method for numerical computation of discontinuous solutions of the equations of fluid dynamics Sergei Godunov, I. Bohachevsky

To cite this version:

Sergei Godunov, I. Bohachevsky. Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Matematičeskij sbornik, Steklov Mathematical Institute of Russian Academy of Sciences, 1959, 47(89) (3), pp.271-306. ￿hal-01620642￿

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. MATEMA TICHESKII SBORNIK

Finite Difference Method for Numerical Computation of Discontinuous

Solutiona of the Equations of Fluid Dynamic•

1959 v. 47 (89) No.3 p. 271

S, K. Godunov

Translated by 1. Bohacheveky

Introduction

The method of characteristics used for nume_rical computation of solutions of fluid dynamical equations is characterized by a large degree of nonetandard­ _ nesa and therefore il not suitable for _automatic computation on electronic computing machines t especially for problems with a large number of shock waves and contact discontinuities.

In 1950 v. Neumann and Richtmyer (1) prOp

In 1954 Lax (Z) published the "triangle'' acheme suitable for computation across the shock" waves. A deficiencfof this scheme is that it doea not allow computation with arbitrarily fine time steps (as compared with the space steps divided by the sound speed) because it then'tranaforma any initial data into linear functions. In addition this scheme smears out contact discontinuities.

The purpose of this paper is to choose a scheme which is in some sense best and which still allows computation across the shock waves. This choice is made for linear equations and then by analogy the scheme is applied to the general equations of fluid dynamics.

Following this scheme we carried out a large number of computations on Soviet electronic computers. For a check, some of these computations were compared with the computations carried out by the method of character· istics . The agreement of results was fully satisfactory.

1 I have found out through the courtesy of anenko that he has also N. N. Y i es ated scheme for the so ution of equations of fluid dynamics which nv tig a l osely related to the scheme proposed in this paper is cl .

io Chapter I Finite Difference Schemes for Linear Equat ns

. on difference schemes § 1 A new requirement

s olve the differential equations of mathematical physic s one To often use s the method of finite differences. It is natural to require of the solution obtained by an approximate method that qua itativ behavior sh_ould be similar its l e to the behavior of the exact solution of the . Such a require­ m nt however, is not always satisfied. e , For example, consider the

�!.t. ----i1za �t � "/.�

initially the te pera ture li.. a monotonic funct ion of � then, clearly, it If m is will remain such for all later times. When solving this equation by a finite difference scheme, even though it be stable and sufficiently accurate, �t may happen that the tempe rature u.. which is monotonic initially will dev lop a e maximum or a m nimwn at some late r time. i As an 'example c ons er the scheme: id

where u,,� the value of temperature at the point is u. whose coordinates are rt l" • scheme is for po sit ve • :1'-.rmh, = t This stable all i I' •tjh2 Prescribe the following initial conditions:

• m a,� 0 for > 0, (/ m r 1/l" .. ! lor "' 0.

2 After the first time step we obtain for the quantity t./ an infinite system m ·of equations which when solved yields:

1 t ye + f 1 21• r r r - m < 0, a m f' - . 9-"'or ., f' .,. + 2 t + (21' l r ' + f' (21'-�' 1")"'-1 = 2ft T()f' m o. lJ..'"' ( - � 2r.;.t+f2f'+f r 7

0 For m ten ding to t- ooJ tends to and f or m tendin to u.;, g tend s to l. It is not difficult to show by, an analysis of �e above -oo, tl.� solution that its monotonicity will be always violated for f' > 3/2 . s It is na tu al that f or this cheme should not cons idered r r > 3/2 be as a satisfactory one. However it must noted that the effects connected be with nonmonotonicity will appear only in the solution of problems with sharply varying

initial condition s. S ooth solutions will be computed by this scheme with m suffi cient accuracy with a s icien tly fine mesh. uff Analogous fac ts obtain also for difference scheme s devised to

solve the equation

au. au. ----· at b% It is well known that the. solution of this equa tion has the form of a stationary * wave ll..= and if was monotonic for will remain so u..(zrt), U.. t •0 it afterwards.

Let us examine examples of difference schemes for thi s equation and verify whether they preserve monotonicity of solution.

The "triangle" scheme of first-order accuracy: 1.

