesh configurations. We

then show that conservative schemes are entropy stable if and only if they

contain more viscosity than the mentioned above entropy conservative ones.

Research was supported in part by the National Aeronautics and Space

Administrationunder NASA Contract No. NASI-17070 while the author was in

residence at ICASE, NASA Langley Research Center, Hampton, VA 23665-5225.

Additional support was provided in part by NSF Grant No. DMS85-03294and ARO

Grant No. DAAG29-85-K-0190while in residenceat the Universityof California,

Los Angeles,CA 90024

Bat-ShevaFoundationFellow.

i

1 ° INTRODUCTION

The systems of conservation laws referred to in the title are of the form

(i i) 8 d • B'-'t"u + _ rILf(k)j = O, t_,tj__ RHx[0,_j; r _

k=l Bxk

here f(k) _ f(k)(u ) = (f_k),...,f_k))T are smooth nonlinear flux mappings of

the conservative variables u _ u(_,t) = (ul,...,UN)T. Owing to the

nonlinearity of the fluxes f(k), solutions of (I.I) may develop singularities

at a finite time after which one must admit weak solutions, i.e., those

derived directly from the underlying integral conservative relations•

Yet, such weak solutions of the conservative equations are not unique•

Additional criteria are required in order to single out a unique physically

relevant weak solution, the latter being identified as, roughly speaking, a

stable limit of a vanishing viscosity mechansim. Entropy stability is then

sought as the usual criterion to identify such vanishing viscosity

solutions. Lax [I0] has shown that entropy stability is in fact equivalent to

a vanishing viscosity mechanism, at least in the small -- in the large for

scalar problems, e.g., [12], [8].

We study entropy stable approximations to such systems of conservation

laws. Entropy stability manifests itself here in terms of a conservative cell

entropy inequality. We note in passing that, if holding for large enough

class of entropy functions, such cell entropy inequality is intimately related

to both, the question of convergence toward a limit solution as well as the

question of this limit solution being the unique physically relevant one,

e.g., [I], [4], [16] and the references therein•
-2-

Starting with Von Neumann and Richtmyer [13] it has long been a common

practice to ensure the entropy stability of conservative schemes by tuning

their numerical viscosity. In this paper we provide a framework for designing

entropy stable schemes by quantifying precisely how much numerical viscosity

is to be added. As in [18] this is accomplished by means of comparison: we

show that entropy stability is achieved if and only if there is more numerical

viscosity than that present in certain entropy conservative schemes. To this

end we proceed as follows.

In Section 2 we begin by discussing the entropy variables associated with

the conservative systems in question. As observed by Mock [12], [5], such

systems are symmetrized w.r.t, these variables. Such symmetrization provides

us with a natural order which then fits our goal to compare between the

numerical viscosities of different schemes. Expressed in terms of these

entropy variables, we then turn to construct the mentioned above entropy

conservative schemes. The entropy conservative schemes -- discussed in

Section 3 and 4, are second-order accurate. Moreover, they are particularily

attractive in view of their interpretation within the finite element framework

which enables possible generalizations to various mesh configurations. In

Section 5 -- the main one in this paper -- we compare between the viscosity

coefficients of different conservative schemes given in their appropriate

viscosity form. It is shown that conservative schemes containing more

viscosity than that of an entropy stable scheme are also entropy stable. In

particular, when compared with the previously discussed entropy conservative

schemes, we arrive at a necessary and sufficient criterion for entropy

stability. Since our entropy conservative schemes are second-order accurate,

such a comparison can still entertain the construction of entropy stable
-3-

schemes which are second-order accurate ones. Finally, in order to simplify

the notations, we keep our presentation within the one-dimensional case; the

multidimensional extension can be worked out dimension by dimension.

2. THE ENTROPY VARIABLES

We begin our discussion with the one-dimensional model

(2.1) _--_u_+_x_ If(u)] = 0.

We assume that the system (2.1) is equipped with a generalized

Entropy Function: a convex mapping U = U(u) augmented with entropy flux

mapping F = F(u) such that the following compatibility relation holds

(2.2a) uTA = FT.

u u

Here A = A(u) is the Jacobian matrix

(2.2b) A(u) = f .

u

We note that the entropy functions, U(u), are exactly those whose positive

Hessians U > 0 symmetrize the system (2.1) upon multiplication on the left uu

[2]

(2.3) Uu A = [UuuA]T.
-4-

Mock [12], [5] has pointed out a more fundamental symmetrization of system

(2.1), preserving both the strong as well as the weak solutions of the

system. To this end one makes use of the entropy variables

_U

(2.4) V _ v(u) ='_u (u).

