Equivalent definitions of conservative finite-difference schemes

100 Continuing this process, we finally construct the entire optimal trajectory from (3.4).

Eq.

5. tit ezumptes. Let the motion of a point be described by system (l.l), where a~@?', with the initial condition ~(t~)-m.,t~-~~-0;the resource constraint is a-1. Consider the problem of the shortest-timetravel of a point of the system (1.1) through the goal sets (3.11, where K,-&(~,,a,),~==l, Z,...,m, is a sphere of radius ck in the threedimensional space centred at the point r,. In order to evaluate the Bellman function, we in each setS,.By (3.2) and (3.31, it choose by some rule I points {a?},k=i, 2,...,1, I<-, K,, i--l, 2,...,m.; we suffices to construct a grid of 1 nodes on the surface of each-sphere The numerical values for 'R-(~,~,T~~, r,J and a,,i-&_&..., m,m-5, are given in take 1498. Table 1. First consider the version when the system ,?i!' consists of four (m==4) first objective sets Kc, i--i,..., 4 (see Table 1). Find the value of the function (3.4) and the optimal sequence of visists to the sets of the system 2 for various initial positions z, in the space C?'. The results of computer calculations are presented in Table 2. Here &(H,z.),His the optimal traversal (I,2, 3, 41, is the value of the Bellman function; J=(r,,ra,rr, rr) sequence of the system of sets X, r,E(1,...,4), i=1,...,4;z,, i=&...,4, are the times when cp(Q m*,U) EaK.4. The optimal trajectory is the polygonal line formed by straight segments successively at The values of the vector function (P(.,x., U) joining the points (P(T,, x.,U), i--l, 2,...,m. the times r=r,,i=l, ....4, are given in Table 3. The values of Let us now consider the complete (m=5) system of sets from Table 1. the function (3.4) and the optimal trajectory visiting the sets KC of the goal system.% for are given in Tables 4 and 5. m=5, i=l, . . . (5,

REFERENCES 1. KUROSH A.G., A Course in Higher Algebra, Nauka, Moscow, 1968. 2. BELLMAN R., Application of dynamic programming to the travelling-salesmanproblem, Kibernet. Sb., 9, 219-225, Mir, Moscow, 1964.

Translated by 2-L.

U.S.S.R. Comput.Matks.~atk.Pkys.,Vo1.29,No.4,pp.100-110,1989 Printed in Great Britain

0041-5553/89 $lO.OO+O.OO 01991 Pergamon Press plc

EQUIVALENT DEFINITIONS OF CONSERVATIVE FINITE-DIFFERENCE SCHEMES* V.V. OSTAPENKO The equivalence of three definitions of conservativity is proved for a homogeneous finite-difference scheme on a uniform grid without a The first of these definitions is associated with boundary. "summation" of the difference scheme over the grid domain, the second is associated with the representation of the scheme as the sum of first-order finite differences (in canonical form), and the third with the divergence of its differential representation. The conservativity of the difference schemes in magnetohydrodynamicsis analysed, and the results are employed to construct a new explicit conservative scheme using a simple approximation of the internal energy equation. 1. Basic definitions. 1. The property of being conservative is one of the most important properties of finitedifference schemes approximating conservative systems of non-linear differential equations /l-4/. If the conservativity property does not hold, the discontinuous limit solutions of the difference scheme cannot be expected to satisfy the laws of conservation of the approximated system /5-0/. In this paper, we prove the equivalence of three definitions of conservativity for a homogeneous difference scheme of fairly general form, defined on a uniform grid without a boundary. Let us state these definitions. In the plane of the variables I, t, we introduce a uniform rectangular grid without a boundary Wh'=={x,=ih, &jz; i,j=Z; h,z=const), %?k.vyckis'l.Hat.mat.Fiz.,29,8.1114-1128,1989

101 where k and T are the grid spacings in space and in time, and H Define a homogeneous difference operator on the grid,

is a set of integers.

where Vb=-V(z,, t’), V=(v, ,..., ur) is the grid vector function, A is a given vector function, and n, and nI are non-negative integers. We assume that the function A may be independent of some of its (Zn,+l) X (b,-l-l)-vector arguments V/z,, where -~
-n6kCn,,

--n,c%h,}.