A "stationary" waves is defined as one which is stationary a coordinate system moving with th e wave velocity. in

3 * o _u.....;, r- u_._t _"t"_ • u IJ. • __ + {u. t - -I) . 2 2h

� It cleaily ca n be rewritte n as follows:

-· U··l' 2

where the stability condition this scheme: Consider the r =t:/A ( for r '1) . conditions for in the form of a step initial t = 0 function:

u.k. - 1 toT' l > I,

compute We obtain and � for t • t: .

ali = 0 for i � - t,

I f ., , 2

lt"1' uf = z

ty in thi s case Since for , we conclude that the f'.( f, I r 11 ( I monotonici z is not violate d.

we U0 Here and in the following shall denote • t.L{t0, X0), U.IJ=t.L{t0-t- z:, �.J, = LL.1 etc.; steps "-t u(t0• �0 rh), �u(t0, X.-h) � and h =time a.nd space respe ctively,

4 An arbitrary mono on function on mesh of size can t ic the h be represented as a sum of step functions each of which changes its value only

in one mesh interval and such step functions are either increasing or

decreasing Using is fact we may conclude that a le . tq the " tri ng " scheme transforms an arbitrary monotonic function into another monotonic function.

The scheme tripod of second order accuracy: 2. " "

This scheme is stable for again take the step function f' ' I . If we

f tli. = 0 . fo,. � " 0

u,� - t lor � � I

initial data at then from this scheme obtain at for t = 0 we t = t:'

, 1- .,..z u,O - , 2

r- f'2 , u,'- I+ ---- 2 ,.

· · · · Sl'nce r#!2 for ' · , th en "1"<- and th e monotomc1ty 1s v1olate d . 1' ) 41 < I > I Note that the scheme of second order accuracy expressing

the value u./' in terms of IL-l is unique; i.e., among these u,, u.0, scheme s there are none which wculd t rans form every monotonic function into other monotonic one s .

to m n to cit condition Criterion verify the o o i y § 2. n

We noting that difference ch e me can be either begin by s s or explicit implicit.

5 An explicit scheme expresses the value of tL. at desired the in e ms known ues of U. at the precedin g time interval. point only t r of val For a linear eq a ion wi.th constant coefficients uch sche e has the form u t s a m

Here the sum can be or infinite. latter case the differe nce either finite In the scheme will be defi e not for all mesh !unc�ions but only for those n d { ""'} which t increase very rapidly with the increase of m ; the allowable do no of growth is determined by the rate at which the coeffic en s decrease: rate i t Cj It is necessary that the sum o er ge L c,.rf:_ (1.." should c nv . implicit scheme is a system uations for the determination .An d. eq of the unknowns i.e., it has the form ll111 ,

We assume that the left hand sum is finite.

An example of an implicit scheme the dif!e ren e scheme for is e he t equation e mine at the be n i of Section 1 of this chapter. the a xa d gi n ng Implicit schemes are of value to us only because they determine uniquely. l.LJ, We shall seek in the clas of sequences bounded for { u.A-} s this class uniqueness holds obviously for all schemes for which oo -+ . In theI -tl difference equations

have a nontrivial well known the general do not bounded solution. As is solution of these difference equations has the form

6 where are the roots with multiplicity of the equation A;. -k·'

From the examination of the expression for the general solution it is clear that in order to uniqueness it is necessary and sufficient that the ensure equation

not have roots of modulus one. the following we shall assume that all In difference schemes with which we will deal �atisfy this condition.

It is not difficult to show that each such difference scheme can

be solved and written in the form for U.l.,

'd conver d into an explicit scheme. Therefore, even though in thiS thus te following paragraph we will consider only explicit schemes, the and the results o btained can be applied directly to implicit schemes.

We shall not consider schemee.-.which connect more than layers. two We shall now give a imple criterion allowing one to verify s easily whether arbitrary differene e scheme transforms monotonic functions an into monotonic ones or not.

order that the differenee sch�me should In U, � = 'l.,c,_,u,, transform functions monotonic ones with the same sense all monotonic into of growth is nd l Cm be nonnega tive. it necessary a sufficient that al and monotonic. For the sake of definiteness Proof: > 0 Suppose Cm { Un} inc reases, i.e., that all are assume that {u,} u., -u.n-f nonnegative.

7 Then

tL�-u.�-1-Lc,_Itu, -Lcn-rf.t-t"IJ • Lc,_A:u., -·Lcn-l!l.l.n-1-

In this way the sufficiency of the condition is i.e., u.A.-u.A-t� 0 . established.