Thanks to the convexity of U(u) the mapping u + v is one-to-one. Hence we

can make the change of variables u = u(v) which puts the system (2.1) in its

equivalent symmetric form

(2.5a) __--f[u(v)] + 3-f _ [g(v)] = 0, g(v) _ f(u(v)).

The system (2.5a) is symmetric in the sense that the Jacobians of its temporal

and spatial fluxes are

(2.5b) H _ H(v) =-_ [u(v)] > O, B _ B(v) =-__ [g(v)] .

This follows from the compatibility relation (2.2) equivalently expressed as

(2.6) vTB(v) = GT

v' G(v) _ F(u(v)),

which in turn implies

[vTu(v) - U(U(V))], g(v) (2.7) u(v) =-_ =-_v [vTg(v) - G(v)].
-5-

Indeed, the Jacobians H(v) and B(v) in (2.5b) are the symmetric Hessians

of the corresponding expressions inside the brackets in the right of (2.7).

Finally, we note that in contrast to the symmetrization on the left quoted

in (2.3), the use of the entropy variables symmetrize the system (2.1) on the

right [17], i.e., the original Jacoblan A = f is replaced here by the

symmetric one B = gv

3. ENTROPY STABLE SC}{EMES

We consider conservative discretizations of the form (1)

d u(t) = 1 (3.1) -_ -_-_ [fv+I/2- f9_i/2]

serving as consistent approximations to the system of conservation laws

(3.2) _ -Fu [f(.)]= 0.

Here u (t) denotes the approximation value along the gridline (x9 _ 9Ax,t),

Ax being the spatial mesh size, and

(3.3a) f9+1_ = _f(Uv_p+l,''',uv+ p)

(1)Both the differential and the discrete formulations will employ the same

notations. The distinction between the two is made by the use of Greek

indices in the discrete formulation.
-6-

is the Lipschitz continuous numerical flux consistent with the differential

one

(3.3b) +f(w,w,...,w) = f(w).

To discuss entropy stability, we let (U,F) be any entropy pair

associated with the system (3.2). Multiplying by UT and employing (2.2) we u

conclude that under the smooth regime we have the additional conservation of

entropy

(3.4) _U _F

Taking into account the nonsmooth regime as well, following Lax [i0] and

Kru_kov [8] we postulate as an admissibility criterion an entropy stability

requirement, expressed in terms of the following

Entropy Inequality: We have, in the sense of distributions

_U _F

(3.5) _-_ + -_ <__0.

Similarily, for the scheme (3.1) to be entropy stable, a discrete cell entropy

inequality is sought [4]

(3.6a) d U(uv(t)) + i _ d--t _-_ [Fv+l/2 Fv_I/2 ] --< 0;

here Fv+ 16 is the numerical entropy flux

(3.6b) Fv+16= ¢F(Uv_q+l,...,Uv+q)
-7-

consistent with the differential one

(3.6c) *F(W,W,''',w) = F(w).

An approximate entropy equality of the above type was derived by Osher in

[14, Section 3]. A slightly different version of such equality proved in the

Appendix employs the consistent numerical flux.

* T

Fv+i/2=I/2[F(u v) + F(Uv+l) ] +I/2[Uu(U _) + Uu(Uv+l) ] f_+I/2

(3.7a)

The referred equality reads

Lemma 3.1: [Osher]

Le____tuv(t) be the discrete solution of the conservative scheme (3.1)

which is consistent with the system (3.2). Then, for any entropy pair (U,F)

the following equality holds

d U(--(t)) v u + 1 * * d-_ A-x IF+ 1/2- Fv_I/2 ] =

(3.7b)

_ 2Ax 1 duTUuu[fv - 1/2- f(u)] + f duTUuu[fv+ 1/2- f(u)] "

U__ I Uv

Hence the scheme (3.1) is entropy stable if the integrals on the right of

(3.7b) are shown to be nonpositive. In order to examine the entropy stability

question, we first study the case in which these integrals vanish, i.e., we

study entropy conservative schemes.
-8-

4. ENTROPY CONSERVATIVE SCHEMES

In this section we identify particular schemes which satisfy a cell

entropy equality. The numerical viscosity present in such entropy

conservative schemes will then be used as the building block for the

construction of entropy stable schemes•

In order to carry out the program above, it will prove useful to work with

the entropy variables rather than the usual conservative ones. Thus,

associated with an entropy function U(u) are the entropy variables

v = Uu(U). Expressed in terms of the latter, the system of conservation laws

considered is, see (2.5a)

(4 •i) _a-F[uCv)] + __ [gCv)]= 0, gCv)= fC.Cv)).