U.2)

We introduce the following basic definition. Definition 1. The difference operator (1.1) is called a divergence difference operator, and the corresponding scheme (1.3)

A,,"*[ VII-0 is called conservative if: 1) A.,"~[V]-0 for V=const; such that N,-N,>2n,,

2) for all integers N,,N,,M,, ME

MI- M,>2n,,

the sum

(1.4) is independent of the values of the function V,’

from the grid rectangle

n=(~~,t'; Nl+n,
~l-hr~ZGW,-n,}. (1.5) From this definition it follows that, like a differential divergence operator, whose integral over a region is expressible as a line integral over the boundary of the region, summation of a difference divergence operator over a sufficiently large grid region retains only the terms that are evaluated on a closed band of finite length along the boundary of this region. The summability property of a difference scheme, as one of the basic properties of conservative difference schemes, is given, e.g., in /l-4/. This property is the basis of Definition 1. Also note that the definition may be regarded as a formalization of a common requirement (see /l, 4/j that a conservative scheme (1.3) should satisfy some difference conservation laws. 2. Denote by T,1 the shift operator whose action on each difference operator (1.1) is defined by the identity T,‘oA[V,tl=A[V:,:‘].

Introduce the shorter notation T,=T,O,

T’=T,‘,

E=To’.

Definition 1’. The difference operator (1.1) is called a divergence difference operator, and the difference scheme (1.3) is called conservative, if the operator (1.1) can be represented in canonical form as An,(ll[ V,‘]=(T,--E)d,[V,‘]+(T‘-E)oAz[ V,‘], (1.3) are difference operators of the form (1.1) such that A,[MJ where A,,& is independent of for all k such that --n,dk
%b(W))+ az

Wu(z,t)) at

=.

(1.7)

in the sense of the conditions A,[Vl=cp(V)/h,&[Vl=$(V)lzfor V=const,

(1.8) then its discontinuous limit solutions satisfy the conservation laws of the approximated system (1.7). Subsequantly, some authors (see, e.g., /lo/) started referring to difference schemes (1.3) with operators satisfying (1.6), (1.8) as Lax-Wendroff conservative. One of the main results of this paper is the following. Theorem 1.

Definitions 1 and 1' are equivalent.

102

For the special case when the operator (1.1) is a polynomial in the components of the function V, the equivalence of Definitions 1 and 1' was proved in /0/. 3. Wow assume that the operator (1.1) has the form

(1-Q) Lb, jti, rtnr rd=Z, %'-con&, -n,
The differential representation &,l.‘[V(x, formal differential operator

t)] of the difference operator (1.1) is

the

(1.11)

obtained from the operator (1.9) by replacing the shift operators with their differential representationsfrom (1.10) and also replacing the grid functions (v,,,)/ with infinitely differentiable functions u,,,.(z, t) of continuous arguments x, t. Transforming the operator(l.11)by the nultiplication and summation rules for formal series /12/, we represent it in standard form as

where Pas are differential operators independent of h and T. Some of the operators F., may be identically zero. The differential representation (1.11) is said to be divergent if all the differential operators F,p occurring in its standard form (1.12) are divergent. The following definition is based on an idea of N.N. Yanenko. Definition I”. The difference operator (1.9) is said to be divergent, and the corresponding scheme (1.3) conservative, if the differential representation (1.11) of the opertor (1.9) is divergent. We have the following theorem. Theorem 2. For difference opertors (1.1) of the form (1.9), Definitions 1 and 1' are equivalent to Definition 1". This theorem implies that for difference operators of the form (1.9) all the three definitions, being equivalent, simply represent different aspects of the same property of being conservative. Note that the equivalence of Definitions 1 and 1" is formulated in a somewhat simpler form in /7/. 2. Proof of Theorem 1. 1. 2a)

Lemma 1. the sum

Condition 2 in Definition 1 is equivalent to the following condition:

is independent of the values of the function V,’

at the node k--1-0.

Proof. It suffices to show that condition 2a implies condition 2 of Definition 1 (the converse is obvious). Let (k, 1) be an arbitrary node in the rectangle (1.5). Then the sum (1.4) can be written in the form S=T,‘&+(S-T,‘S,).