We no prove the necessity. Suppose for example, a, 0 w 0 < Let

u � """ 0 fol' � /. < m0 -

Then , which is not possible because of the hypothesis u�- u-1 = c,0 < 0 that the scheme transforms monotonic se ence into monotonic ones with qu s the same direction of ro th Thus the necessity is demonstrated. g w . It is not difficult to s o that if all and c, -1, h w ,Z then d renc scheme necessarily stable. Indeed the iffe e is

But because of our assumptions ( f ; therefore maz lem L:luml.( L',luJ and this means that the stable.l scheme is The c ti n appears be em ""'I to quite natural for ondi o L the to solve, for example, the followin equations: schemes devised g

and means that equations is also a solution the solution of these U. = const. of the di e e ce ff r n equations.

8 As an application of the monotonicity criterion listed above we shall now give the derivation of a most accurate scheme of first order accuracy for the equation which expresses the value • ' ou/Dt Juj�-x (L0 and satisfies the monotonicity condition (as we in terms of u..,, u1, "'--! remarked at the end of Section 1, there are no such second order schemes).

It is easily verified that the general form of a first order scheme -- connecting only the above listed points -- is the following:

-r • k = U4 11.0 f (u1-"-t} + (u., - 2u� + u_,)

• (-;f -k)lit -1- (f-Zit)�o +(A-;J..�-t· For this scheme is of second order a<;=curacy and for an arbitrary -k • r;:f2. i its last term is "Jf! h 2 { - �.i:!) U.r.�·· this way the problez:p reduced to the determination of In is � which differs least from f'2/2 and such that all coefficients of the scheme

are nonnegative this last requirement is necessary order that th� scheme { in satisfy the monotonicit condition). Clearly it is necessary to take y ,t � r/2 . Then the scheme becomes

,.. -2ti.0+ +(t-r)u0• U.0•1.l,; +; (u1-u_1)+; (u1 tL1} =f'tt.1

As one can easily check, the stability condition for this scheme is It /1 (I . is of interest to note another way of obtaining this formula. from the point If at which we we draw a straight line which is the (tc + 7:, 0) seek tL0 ti it characteristic of the e ua o �u./1Jt= then will intersect the initial q n 8u/O.X, layer to at the point which lies (for t = (to J rh) ,. < f between p ints and at hich the values of and o (to} o) (to J h) w t.lo are given. The value of at this point is obviously since (J 1 tl. a" u remains constant along t e characteristic, h

9 Consequently we will obtain our scheme if we compute at the point IJ.. (t4, rh) by nea r interpolation between the values and "' at the points a li U.o (to, o) and {to, h) and then transfer this value along the characteristic to the point (t0 f 7:, OJ.

§ 3. Among schemes of second order accuracy for the equation • ere is none which satisfied the monotonicity ou./at 8u./1,: th condition

Section we remarked that for the equation �u./at•au/ltthere In 1 a.re no difference schemes of second order accuracy expressing in terms a.• monC'tonic functions i to otoni of 1.L and transforming n on c f, (,(,OJ f.t_1 m ones. Now we shall generalize this stat ement and prove that for this equation with •.. in general there are no explicit or implicit schemes f' •t:/h f 0, I. 2, of second order accuracy connecting an arbitrary number of points at two successive time steps and transforming monotonic functions into other monotonic o es n . As we noted at the be ginning of Section 2 it is sufficient without loss of generality to consider only schemes of the form

, : We shall say that this scheme is of second order accuracy if it is exact for

.. initial data that are polynorninal of second degree, i.e., if for such initial .·. a conditions the result of computation acc rding to the scheme agrees with the o solution of the differential equation c n de ed Prescribe si . at the� point)2 o r t. I --I · u.1o z) = --- (t , h 2 4-

Then inte ger points at

• f • '1 f./.. II 1)2. (0, h) - ( - - 4- U t) Z

10 The solution of equation wi these initial conditions is f)u./&t - ()uI a� th

T t I It,/ 't) - ("! ')� tL h - 2 - 4- •

Suppose now we wish to compute the value by the diffe renee u.P • tL (r:,ph) scheme. Since we assume that the scheme has second order accuracy we should obtain the exact val�e of the differential equation because the initial function is a second degree polynomial; i.e., we obtain

1 t)� I · a�' • l P t- -r - 2 - 7

Using the difference scheme we arrive at the equation

this scheme satisfied the monotonicity condition then all would be If a,_p nonnegative and since we would obtain that for all ( n -1/2.)2-I /4-) 0 f'1 Actually it is not so. Indeed {p + f' -1/E)z-1/.f..? 0. if .t > -I' > L -1, where ./!, is an integer, then

I I / 1);: • .{, = •· 7 - 4 0 . /(.I. ( / /' ( This contradiction pro.ve s the original statement.