It is augmented by the corresponding entropy inequality

(4 2a) _V _G • _F + _-f< 0

where (V,G) is the appropriate entropy pair

(4.2b) V _ V(v) = U(u(v)), G _ G(v) = F(u(v)).

In a similar manner, we interpret the conservative scheme (3.1) in terms of

the appropriate entropy variables v = Uu(U ). Thus, (4.1) is approximated

by •

(4.3a) dd__[u(v (t))] = --_IAx[gv+l/2- gv-1/2 ]
-9-

where g_+i/2 is the numerical flux

(4.3b) g_+l_ = Cg(V__p+ I' "",vv+p), _g(...vv, •..) _ +f(...u(vv)...)

consistent with the differential one

(4.3c) Cg(W,W,...,w)= g(w).

The corresponding all entropy inequality takes the form

(4.4a) d V(v (t))+ i d--t _ Icy+ 1/2- Gv-1/2 ] --<0.

Here Gv+ 16 is the numerical entropy flux

(4.4b) Gv+1_= _G(V _q+l ,...,V+q)

consistent with the differential one

(4.4c) _G(W,w,...,w) = G(w).

Such a consistent numerical entropy flux suggested by (3.7a) is given by

* i 1 ]T G_+i=/_2 [G(%)+G(%+I)+]_ [% +v+_ g_+v2

(4.5a)

- _ Iv g ) + v v+ 1) .
-10-

The equality (3.7) now simplifies into

Lemma 4.1: (An approximte cell entropy equallty-revisited).

Let vv(t ) be the discrete solution of the conservative scheme (4.3).

then, for any entropy pair (V,G) the following equallty holds

d V(vvCt)) + 1 * * d-_ _ [G +I/2- G9_1/2 ] =

(4.5b)

,) 1 T

_ 2Axi dvTEgv_ I/2- g(v)] + f dv [gv+i/2 - g(v)] "

1 v

Our entropy conservative schemes will be determined by setting the

numerical flux gg+ 16 to be

, 1

(4.6a) _+ 1/2= gv+ i/2= f_=0g(vv+ 1/2(_))d_,

where v + l_(g) denotes the segment connecting v and v_+ 1

(4.6b) vv+i/2(g) - vv + gAvv+ I/2' Avv+1/2= Vv+l - v .

With thls choice of numerical flux we have

W+I 1

* = AvTv+ _Avv+ )dE •

T * AvT_+1/2 gv+ 1/2 _=0 1/2 1/2

(4.7) / dv gv+ 1/2--- f g(v + V

It is here where we make use of the entropy variables formulation: thanks to

the symmetry of gv' the expression on the right equals the path independent

integral, see (2.7),
-IiW+I T * 1 Vv+l T

(4.8) vf dv g_+1/2= _f=0 AvT+I/2g(v + _Av +1/2)d_ = vf dv g(v)

and therefore each of the integrals on the RHS of (4.5b) vanishes in this

case. Let us summarize what we have shown in

Theorem 4.2: (Entropy conservative schemes).

The conservative scheme

1 * * , 1

(4.9) dd__[u(v_(t))] = _x [g_+i/2- g_-i/2 ]' g_+i/2= _=0 f g(vv+I/2(_))d_

is also entropy conservative, i.e., it satisfies the following cell entropy

equality

(4.10) d V(vv(t)) + 1 * * d-_ A-x [Gv+I/2- G_-I/2] = 0.

We close this section by noting that the scheme (4.9) besides being

entropy conservative is also a second-order accurate one. Both the secondorder accuracy and in particular the entropy conservation of the scheme (4.9)

can be directly verified within the framework of a finite-element formulation

advocated by, e.g., Mock [Ii] and in particular by Hughes and his

collaborators, [7] and the references therein. To this end one considers the

weak formulation of (4.1)

(4.11) f [_T L_t u(v) + _T g(v)]dxdt = 0.