The operator T.W, is independent of V,’ by the assumption of the lemma, and the operator in parentheses is independent of VI1 because its pattern does not contain the node (k,l). Since (k,Z) is an arbitrary node in the rectangle (1.5), the sum S is independent of the values of the function Vi from the entire rectangle (1.5). The lemma is proved. It follows from this lemma that condition 2 in Definition 1 can be replaced with the

103 simpler condition 2a. 2. That Definition 1' implies the conditions of Definition 1 is obvious. In order to prove this, it suffices to sum the difference operator (1.6) over the corresponding grid region. We will now show that condition 1 of Definition 1 and condition 2a of Lemma 1 imolv - _ the condition of Definition 1'. We will first prove this for the one-dimensionalcase. Lenmm 2.

Let the difference operator

VFV(k),

A.[V,l~h(V,,V,+,,..., V‘,J I

xc-ih,

(2.1)

satisfy the conditions Am [VI -0 npu V--con&

(2.2)

and let the sum

(2.3) be independent of the values of the function V, can be represented in canonical form

at the node k-0.

Then the operator (2.1)

A”[v&=v,-m&v,1, where &[ V,]

(2.4)

is an operator of the form (2.1) independent of V c+.?.

The proof is by induction on n. to the following:

For

n-l, conditions (2.2) are (2.3) are equivalent j2.5)

A,(z,z)=O, the function S(s, y, z)=At(z, y)+A,(y,a) for all I, y, z is independent of y. The second of these two conditions can be rewritten in the form Al&y) -A,(z,O)+At(g,z)-A1(0,z)-0.

(2.6)

Hence it follows that the difference A,(s, y)-Ar(x,0) is independent of I. A,kO)-f(r), we obtain A,(r,~)=f(r)+g(y), where g(y) is some function. Substituting (2.7) into (2.6), we obtain 0. Putting Chg(O)+f(O), we obtain

Therefore, setting (2.7)

g(y) -g(O)+f(y)--f(O)=

f(y)==-g(y)+C. Substituting (2.8) into (2.7) and using (2.5), we obtain C=O, A1(z,y)=g(y)-g(s). z-v,,y-v,,,, we obtain the formula

(2.8) Setting

A,(V,, V,+,)=g(V<+,)-g(V<)-(TrE)v(V,). We have thus proved the lemma for n=l. In general, condition (2.3) can be written in the form ~I~~l=Sw~lIVI-O.

(2.9)

From (2.9) we obtain that the difference

A.(Po,..., V,)-A.(O, is

independent of

V,,

V,, . . . , v,,)

and therefore &(I',,..., V.)-At(Vo,...,V.-J+Ar(V,,...,Vm),

(2.10)

where A*(V,,...,V.)=h(O, Vt,...,V,), and A, is some function. Substituting (2.10) into (2.9), we obtain after some reduction S[V,l=S[ VA L-0,

(2.11)

where 0 s[vJ=

r,

An-,(J’t, V,,,,

I--"+,

. . . , V,+.-,),

(2.12)

An-,(Vo,...,V,-,)=A,(V~,..., V,-,)+A,(V,,...,v,_,). The identity (2.11) is condition (2.3) for the operator

A.-,[V,l==A,-,(I',, Vs+,,..., V,+.-,).

(2.13)

Identities (2.10) and (2.12) imply that condition (2.2) is also satisfied for this operator. Assuming that the lemma has been proved for the operator (2.13), we obtain An-l(Vo,. . . , VW.,)=Li(V‘, . . . , v.-,)--Li(V0,. . . (V”_.,), where I,,-, is some function. Wow substituting A,=-AZ-!-An--( into (2.10) and using (2.14), we obtain

(2.14)

104 h(V*,..

.,

V,)-YMV,,.. .* v&i.rVo,.. . , v*-a),

where h(V,,...,v,-,I-AdVo,.

. . , V,-I)+&-00,.

. . , V,-2).

Induction with respect to n completes the proof of the lemma. is an even number, and consider the difference operator Now assume that n-2m

e,[V,l--h~V~,l~h~V,-,,

Vf-In+,, . . . , v,+m.).