0 4. Construction of the best scheme for a system of two equations

We shall now investigate the system of equations

� 11' � ?I' ()u -.-=tJu --,- A t_) -H· -. ( 1 ) rd I tJf .1/

(Coefficients A and will be assumed constant.) Multiplying the second B equation by A. and adding it to the first we obtain

i · ;: t (I ( -( ;) ) ,; {A-t�) da . = 1_ A fJ ijf (} 1- I)y

11 we obtain Choosing

of the equations has general solution in the form of a stationary Each above a wave.

Fr (1- liB t), /,( + ffv-- +

11"' • F_ ll.- ff ('j,- fAB t).

were onotonic initially then if � V"' Obviously lL + fA /B or lL- -{A/8 m they would remain so for all later times. It is natural therefore impose on to the diffe renee scheme for equations (1) the requirement that it preserve this monotonici ty.

It is not difficult to verify that any linear difference scheme

the system (1) expressing the values u" in terms for , t/"0 of t.LrJ Vf, a.tJ, will have the form

(2)

We not consider schemes use for the tl.0 shall which computation of , 'rrD ·values at initial time in more than three points because in of U and � the solving problems boundary values such schemes require considerable with modification near the boundary; this when standard machine is awkward computations are used.

l 2 Multiplying the second of equations (Z) by ± (A/B and adding the result to the first we obtain

L+M±N (f±Kff + 2 [{a±H7 + -z(utlf�l (3) L -M ± N - + K - +\ll( ± {f ff f v + �(.tJ.. {f8 -v-). ( . ff8 2 1-f J

- 2 tiL"rr v-J. :1- r� f If"" In consider firstJ_J the upper sign. Suppose initially these formulae everywhere and e eryw r 11" •0 V'•l v U..+ fA/8 u- fA/B he e except at one 11" . - -r - point where u. -/A/8 + I Obvioully if L M N fA/8 lf..'fA/8 =f:0 then will di f e t from zero at t ee values be r the of 1./., +tA/8 v-- f e n hr points and therefore the monotonicity of be violated. t.J.. + /A/8 Zl" will From we conclude that this necessarily

Choosing in (3) the lower sign and carrying out analogous considerations we obtain that, also necessarily,

K L+M-N{f- ff•0.

Introducing notation the

L r M g, T Aliir K ff- L -M -tN (f- X ff=rJ, equations a u e (3) ss m the form:

13 = {u + {u+ {-fv)o +�v2 ';:e[(�+ /4v�­ -(uff.-J,]+g[(HffvJ -z(U+{[�).+ +(a+/1vl,]. (3a) (�-!Jv/-f-/Jv-l- �::[(�-�v),­ + G 2 (u- -(u,_ff�' J + fu-/iv),- {-! vl +{u-ffvL].

As we have at the beginning of this section satisfies shown U + /A/B 'V' the equation

t e same way in whic we chose the most accurate scheme for the equation In h h e h ch transforms monotonic functions into monotonic bu/8t- aujar- ( s e § 2), w i we can convince ourselves that for 11-' the most ones, a. of -/A/8 accurate sa f ing the monotonicity conditi n o • scheme tis y will be one with !] · rj{B B for u- 1.1'. one with . By substit tin these expressions tA/8 -· t"/{8 u g for and in (3a) and adding he equations we obtain the expression for g G t ; by subtracting one second from the the U..0 first multiplying difference and by we get the expression for {B/A 7/"0 :

o -t'A ( ) u =(.10 - V::1 _,-tt: + 2h (4) o t:B ( .. vo + 1Y "' ) 2h -(.{_, +

14 interpretation of the constructed scheme §' 5. Physical

We shall now give a physical interpretation for the difference - scheme (4). Consider the equations of fluid dynamics in Lagrangian coordinates:

a''- f 8 • ofJ(v) o • DZ �t (5) 811' _ 8 au • o. at a�

Here a =velocity, =pressure, = specific volume. p '11-' the case when is a linear function In p (zr)

(This may be assumed the case of acoustic waves. System is identic l for ) (5) a with (1).

au _ 071' , A at 'I% (1) 'II' -;I •B -� u.. , » t a-r. for which the scheme (4) was constructed. Using the equation of state (6) this scheme can be rewritten as follows

_ _ _ _ o �a. p, "' . "-t u �8 [(-Pt; ({ ;"•) {-�'•:1'-L j� u,; )] ,

- -I _ Po - 1't ,b0_) -("-o+ U. P-t)ll· 2{f) 2 zf[J

the notation: Introducing

(7)

15 thi s scheme assumes the fo rm:

(8 )