If now the trial solutions v . v(x,t) and the weighting test functions

. _(x,t) are chosen out of the typical finite-element set spanned by the

C° "hat functions"
-12-

x - x_ 1

xv- xv-i x_ l<x<x v

x +lil x <x<

x+ I - xv _ -- _ x_+ I

we are led to

%+I _H(x)

I ix + 1 - x _i] _ [u(v (t)] = - _ V_x g(v(x,t) - I v (t)H (x))dx,

x__ 1

or, after changing variables on the right

El i 1

[u(w(t)] = -2 f g(vv+ (_))d_- f g(v_I/2(_))d (4.12) _-_ Xv+l - xv-I _=0 1/2 _=0

Observe that (4.12) also suggests the natural extension of the entropy

conservative schemes (4.9) to non-uniformily spaced meshes. The entropy

^

conservation follows, at once, by selecting the test function m to conicide

A

with the trial function v, so that in view of (2.6), (4.11) yields

AT

0 = f [_t _ g(_) ]dxdt - f [_ v(v)+

5. NUMERICAL VISCOSITY AND ENTROPY STABILITY

The essential role played by the numerical viscosity has long been

recognized starting with Von Neumann and Richtmyer [13]. In this section we

quantify the amount of numerical viscosity required in order to guarantee

entropy stability. As in [18], this will be done by means of comparison.

To start with, we consider the entropy conservative scheme (4.9).

Integration by parts of its numerical flux formula (4.6a) yields
-13-

g,_x+/2={=og(_+_/2(€))d=€(€-7)g(._ X/(€))1 2 (=0

(5.1)

I

d

Recalling the notation for the Jacoblan gv' see (2.5b),

(5.2a) B(v) = gv'

the RHS of (5.1) can be rewritten as

* 1 1 1

(5.2b) gv+l_=_ [_(_v) + g(_+l)] - _ f_=0(2_- l)B(v +i/2(_))d_Avv+16 .

Inserting this into (4,9), our entropy conservative scheme assumes the

viscosity form

d 1

d-_[u(_+l(t))] = -_2Ax [g(_+l ) - g(_+l)] +

(5.3a)

+ 2-A_x[Qv+I/2Av_+16 - Q_-l_AV_-i/2 ]"

Here Qv+ i/2 is the numerical viscosity coefficient matrix given by

. I

(5.3b) %+ 1/2= f_=0(2_- l)B(vv+ 1/2(_))d_.

We note that the second brackets on the right of (5°3a) mimic a diffusive-llke

term Ax(Q Vx) x. Yet, though the viscosity martfx Q_+ 16 is symmetric, it is

not necessarily a positive definite one; rather, it is determined so as to
-14-

counterbalance the dispersive flux central differencing inside the first

brackets on the right of (5.3a).

Motivated by the discussion above, we would like to consider schemes given

in a similar viscosity form [18]

ddt [u(_(t))] : _i2Ax [g(v+l) _ g(v - i)] +

(5.4)

1

+ _ [Q_+I/2Av+l/2- Qv-1/2Av-I/2 ].

The matrix Q_+I_ on the right will be referred to as the numerical viscosity

coefficient matrix.

Remark: We observe that the above definition of the numerical viscosity

coefficient depends on the specific entropy function under consideration

2

U(u). The special choice U(u) = u corresponds to our earlier viscosity

definition [18] in the scalar case.

What schemes admit a viscosity form-like (5.4)? To answer this question

we observe that the numerical flux determined by (5.4)

I

4+1/2 = _g(W_p+l'''''V_+p ) =_ [g(_) + g(vv+l) ]

(5.5)

1

- _ Qv+l/2 Iv + I - v ]

satisfies the consistency relation

(5.6) _g(Vv_p+l,...,Vv_l,W,w,v +2,...,Vv+p) = g(w).
-15-

The consistency relation (5.6) is slightly more stringent than the usual one

(4.3¢). It characterizes the class of essentially three-point schemes, a

class which includes, beside the standard three-point schemes, most of the

recently constructed second-order accurate TVD schemes, e.g., [3], [15],

[6]. In the Appendix we show that the converse of the above implication

holds, namely we have

Lemma 5.1: The consistent conservative scheme (4.3) can be rewritten in

the viscous form (5.4) if and only if it is an essentially three-point scheme.