Conditions (2.2) and (2.3) are equivalent to ttiefollowing conditions: Q,[V]-0

for V--con&,;

(2.15)

the sum II) (2.16)

&uV‘l

s-

‘--

is independent of the value of the function V, at the node k-0. satisfying conditions (2.15) and (2.16) can Lemma 2 implies that the operator Q,[V,] be represented in canonical form as n,[v,l-h[~-,I=(T,-E)~T-,~~.~v,l. This proves Theorem 1 for the one-dimensionalcase. V.) are continuous differentiable functions of Remark. Assuming that A\.(VO,..., arguments, condition (2.3) of Lemma 2 can be rewritten in the form

their

0

8

z

sv,

b(Vi,

Vd,i,. . . v<+fJ-o.

(217)

I

,---I

For the case n-2, i.e., for operators defined on a three-point pattern, the equivalence of (2.2) and (2.17) to condition (2.4) was proved in /13/, where the corresponding equality (2.17) was proposed as a test of the divergence property of particular difference operators. 3. Let us consider the essentially two-dimensionalcase. We will show that if the operator (1.1) satisfies the conditions &,“*[V]=O

it 1--l%, I-4.

for V=const, (lin,“[

(2.18)

VA-A?[ VA IV&-o) -0

(2.19)

(where (2.19) is equivalent to condition 2a of Lemma l), then the difference operator (1.1) can be represented in the canonical form (1.6). Direct application to the two-dimensionalcase of the formal method of proof previously used for the one-dimensionaloperator is extremely complicated. Therefore, for the twodimensional case, we will use a more constructive proof, which can be extended to any multidimensional case without significant modifications. Denote by V[W(J the set of values of the grid function V,’ on some grid patternWi=X,? Definition

2. TWO pattern

W&

and

W,:,,

will be called analogous if they can be

made to coincide by parallel translation, i.e., if integers k,Z exist such that W,:~:+h-W,:,,. Definition 3. A function of m arguments is called decomposable if it can be represented as the sum of functions each dependent on at most m-l arguments. A function of a single

argument.is assumed to be decomposable if it is identically equal to a constant. Lennna 3.

The difference operator (1.1) can be represented in the form

(2.21) (2.22)

where all the functions & Proof. Ti,=~l.l,~,-O,

in (2.21) are non-decomposableand no patterns W{,,,

If the operator (1.1) as a function of

V[X,l]

are analogous.

is non-decomposable,then setting

we can directly represent it in the form (2.20). If the operator (1.1) is

105 decomposable, then it can be written in the form

the where the patterns of the operators A, are proper subsets of the pattern (1.2) of operator (1.1). If some of the operators A, are decomposable, then we also decompose them into a sum of the corresponding operators, and so on. If the process of decomposition produces operators defined on equal patterns, then their sum is subsequently treated as a single operator A,. defined If the decomposition process produces the operators A,(V[I+'t'l), A#'[$+'& I), on analogous patterns, then their sum is transformed as follows:

The operator A, is excluded from further transformationsand & continues participating as a single opertor. As a result of successive application of these transformations,the operaboor(1.1) will eventually take the form (2.20)-(2.22).The lemma is proved. Lennra4. The operator (2.22) can be represented in canonical form (1.6). Proof.

Using the identity l-i

T,-E-

(T,-E)

ox

T,,

h-0

we transform the operators Thb-E

in (2.22) to the form

im-1

f-1

(T' - E) 0 Tim0 y

Tk+ (T, -

E)

0

7

T,.

Substituting (2.23) into (2.221, we obtain after simple reduction the operator (2.22) in canonical form. The lemma is proved. Lemma 5.

If conditions (2.18) and (2.19) are satisfied, the operator (2.21) identically

vanishes. Proof. Let n, n-mu

n,.

I

be the number of nodes in the pattern

Assume that n24.