It out that the qua tities have a definite turns n and tJ,::i physical meaning. Let imagine that in the interval between the points us i and (i.e. between the point and the value s of � s �= i h '% = � h U.. and are initially constant and equal u., p, between points and p , f they are also constant and equal U.0, Since the point is - f Po i a poin of contact for two regions occupie by a gas with, general, different t d in velocities and pressures, the so-called resolution of the discontinuity will

take place at this point. Namely. sound waves will spread to the right and

left of the point with the velocity .., (this is the equation of i :1 ± fAS characteristics !or the system . front of these waves u. and will (1 )) In p the remain constant and equal to u.,1 , Pt before right-travelling wave and

to Uo, Po before the l ft - travelling wave . (Obviously such a state will be e preserved only until the waves gene rated by the resolution of discontinuity at

thepoint collide a es generated at thepoints and .- . J with those w v � � ) Between the waves spreading from the point the values of and p will f u. be constants which can be computed using relations satisfied on a s ound wave.

Consider the first of bur equa tiona ·

From which it follows that for an arbitrary contour

¢ udz - Bp(r)dt -=0.

It is not difficult to obtain as a consequence of thi s integral identity that the discontinuitie s in u. satisfy the condition a.nd p must

(u)dr- 8 (p}dt =0.

16 On the wave ro agat to and p p ing the right, ;; = fAB we obtain

(u.) fl- (p)- o, and on the wave propagating to the left, and ::= - (AB

(u.) if+(p) •0. D oting the values of and between the pro agating waves by and en U p p tl P respectiv ly we a ri e at the system of equat ons e r v i (v- a1) {-f- (P-;Dt) - o,

• u P • 0 ( - u�) {f+ ( -P-i) Solving it we find

p "Pt +I'D - /A. u.,- uD ' = 2 18 2 u., + Uo l't- Po IJ= --- z. z{f We observe that and ag ee with and V P r Vi � det min d from (7) and ente ng into ou wa er e ri r difference scheme (8). In this y we see that and are the valu�s of velocity and l1 If pressure obtained as a resul of the re olut on of the disco tinuity in the eg on between propagating t s i n r i waves and consequently also at the point i from which the waves emanated.

It is of n erest to note that ob i ed values and i t the ta n {It If will ema n constant until the bo nd r considered is ached by the waves r i u a y re generated from the resolution of the discontinuities on neighboring the boundaries, i.e., at th points and y tem considered e - ; ! . For the s s the disturbances propagate with the sound velocity. Therefore during iAB . if the time n r al values and are to remain constant it is i te v 1: the lli f1 necessary to have Fortunately this in quality agrees with the t" < h/fAB . e stability condition for the scheme (4).

17 Clearly, after the ti e he values of nd 7r' m interval 't' t t.i. a will no longer be constant. Let us denote be tween the points � and - ; their mean values by and 71'" • For t ei computation we shall use u.<' h r the law of conservaticrp of momentwn which yields the first of equations (8)

and the conservation of volwne which gives the second one.

The physical interpretation of the difference will scheme (4} serve as a ba i for the construction of a scheme for the general s s computation system of equations of fluid mechanics.

Chapter ll An Approximate Scheme for the Computation of Generalized Solutions of the Equations of Fluid Mechanics

Formulation of he problem § 1. t

Our will be to construct the difference scheme for plane aim (non-axisymmetric) one-dimensional unsteady equations of fluid mechanics (in Lagrangian form)

()«. ap (?1;E) .., 0' at f- B 8%

�v--- B -au _, 0 ' at � � (1)

a(£ +f) iJp� + B --;;-- • o. �t As is well known this system of equations does not always have a smooth

solution even for data. Therefore one also must consider smooth initial generalized solutions with discontinuities -- ock wa ves sh . After S. Sobolev, we shall call the system of functions L. a generalized sol u ion of for each i ini ely differ­ ( V., 'Zt'; E) t the system (1) if nf t ential function "1 (x, t.) which differs from zero only on a finite subdomain of the domain on w ich functions are defined, the following 6 h the tt, � E equ alities are satisfied:

18 �<::r Bp(v;E)::]drdt•O, if[G O, �t • B t - d Jj [v� ::Jdz Bp(v,£) 0. u [(E r ;):r + • u. ::]tirdt • functions a, "21; are pie�e...))�'� continuous then these requirements are If E equivalent to the fact that around an arbitrary contour