Granted the viscosity form (5.4), we now turn to discuss the question of

entropy stability. We first note that there is only one degree of freedom in

setting up the viscosity form (5.4), that is, setting up the viscosity

coefficient Qv+ i/2" Motivated by Lemma 4.1, we shall examine the quantity in

light of the integrals on the RHS of (4.5b). Thus, using (5.5) we find

%+1 T I ArT+ [g(%) + )] fv d,,[ i/2- ] = 1⁄2 v

Vv+l T i AvT+ - f dv g(v) - _ 1/2Q_+1/2Avv+I/2 .

v

v

The first two terms on the right are fixed by the differential flux;

abbreviate their difference by KV+I/2 . Then, Lemma 4.1 tells us that

d V(v (t))+ 1 * * d--t _ [G +1/2- G 1/2] = KX)- 1/2 + K_)+ 1/2

(5.7)

I ArT- Qv Avv_ 1 AvT+ Qv+ Avv+ -_ 1/2 -1/2 1/2- _- 1/2 1/2 1/2"
-16-

Hence we conclude that the cell expression on the left decreases whenever the

quadratic forms AT± 1/2Q ± 1/2Avv± 1/2 increase. This suggests to compare

between different schemes in terms of their numerical viscosity coefficients

used as our scale.

We shall say one scheme contains more viscosity than another scheme if the

^(i)

viscosity coefficient of the first scheme, say _ +1/2 , dominates that of the

^(2)

second one, _ +1/2 , i.e., if we have

r q(1)

(5.8a) AvT+I/2Q_2)I/2Av +i/2_< Avv+i/2 v+i/2Av +I/2.

An strengthened formulation of this, in terms of the natural order among

symmetric matrices, reads

(5.8b) ReQ (2)i/2_<R eQ(vl)i/.2

In case the entropy variables were used to begin with, they would lead to

symmetric viscosity coefficients very much the same way they led to symmetric

Jacoians, e.g., (5.3b), and we would arrive at the natural hierarchy

Q(2) Q(1) (5.8c) "_+i/2< 9+i/2.

The last three inequalities mean the same when dealing with the scalar case.

Equipped with the above terminology we now turn to a particular comparsion

with the entropy conservative scheme (5.3). Thus, we consider a conservative

scheme in its viscous form (5.4) and we let D +i/2

(5.9) D+ i_ = Q_+ I12- Q_+ I_
-17-

denotes its viscosity deviation from the entropy conservative one (5.3). Then

we can write the scheme under conservation (5.4)

d 1 1 * *

d-_ [u(_(t))] - max [g(_+l ) - g(vv-l)] +_-_x [qv+ i/2Avv+i/2 - Qv-I/2 Ave-i/2 ]

i

+ _x [D i/2Avv+I/2- Dr_I/2Avv_I/2]

_ AX1 [gv+* i/2- g_-* I/2] + _x1 [D 1/2Av+ 1/2- D_- 1/2AV_ I/2].

Multiplying the last equality by uT(uv(t)) = vTv(t) on the left yields

d V(v (t) + 1 T * *

d-_ v _xVv[gv+i/2 - gv-i/2 ]

(5.10)

_ max1 v_[Dv+ i/2hvv+ I/2- Dv-i/2 Avv_ I/2]"

By Theorem 4.2 the second expression on the RHS of (5.10) equals the

conservative difference

_xI vv[gv+ T * i12-gv-* iI_= _ vvT(t) _-_ d [U(Vv(t))]

(5.11)

_ d V(vv(t)) = I * * at _x [Gv+ 1/2- Gv-i/2]"

Regarding the RHS of (5.10), the following identity puts it as the sum of

familiar quadratic terms plus a conservative difference
-18-

12Ax vT[Dv+ i/2AvV+ i/2- DV-I/2Av_ 1/2]

(5.12) - 2Ax [_ I/2 Dr+ 1/2 I/2 _ - 1/2 1/2

i 1 T 1 TD

+2T.[_[%+%+i]%+I/_v_ 2 i/2_ [%1 +%] _ i/_2 I/]2

Using the last equality we conclude the main result of this section,

asserting

Theorem 5.2: (Entropy stability).