.WU:( in the operator (2.21) and Without loss of generality, we may assume that the pattern

of the operator A, contains n nodes. Since the operator (2.22) satisfies the

Q,‘

con-

dition (2.19), the operator (2.21) should satisfy the same condition, i.e.,

Consider this identity on the set of functions V?

whose support is the pattern W-W&,:_,,

where (9, t') is some node of the pattern W(,,i. In other words, the functions that V&=0 for all (z,,P)O!V. such functions in the form

V/ are such

Wow rewrite the identity (2.24) using (2.21) on the set of

A,(Vt~l)-A,(V[Wl)Iv,-,+ll’[V(l,

(2.25)

where A is the sum of functions each dependent by construction on at most n-l arguments N: (zg, &EW. The last point follows from the fact that there are no analogous pattern in (2.21), and also from the fact that each of these patterns contains at most among W&, as many nodes as the pattern W-T_,-‘oW,&,. Since

of n-l

the node (O,O)EIV, the first term on the right-hand side of (2.25) is a function arguments, VI': (zr, t+w, (i,I)#(O, 0).

Thus, formula (2.25) is a representationof the function A, of n arguments V,’ in the form of a sum of functions of fewer arguments. This contradicts the assumption that all the

106 &in

(2.21) are non-decomposable. has led to a contradiction. Therefore, n-0. In this Thus, the assumption that n>l degenerate case, the operator (2.21) is independent of the values of the function V? and is a constant X,=C.

Since the operator (2.22) satisfies the condition (2.181, the operator

The lemma is proved. (2.21) should also satisfy the same condition. This means that x,=C=O. Successively applying Lemmas 3-5, we obtain that condition (2.181, (2.19) imply that the operator (1.1) can be written in canonical form. We have thus proved the equivalence of Definiton 1 and 1'. 3. Proof of theorem 2. 1. That the condition of Definition 1' implies the condition of Definition 1" follows directly from the fact that the standard form (1.12) of the differential representationis independent of the form of the difference operator and that the differential representations and T’-E are divergent. of the operatorsT,-E 2. We will now show that the difference operator (1.9) whose differential representation is divergent can be written in canonical form (1.6). This will require an auxiliary definition. is called minimal if it is obtained from Definition 4. The difference operator A[ Vi] by replacing all the elementary differential operators some differential operator P[V(x,t)] with the corresponding elementary finite-differenceratios P/al+,ayat’, k, 01, T,‘~(T’-E)[/rl

T$ ( T,-E)L/hk,

with the grid function VI-V(z,,tf). and by replacing the function V(r,t) The minimal operator h is clearly defined on the minimal pattern which admits of difference approximation of the differential operator F. The follow lemma is of importance. Lennnd6.

a

Each difference operator (1.9) can be represented in the form

(3.1)

N-l; for all a, S, the operator hs where k,EZ, k,+,>k,+i for i=O, 1,..., ically zero or is the minimum operator for some differential operator % and .T.

is either identindependent of k

Proof. We will construct the operator (3.1) using the standard form (1.12) of the differential representation (1.11) of the difference operator (1.9). Denote (1.9) by &, and consider the minimum difference operator

(3.2) where &s are the minimum difference operators approximating the corresponding differential operators F;, in the operator -

Fo-

L

h”?F

all,

which in turn is a component of the operator (1.12). NOW construct the operators &=Q,-A,,. Since the operators Q/h* and ,AJhb both conditionally (given z=h) approximate the differential operator F,=F,lha

1r_,, = r, FOrb, 0+f.-5

the differential representationof the operator 8, has the form

where kf>ko+l,F$ are differential operators independent of h and z. Since the operator is minimal for FO, its pattern can always be chosen so that it is included in the pattern & (1.2) of the operator (1.9). Choosing the pattern of the operator & in this way, we obtain that the pattern of the operator 52, is also always contained in the rectangular pattern (1.2). Now construct the minimum operator A.1 = r, ha?&, S+b-*,

(3.3)

107 where hs

are the minimum difference operators for the differential operators F%

occurring

in the operator

Choose the pattern of the operator (3.3) so that it is contained in the pattern of the operator 62,. Then construct the operator 61,-&-A, whose pattern is contained in the pattern of the operator f& and therefore in the pattern (1.2) and whose differential representation has the form

where k,>k,+l,

Fj$

are some differential operators independent of h and 7.