¢ u.d� -Bpdt -o, o, rdz • ¢ + Butft (2) ¢ (£+:)tiz-Bptdt -o, is the usual formulation of the conservatio laws. From the conserva- and this n tion laws it is easy to establ i h as is done in every course of gas dynamics, s , the relations ac ross the discontinuities (shock waves):

(w-u.- BpJ • 0,

[ttrV" -1-Bu] • o, (3) o. [ fU{E+ �'1-epu] = Here is the velocity of the shock wave and m ans the jump w- = :: [ ] e of the quantity the difference of its values on the right and left of the wave ( ). We must note that order to ensure uniqueness it is necessary in to exclude rarefaction shock waves; for this it is sufficient to require that around any contour the following tegral inequality is satisfied: in

where J is the entropy determined by the known methods of thermodynamics as certain function a of p and 1/-' .

19 We propose t or an approximate scheme to cons true f system (1) with the property that as the size of steps diminishe s the solution obtained by this scheme c o erge to the generalized solution of he stem will nv t sy . shall construct the difference scheme in uch a way that We s for the linear case of sound waves it will coinc de the scheme considered t i with in the previous chapter which transforms monotonic waves into other monotonic on . do not this property d e not es The application of schemes wh ich possess o a appear to be intell gent since the effect of non monotonicity appears prec el i is y in gions where the solution varies sharply which are the shock waves. re In attempts to compute shock waves using schemes which do not satisf.y the monotonicity condi ion one obtains for them h ped profiles and the humps t " um 11 pulsate from one time step to the next (see, for example, the graph in

Applica ion 1 ) t .

Sometimes instead of 1) on e has to consider the following ( system of e a io s of fluid dynamics: qu t n

(l a) /}tr' �tl.. --a - =o ot (Jz (Such equations describe, for example, the flow of water in shallow channels.) The reader will have no diff iculty in transferring all our con­ siderations to th is si�pler case. Let us state only that as the uniquenes s c ndition, the entropy .)01 t is case should � law of increase of c}Stiz, in h be replaced by la of dissipation of energy the w

The d c r ipti of the computational § 2. es on scl'-e_!!!!

We w l now de cribe he propo ed sche m e i l s t s . ima ine that beha ior m te g v to co p Let us the gas whose we wish u d ded into iv a sequence of l a e r by points integer labels is i y s with 0, 1, 2, 3, and by 'half integers" 4, ... .! , �, the layers themselves are numbered 2. !:..2. 2. .,. We shall assume that th are e quantities t&,VJE,p = p{v;£) initially each constant inside m two jac ent layer. At the boundary between the ad

20 f f layers a.nd the discontinuity is resolved, in consequence m- 2 m r 2 of which at the point the pressure and velocity become and m � u, 5 (unlike § of Chapter I, points at which the discontinuity is resolved are now labeled by integer� and layers between them by hal! integers).

The rules for computation of P and (/ a.re derived in any course of gas dyna mics (see e. g. (3)). We shall list the formulae for !', and in a. form convenient for us in below. (/117 Section 4 After the values of fin are determined at all in eger points we determine the values which a, z.) E will assume when the t time interval t' has elapsed by formulae analogous to (8) of Chapter I:

(4)

Here � denotes the mesh size of the scheme, the difference in i.e •• Lagrangian coordinates for any two adjacent integer points.

As in Chapter I we must note that after the time interval t', 'Z' has elapse d the values be tween two successive integer points will no longer

_f be constant and the computed values (J,/I?+J., v"�'+"S1I I £"''�-r. represent only avera ges over the layer which replace their true distribution with a certain accuracy that is characteristic of the approxima te method described above.

with the diminishing mesh size the difference solution con­ § 3. If verges, then it converges to the generalized solution of the dilfe rential equation

as suming that as the mesh size decreases u., v; Now , E computed by the difference scheme (4) converge to certain piecemeal smooth

21 shall show that for these limit functions the conservation limit functions, we laws (2) re satisf ied, i . e ., these limit functions are generalized solutions a * of ) or a (1 ( 1 ) . t Let us consider simple rectlinear contours of the form 1. represented n g re On this figure, crosses denote the half-integer i F i u points located inside the layers and circles, integer points.