A conservative scheme is entropy stable if and only if it contains more

numerical viscosity than the entropy conservative scheme. Moreover, the

entropy dissipates in this case at rate governed by the cell-estimate

d V(vv(t)) + 1 d--_ _x [Gv+i/2- Gv_i/2 ]

(5.13a)

- 4Axl [Av_+ i/2Dv+ 1/2Avv+ i/2+ AvTv_I/2Dr_ 1/2Avv_ 1/2] _< 0.

Here Gv+i/2 is the consistent numerical entropy flux

I T I

Gv+l_= 71 [G(vv) + G(Vv+l)] + 7 [vv + Vv+l] [gv+l_- 7 Dv+I/2Ave+I/2 ]

(5.13b)

i

Proof: Inserting (5.11) and (5.12) into (5.10) we obtain
-19-

d V(_(t))+ 1 d--_ _-_ [Gv+ 1/2- Gv_I/2 ]

(5.14)

- 4Axl [AT+ 1/2Dr+ 1/2Avv+ 1/2+ ArTy-1/2Dr- I/2Av_ 1/2]

where Gv+l_ is the consistent numerical entropy flux given in (5.13b). Now,

if the scheme considered contains more viscosity than the entropy conservative

scheme, then by (5.9) the RHS of (5.14) is nonposltive and entropy stability

follows. Conversely, assume that our scheme is entropy stable, i.e., that it

satisfies a consistent all entropy inequality of the form

dd-_ V(%(t))+A-xi [Gv+ 1/2-_v- I/2]--< 0.

Subtracting this from (5.14) we get after multiplication by Ax

(5.15a) H+i/-2H la->i AT+ i/2Dv+ i/2Avv+ 1/2+ AvT I/D2_ 1/2Av_ 1/2]

Here Hv+i/2= H(Vv_q+l,...,Vv+q) stands for the entropy flux difference

Gv+ 1/2- _v+ 1/2; by Lemma 5.1 the entropy flux Gv+ 1/2 is essentially a threepoint one and a similar form of Gv+I/2 implies that Hv+I/2 satisfies the

essentially three-point consistency relation, see (5.6),

(5.15b) H(Vv_q+l,...,Vv_l,W,w,v +2,...,vv+ q) = 0.

Choosing vv_ I = vv in (5.15a) we obtain, in view of (5.15b)
Similarly, taking

(5.17)\I

V

\I

-20-

in (5.15a) yields, in view of (5.15b)

Adding (5.16)" together with (5.17)v+l implies

or, according to (5.9),

(5.18) T

*T' Iiv 1 Q 1 liv 1 < liv 1 Q 1 liv 1 • \1+ /z \1+ 12 v+ 12 - v+ 12 ,,+ /z v+ /z

Thus the scheme considered, (5.4), contains more viscosity than the entropy

conservative one, (5.3), as asserted.

We recall that the entropy conservative scheme (4.9) is a second-order

accurate one. Hence, Theorem 5.2 allows us, in particular, to tune additional

numerical viscosity so that we retain both the entropy stability'and secondorder accuracy [15], [16].

Finally, using arguments similar to those employed in the proof of the

last theorem, we conclude with the follOWing extension of [18, Theorem 6.1],

dealing with systems of conservation laws.

Theorem 5.3: (Entropy stability by comparison).

Conservative schemes containing more viscosity than an entropy stable

scheme are also entropy stable.
-21-

The last theorem provides us with a framework for designing entropy stable

schemes. One could start with any scheme which is already known to be entropy

stable and then tune in additional numerical viscosity according to the above

guidelines so as to obtain a better performance in terms of, e.g, simplicity,

controlled variation, avoiding entropy "glitches" etc. Virtually all the

first-order entropy stable schemes fall within that framework when compared to

Godunov scheme u these are the E schemes discussed in [14], [18].
-22-

APPENDIX

A. AN APPROXIMATE CELL ENTROPY EQUALITY

We consider the scheme

(a.l) d 1

d-_ u (t) .... ]. v hx [fv+ i/2 fv- i/2

Multiplication by U_(uv(t)) on the left gives us

l * *

Adding the term _-_ [F +i/2- Fv_I/2 ] tO both sides, we find on account of

(3.7a)

(a.3)

d I * * 1

_U(u (t)) +_x [F_+I/2- Fv_I/2] - 2hx [(F(Uv+l) - F(U_-l))