We have thus obtained an iterative process for constructing the operators 9,, F,, and A. Mow note that an unlimited increase in i will produce an unlimited increase in the order of the operators Fc and hence the size of the minimum pattern for the difference approximation of the operators (3.4)

i-+00 the pattern of the operator 62,should also increase without Hence it follows that as limit, because f&/h' for all i conditionally (for s-h) approximates the operator (3.4). This is impossible, however, because for all i the pattern of the operator Q, by construction such that I 62n+r-0 is contained in the pattern (1.2). This means that there exists N>O and therefore

The lemma is proved. Now assume that the differential representation (1.11) of the difference operator (1.9) Fas in (1.12) are is a divergence. This means that all the differential operators divergence operators. But then the operator A0 by construction can be represented in canonical form (1.61, and therefore, using the previous results, we see that its differential representation is also divergent. Thus, the differential representation of the operator S& is divergent. Repeating this argument successively for i=i,2,...,N, we obtain that for all i the operators A, can be represented in canonical form (1.6) and therefore the operator (1.9) whose differential representation is divergent can also be represented in canonical form. This completes the proof of Theorem 2. 4. The coneeruatiuity of difference schemes in mugnetoh@odpamics. 1. Definition 1 was applied in /a/ to obtain a fairly simple test for the divergence property of particular difference operators. We will apply this test to identify all the conservative schemes in a given family of MHD schemes that use a simple standard approximation of the internal energy equation. The system of MHD differential equations for the one-dimensionalplane case under the assumption of an ideally conducting gas with the magnetic field vector always lying in one plane can be written in the form /4, 14/

(4.la)

all -a-

at

au a8

aw-4 _ be at

as'

(4.lb) (4.k)

where 8 is the Lagrangian mass coordinate, v,b=constare the longitudinal components and u, H are the transverse components of gas velocity and magnetic field, respectively, p is the pressure, n-l/p is the specific volume, e is the internal energy of the gas, and ~-(8x)-'. From (4.lb) we can obtain the equation /4/

(4.2)

108 which describes the variation of the specific magnetic energy with time. Adding the Eqs.(4.la) multiplied by v and u respectively and the Eqs.(4.lc) and (4.2), we obtain the total energy equation

a

V’fU

-++aqEP at(e+ 2

) - - $1

(p+aIfy v-2abHu1,

whose conservative approximation is a necessary condition for the MHD difference scheme to be conservative. 2. Consider the difference scheme which provides the standard approximation /15/ for system (4.1): (4.3a) (4.3b) (4*3c) where a, are temporal weights. This scheme is written using the standard notation from /4/. Eq.(4.3b) leads to the difference analogue of Eq.(4.2) (anET),=-aHZi%~' +2abHio%!a’. Summing this equation with Eq.(4.3c) and Eqs.(4.3a), multiplied respectively by &XI), we obtain a total energy equation which is conservative if a2=0,

aJ=0.5,

as=lr

[email protected]) and

(4.4)

and also if one condition from each of the following pairs is satisfied: ai-a7,

as-O.5 and al+aa==l,a,-0.5,

a,==O.5, a&=0.5,and a,fos-1.

(4.5a) (4.5Li)

We thus obtain four families of conservative schemes (4.3). The first conditions in (4.5) correspond to the fully conservative schemes constructed in /4, 141. 3. From (4.4) it follows that none of the conservative MHD schemes constructed in this way has an explicit realization. Therefore, consider a non-conservativescheme (4.3) with weights ai=an=as=a,-0, a,=as==aa=l,a,-0.5. (4.9) This scheme satisfies the second conditions in (4.51 and does not satisfy only condition (4.4). As a result, the corresponding total energy equation is non-conservative. It may be written in the form e+-+a+

1I -_[(p+&)

LabH@.‘)l, - f[

(p+afP)v,--2abHu.

(

v’+u’ 2

y(@.‘l _

(4.7) Jr+EE,

where &E* $(H,)'v:",

(4.8)

and h and T are the space and time spacings of the scheme. Therefore, to ?-m-h t) 9 make the scheme (4.3) with the weights (4.61 conservative, the internal energy equation must be taken in the form ~~=--p~~‘)v~~) --6E, where the non-divergent total energy imbalance (4.8) has been subtracted from the right-hand side. As a result, we obtain the conservative scheme v*=-(p+aP)r, (II-V.

(1)

e,=-

I

[

~r=2abH;,

($2) t-but’),

p’05)+ ‘+(H,)‘]

vi”.