From i follows (4) t that

Az � Az A, (Sa) u. - 8 p� 8 P�. h L (.(, - h l [ + [ A1 143 � A� as the mesh size decreases the functions u, Z1 If mesh E, P,(/ converge to some limit £unctions defined in the plane se £unctions ( the limit shall denote by the same letters as the corresponding mesh functions), we then from the difference conservation law (Sa) it follows that for the limit functions around an arbitrary rectlinear contour,

�adz+BPdt -o. (5)

From is satisfied for an arbitrary rectli e contour it the fact that (5) n ar fol ow is satisfied for any contour. l s that (5) From the formulae f r and de ived in fo low ng � P IJ r the l i section it ol o that if in the r i ns where s l tio of the differential f l ws e g o the o u n c on er e to these solutions, equation is smooth the mesh functions, u, g v; E v then in these regions limit functions for P and coincide. U i g this the /' s n and the fact that the discontinuity lines cannot influence the values of the i tegr s we arrive from at the equation n al , (5)

We shall prove in the limit S the law inc ease of (if the syst em considered4 that for is functionsdissipation of r ( 1)) energy (if the sentropy or of of ystem considered is ( la)) hold.

22 Analogously one can demonstrate that the remaining two equations (2) are satisfied.

0 X 0 0 0 X X

0 o- --x---o-- -x---o X X A_, -1(1 I I I I I 0 X 0 X 0 X 0 X I I ' I 0 X 0' X 0 X 0 X I I I I -X- --0-- 0 X b-- -X--.- 0 X A3 A+

0 X 0 X 0 X 0 X

Figure 1

In this way we have proved the statement formulated in the title of this section.

Formulae for computating the resolution of a discontinuity § 4.

We shall now describe the derivation of formulae for P and !or simplicity we limit ourselves to the case of a gas with the equation U ; _ of state

! £- (�-! ) pv.

Suppose that to the right of the point 0 the gas has specific volume Vf, internal energy (per unit mass) and velocity 'i Ef Uf . � z The pressure in this gas is giv n by e

E1. (1- 1) iii

p_,� = ---­

23 Suppo se the state of the gas to the left val es of 0 is defined by the u

We assume to be gin with that the pre s sur e Po which obtains afte r the of the discontinuity greater than .P- -J- and thi s case resolution is 1'-J: . In shock waves will propaga te to the right and left o£ point 0 . we have already noted (see fo rmulae (3) of of this chapte r). As § 1 the shock wave the following conditions are satisfied: on

-Bp 0, [ m-u J - BU.J - 0, ( fU'V't- [tu-( E r ;•�-BJ>ll]= 0.

Introducing the de f tion ( velocity the ini s = b0, tUL = � = Btu.t B - tL0 of wave propagating to the right, -w:t velocity of the wave the = propagating to left) we can rewrite the se relations as

-r = 0, a0 [u.] [p] ILo = O. [v-) - [u.] on the left w ve (6) 1%.[u �']+ [;u] =0 a

b0 -= 0, ( U ] - [p] bo [v-]+(a. ]� 0, on the right wave (7) h0 [E + �]-[P/4]= 0 As is well known, in the region be tween the two wa e s and will be v U. p constant and equal U and P the values on the contact di scontinuity to -- point 0 originating at the The value s of the s pe c ific . volume will be c o s tants be tween the con c t discontinuity and th e wave s but these constants n ta will be diffe rent to the right and left of this discontinuity. We s hall denote by v.t, the s pecific volwne between th e discontinuity.and the right c ontact shock wave and by specific e e n th e contactdi scontinuity and � the volume b twe the left shock wave . 24 The first of equations (6) and (7) can be rewritten in the oll wing way: f o

Assuming � and known, and be determined b� V:, Po can from the above system:

boP- f 1/.oh o - + a,pf + {tt..-f u..;J Po= ' IZo + ho I tl I . #'l f v bo + - -i:J1 (J,,. tJ-r + i' r- "L r� �------�------a, + b0 the other hand, if we knew then the we On Po for determination of a() could use equa tion obtained from the irs t relations after the the f two (6) is eliminated between them: velocity

a,0 =

value can eliminated '!rom thi s formula with the aid of the The VO.t be Hugoniot obtained from by the method de scribed in any course on curve (6) gas dynamics (see e. g. (3)):

(1 -I)Po+( /+f) P.. f

(;r + 1) Pa -f ()' - t) p_ .! 2.

subs tituted into fo rmula for we Afte r this expression for 2-0..t is the t:Z0 obtain

()' + !) Po (/ -I) P_ L +- :l tt,o =

25 Com pl e tely analogously we can derive

To de te rmine Po we can now use the following ite ration n t a we d te e process: beginni g with an arbi r ry f'o e rmin a0 and 60 and then compute the new value for Po Substituting it into the formulae for ·a0 and b0 we find Po again , and so on until the process conve rges. Afte r this we dete rmi ne t{,