+ _ 1[[Uu(Ug+l) - Uu(Uv)]Tfv+ I/2+ [Uu(Uv) - Uu(Uv_ I) ]Tfv_ i/2]"

Integration by parts of the compatibility relation (2.2) implies that the

first brackets on the right of (a.3) equal
-23-

F(u) UTu(U)f(u)[uv+l uv+iT Iuv+l - = f FudU- uT(u)f(U)u

u_ I u_ I u_ I

(a.4)

= f u_+iuU TfU du - UuTf(u) Iu_+1= - f U_+lduTUuuf(U)

u_ i uv- i u_ i

The second brackets on the right of (a.3) can be rewritten as

(a.5) uT(u) U_+l ufv+i/2- UuT(u) ]u_ fV-i/2 = u/ Uv+l duTUuuf + 1/2 - / u_ duTUuuf v -1/2

uv- i _ uv- i

Inserting (a.4) and (a.5) into (a.3) yields the desired approximate cell

entropy equality, see Lemma 3.1,

dd--tU(uv(t)) + _-_1 [Fv*+ i/2- FV_* i/_ =

[ u duTUuu[f _i/2- u + i/2- 1

i v v i

- 2Ax f f(u)] + f duTu u[fv+ f(u)]

uv_ I u v

The above proof essentially follows that of Osher in [16, Section 3]. It

,

differs, however, in that it employs a numerical entropy flux, Fv+i/2,

centered at mesh midpoints.

B. THE VISCOSITY FORM OF ESSENTIALLY THREE-POINT SCHEMES

We consider conservative schemes in the viscous form

d [u(%(t)] = W1 [%+1/2-% i/2]_ 2Ax i [g(%+i)- g(wv_l) ]

(b.1)

1

+ _ [Q_+I/2Av +I/2- Q__I/2Avv_l/2 ].
-24-

The numerical flux associated with scheme (b.l)

gv+ 1/2= Cg(Vv_p+l,...,V+p)

(b.2)

I [g(%) + )] - I = _ g(vv+l _ Qv+l/2 [V_+l - v ].

satisfies the essentially three-point consistency relation

(b.3) Cg(V_p+l,...,V_l,W,W,v+2,...,v +p) = g(w).

Conversely, consider a conservative scheme

(b.4) d-_[u(v (t)] = - A1--x[_g(Vv_p+I'''",Vv+p) - _g(Vv_p'''''V_+p-l)]"

Subject to the essentially three-polnt consistency relation

(b.5) _g(V_p+l,...,V_l,W,W,v +2,...v +p) = g(w)

the scheme (b.4) can be put into the viscosity form (b.l) provided a numerical

viscosity coefficient, Qv+l'-12 ' can be found such that

(b.6) Q_+ i_ AVv+l _ = g(v ) + g(v v+l ) - 2_g(V_p+ 1 '''''Vv+p ).

Using the consistency relation (b.5) we can rewrite the RHS of (b.6) as
-25-

[,_(...,%,v,...) - ,_g(...v 'V_+I ...)]

+ [,Ig, (... ' vv+ 1 '%+z' "") -_ g ('"'%'vv+l'"')]

and equality (b.6) is then fulfilled by setting the numerical viscosity

coefficient to be

i

Qv+l/2= - y _ [@g(... (_) ...)]dE

_=0 _V_+l 'v_'vv+l_ '

(b.7)

i

+ / _ [¢g(...,vv+16 ($)'Vv+l''')]d$" $=0 _v
26  ̧

REFERENCES

I. R. J. DiPerna, "Convergence of approximate solutions to conservation

laws," Arch. Rational Mech. Anal., Vol. 82, 1983, pp. 27-70.

2. K. O. Friedrichs and P. D. Lax, "Systems of conservation laws with a

convex extension," Proc. Nat. Acad. Sci. U.S.A., Vol. 68, 1971, pp. 1686-1688.

3. A. Harten, "High resolution schemes for hyperbolic conservation laws,"

J. Comput. Phys., Vol. 49, 1983, pp. 357-393.

4. A. Harten, J. M. Hyman, and P. D. Lax, "On finite difference

approximations and entropy conditions for shocks," Comm. Pure Appl. Math.,

Vol. 29, 1976, pp. 297-322.