This scheme can be regarded as an analogue of the "crossl'scheme for the equations of gas dynamics /16/. It admits of an explicit realization for the polytropic state equation, it is Courant stable in the linear approximation,and ensures second-order approximation for an appropriate choice of the pattern. The total energy equation corresponding to this scheme is conservative and has the form (4.7) for 6E-0. Using the technique of /16/, we rewrite this equation in symmetrical form [e-i-0.5 (vv~‘)+tzU(‘)) toJ,+aqHP],=--[

(:+a&)

$‘:; rP)-2ab@~,q’tP],,

109 where f~ol,-[f(z+h,t)-f(s,h)]/2. 4. As a test example, consider the problem of a piston pushed into a gas with constant velocity and generating a fast MHD shock wave. In Eqs.(4.9a) and (4.9c) we introduce linear artificial viscosity by replacing the pressure p with C=const; v=Ch, g=p-vpv., we take the equation of state in the form e=pq/(y-4). the following values: v(8,0)--u(s, O)=p(s,O)=e(s,

The gas parameters for t-0

O)=O,

have

q(s, O)=l,

H(s, 0)=-6"/5, b==(2n)".

~(0,t)=-2"/5 is given on the piston. The flow parameters behind The velocity ~(0,t)='/,, the shock are calculated from the Hugoniot relations

vi=‘/,,

u,=-2"/5,

p,=O.Ol,

q,=V.,

H,=8n"l5,

e,=0.01125.

The shock wave velocity is D-l.

(This test example is taken from /4, p.256/). The dots in Fig.1 plot the solution of this problem by the explicit conservation scheme (4.9) with the parameters h-O.&r= The solid 0.06, v=O.O5, and r -"/,. The solution time was T=9. line plots the exact solution. Open circles and triangles plot the results obtained for the same problem by the scheme (4.9) with Eq.(4.9c) replaced respectively by e,=-pw%$r) (open circles) et=-pvf)

(triangles),so that the corresponding schemes were

non-conservative. The test results show that the conservative scheme (4.9) reproduces with sufficient accuracy the gas parameters behind the shock and therefore can be used to calculate discontinuous MHD flows. 5. Let us derive some theoretical limits on the imbalances in determining the gas parameters behind the shock 6f=fc--f, (see Fig.11 and GD=D,-D in determining the wave propagation Fig.1 velocity, associated with the use of the non-conservativescheme (4.3). Here f, and D are the exact solution parameters, and f. and Do_ are the values obtained in the numerical solution. We will only examine the special case b=O,$f=const. Applying the method of /0/, we find that the imbalances 6f and 8D at the shock (assuming that they are small compared with ft and D) are related by the following approximate relationships: D8v+v~8D=8p+2aH,MI,

(qo-qd8D=D8r)+60,

(4.10a)

(r--1)6e=p,6t1+n16p, 6p=-p,%n, 6H=-H,9,8q,

(4AOb)

[e,-ea+v,‘/2-tarluH,(H,-Ho)

(4.lOc)

]8D+D(8e+v18v-!-aHoq,6H)-

(p&aH,?Gv-vt(8p+2aH,8H)=J,

where fo

are the gas parameters before the shock (~~-0). For hu+O, J is determined by the formula

~=l-(ar+o,)/2-a‘,

la

where h=w-a,-a,+0.5,

~(r+lP* -_[hD"(A-q'B)-2(h-~)aH,"p,Z(qoq,B-q'A)],

(4.11)

2v

where A=(Q-qr)‘i6, q’-A,q-B,=O,

B=(~o”-qt”)/2-~~11

A

_

’ For

k=p=O

ln(llo/rlt).