5. A. Harten and P. D. Lax, "A random choice finite-difference scheme for

hyperbolic conservation laws," SINUM, Vol. 18, 1981, pp. 289-315.

6. A. Harten and S. Osher, "Uniformly high-order accurate non-oscillatory

schemes, I," MRC Technical Summary Report No. 2823, May 1985.

7. T. J. R. Hughes, M. Mallet, and L. P. Franca, "Entropy-stable finite

element methods for compressible fluids; application to high mach number flows

with shocks," Finite Element Methods for Nonlinear Problems, Springer-Verlag,

to appear.
-27-

8. S. N. Kru_kov, "First-order quasilinear equations in several independent

variables," Math. USSR-Sb., Vol. i0, 1970, pp. 217-243.

9. P. D. Lax, "Hyperbolic systems of conservation laws, II," Comm. Pure

Appl. Math., Vol. i0, 1957, pp. 537-566.

I0. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical

Theory of Shock Waves, SIAM Regional Conference Lecturers in Applied

Mathematics, No. ii, 1972.

11. M. S. Mock, "A difference scheme employing fourth-order viscosity to

enforce an entropy inequality," Proc. Bat-Sheva Conference, Tel-Aviv

University, 1977.

12. M. S. Mock, "Systems of conservation laws of mixed type," J.

Differential Equations, Vol. 37, 1980, pp. 70-88.

13. J. Von Neumann and R. D. Richtmyer, "A method for the numerical

calculations of hydrodynamical shocks," J. Appl. Phys., Vol 21, 1950, pp. 232.

14. S. Osher, "Rieman solvers, the entropy condition and difference

approximating," SINUM, Vol. 21, 1984, pp. 217-235.

15. S. Osher and S. R. Chakravarthy, "High resolution schemes and the

entropy condition," SINUM, Vol. 21, 1984, pp. 955-984.
28

16. S. Osher and E. Tadmor, "The convergence of difference approximations to

scalar conservation laws," ICASE Report No. 85-28, NASA CR-172614.

17. E. Tadmor, "Skew-selfadjoint form for systems of conservation laws," J____.

Math. Anal. Appl., Vol. 103, 1984, pp. 428-442.

18. E. Tadmor, "Numerical viscosity and the entropy condition for

conservative difference schemes," Math. Comp., Vol. 43, 1984, pp. 369-382.

19. E. Tadmor, "Entropy functions for symmetric systems of conservation

laws," J. Math. Anal. Appl., to appear.

I. Report No. NASA CR-178021 I2. Govefnment Acceuion No. 3. Rt\:ipient's CAUlog No.

ICASE Report No. 85-51

4. T,tle and Subtitle 5. Report Olt.

November 1985

THE NUMERICAL VISCOSITY OF ENTROPY STABLE 6. Performing Organization Code

SCHEMES FOR SYSTEMS OF CONSERVATION LAWS. I.

7. Authorlsl 8. Performing Organization Report No.

Eitan Tadmor 85-51

10. Work Unit No.

9. Performing Organization Name and Addresa

Institute for Computer Applications in Science 11. Contrlct or Grant No. and Engineering

Mail Stop l32C, NASA Langley Research Center NAS1-17070 u "A ?1""<;-<;??<; 13. Type of Report and Pllfiod Covered

12. Sponsoring Agency Name and Addresa Contractor Report

National Aeronautics and Space Administration 14. Sponsoring Agency Code Washington, D.C. 20546

505-31-83-01

15. Supplementary Notes

Langley Technical Monitor: Submitted to Math. Compo

J. C. South Jr.

Final Report

16. Abstract

Discrete approximations to hyperbolic systems of conservation laws are

studied. We quantify the amount of numerical viscosity present in such

schemes, and relate it to their entropy stability by means of comparison. To

this end, conservative schemes which are also entropy conservative are

constructed. These entropy conservative schemes enjoy second-order accuracy;

moreover, they admit a particular interpretation within the finite-element

framework, and hence can be formulated on various mesh configurations. We

then show that conservative schemes are entropy stable if and only if they

contain viscosity than the mentioned above entropy conservative ones.

17. Key Words (Suggested by AuthorIs) ) 18. Distribution Statement

hyperbolic systems

stability 64 - Numerical Analysis

finite differences

finite elements

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19. Security Oassif. (of this reportl 20. Stl:urity On-sif. Cof this page) 21. No. of Pages 22. !'fice

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