Here

tit, q’

are the roots of the equation

where

Y-1

I

r+1

2m+WH, (y+l)D*

B

’

’

=2(2-‘f)aqoHoZ (y+l)D*

’

J is determined from

(4.12)

F,(q)+ where

L=b-ul)(u.+ur-I),

TFiz(q)]

P~=(o~+u,-I)/~-u~u~,

n'+rlorlNlFfi(q). Tables 1 and 2 compare the imbalances

dq,

~~(~)=(q~-q)(q-q,)

X(r)+,

F,(q)=[2r13-_(rlo+7)r+q')

Sf, determined theoretically from formulas (4.10)-

110 (4.12) with the imbalances obtained by calculation 6fo using the non-conservativescheme

where h-0.1,r-0.02, and v-0.05. Table 1 Imbalantes I!;

11

I

8%::

P

I

-0.00636

-0.oc684

P KE:

8 %Z

a

I

-0.0113 -0.0123

Table 2

Imbalances

The shock wave is generated by a piston pushed into a uniform stationary cold gas (qOspecified on with constant velocity (u,-u,-0.5*8v=O) vo=po=Eo=o, &==X'h, 7=2=++=0) the piston. The gas parameters behind the shock and the shock velocity are determined from iiugoniot 1s conditions q,==0.5, P,=‘/~, el=g/,6r H,=2nH, D--1. Table 1 presents calculations for a--l*h-0.5, p==o and with J obtained from (4.19) (J=O.OOl986). Table 2 presents calculations o=O.li=+)t==-p-hi-O, ~~-0.5, and with J obtained from (4.20) (J=-&75x10-'). For the first for case, 8Dt=O.O0313, and for the second 80,~-1.4 x IO-'. Comparison of the imbalances 8ft and Sf, confirms our previous conclusion that conservativity of the difference scheme (4.9) is ensured by augmenting the internal energy Eq. (4.9c) with a non-divergent term (4.8) of second order of smallness in z.

po=l,

REFERENCES 1. SAMARSKII A.A., Theory of Difference Schemes, Nauka, Moscow, 1983. of Quasilinear Equations, Nauka, Moscow, 2. ROZHDESTVENSKIIB.L. and YANENKO N-N., System 1978. 3. GODUNOV S.K. and RYABEN'KII V.S., Difference Schemes, Nauka, Moscow, 1973. 4. SAMARSKII A.A. and POPOV YU.P., Difference Methods of Solving Problems in Gas Dynamics, Nauka, Moscow, 1980. 5. IIKHONOV A.N. and SAMARSKII A.A., On the convergence of difference schemes in the class of discontinuous coefficients, Dokl. Akad. Nauk SSSR, 124, 3, 529-532, 1959. 6. PIKHONOV A.N. and SAMARSKII A.A., On homogeneous difference schemes, Zh. vychisl. Mat. mat. Fis., 1, 1, S-63, 1961. 7. 3STAPENK0 V.V., On the conservativityof finite-differenceschemes, Dokl. Akad. Nauk SSSR, 284, 1, 47-50, 1985. 8. 3STAPENKO V.V., On the convergence on a shock wave of through difference schemes invariant under similarity transformations,Zh. vychisl. Mat. mat. Fis., 26, 11, 1661-1678, 1986. 9. LAX P.D. and WENDROFF B., Systems of conservation laws, Comm. Pure Appl. Math., 13, 2, 217-237, 1960. 10. DI PERNA R.J., Finite difference schemes for conservation laws, Comm. Pure Appl. Math., 25, 1, 379-450, 1982. 11. SHOKIN YU.1. and YANENKO N.N., The Differential Approximation Method, Nauka, Novosibirsk, 1985. 12. CARTAN H., Elementary Theory of Analytic Functions of One and Many Complex Variables, Izd. IL, MOSCOW, 1963. 13. KUZ'MIN A.V. and MAKAROV V.L., On an algorithm for constructing fully conservative difference schemes, Zh. vychisl. Mat. mat. Fiz., 22, 1, 123-132, 1982. 14. PGPOV YU.P. and SAMARSKII A.A., Fully conservative difference schemes for the equations of magnetohydrodynamics,Zh. vychisl. Mat. mat, Fiz., 10, 4, 990-994, 1970. 15. OSTAPENKO V.V., On standard approxmation of differential operators, Dinamika sploshnoi sredy, Inst. Girdodinamiki SO Akad. Nauk SSSR, Novosibirsk, 72, 67-03, 1985. 16. TROSHCHIYEV V.E., On the divergence of the"cross" scheme for the numerical solution of the equations of gas dynamics, Chisl. Metody Mekhan. Sploshnoi Sredy, ITPM, SO Akad. Nauk SSSR, Novosibirsk, 1, 5, 79-91, 1970, Translated by Z.L.