So far we have cons ide red only the case when simultaneously

and i. e. , �hen no rarefaction wave occurs in the fJa ) p_t Po > P.; , resolution of the discontinuity It turns out that when such waves occur the . proce ss of resolution can be c omputed in the same manner only changing the formulae for and • t/,0 b0

Imagine , for e xample that t , that a rarefac tion , Po p < % wave propagat s to right. As is well known, across a rarefaction wave e the the following relations hold:

c, t! ()/f. 2 2 '!" - - Uo f.tf2 1-1 J'-1

- P..1. v-r Po ' � 2 1/(}� where sound speed, and and the value s of = = c - f.J'ptr (JQA, �...t- e and V' to the right contact di scontinuity. The first of the above of the e qualities can be rewritte n in the f ollowing way:

26 Denoting

l-1 1'..1 Po --· - 7 b0 = '

we obtain the relation comple tely analogous to the one obtaining across the

shock wave . in the formula is expressed in te rms of • p If for he (J and and is eliminated with the aid of the Poisson adiabat (the V"" UO..t. second of our equalities holding across the rarefaction wave (. the following

obtains:

Po l- 1 1 - - I'..!�

2)'

the case when the rarefaction wave propagates to the left we should have In set

)'-f

- = tl() 2J'

tha t we mu st In this way we arrive at the result in orde r to dete rmine Po solve by iterations the following system: (/t-t) {J(l-t) + (P -t)p_!.. � P,(i-t) fo r 0 7 __f A ,.t:J 2

ao - A0 d-1) (8) 1-f !- P -i_ pJi-1) P. f fo r () < --

27 !;, (f- 1) b0 = 0 I - - f.! & D rc'- t) n ¥- 1 Lo r 'o P..t < .. (-Po( /-!)) 2¥ t- /JJ.;z

•

After the ite rations have converged and we have dete rmined final value s the for , we find from the for ula Po a0 , h0 Vo �

•

Detailed investigation of the conve rgence of the ite rations this process conve rges if in the resolution of (l) :c (I Po (� (i-f� show.s tha t d iscontinuity e rarefaction wave not of excessive s trength the th resulting is . order to make on r en in all cases, is ne cessary to carry it out In it c ve g t it with somewhat modified formulae, e. g.

.. r

where )" - I 1 - a.,(i-f} .. -----:----=-z.,__;_---=-_, if - I, this expression exceeds 0 z z. 34 �1 �- ::,� ) ·i-t l I c-f �

0 othe rwise - 1) R.0 (i = z..(- I + /J.!.z P-.16

28 We omit the investi gation of this convergence because it follows standard methods for investigating the convergence of ite ration processes of the type

which reduce to compu tion and inves tigation of the complicated Z i • f (l i-') ta expressions for the d'rivatives.

From formulae (8) it is seen t at if " and conve rge h p with diminishing mesh size to bounded continuous functions. Then an t/ d P c on ge to the same limits. We have already used this fact to prove at the ve r th limit functions are generalized solutions of the e at ons of fluid dynamics . qu i Chapte r we derived formulae !or the resolution of a In I (7) discontinuity usin g sound waves. They agree w h the expression of this it paragraph we let if

the computation with sound waves the time te had to be limited by the In s ps stability condition

= ;, a(f

It seem s natural to us to use in the present nonlinear case the following bound on the time step

It is true the above e in only approximately equals the time necessary d f ed 1:' for the waves obtained from th e resolution at on integer point to eac h the e r ad c nt change the values of and ob ine there after the ja e one and (/ P ta d resolution of the discontinuity. Howeve r, a large be r of different com­ num putations using this condition shows convincingly that with such a bound on

co ta tion is stable. addition, thi s co d tion coincides with 1: the mpu In n i the one given above for the linear scheme wh n o n waves are being e weak ( s u d ) computed.

29 Le t us add that i£ we wished to know the true dis tribution of

quantities tt, t1-; the end of time inte rval the resolution E at 7: afte r of the discontinuity we could obtain it by solving the elementary problem of

gas dynamics inside each la e r . (It is only necessary that should not y r exceed the ti e necessary for the wave fr om one intege r p i t to reach m o n the adjacent one . )

An especially simple case and the one which always admits closed-form s olu tion is the case when is smaller than the time necessary t' for the waves emitted from the two adjacent points to collide.

As is known from elementary gas dynamics the entropy S which obtains inside the laye r afte r the time nte al will all i rv 1:' be for large r than the initial value: 11 we examine the layer S (-,:) > � (if numbe red i ). Recalling that 2

and using the following simple inequalities:

z (;c.}dz (r.) dz {'"t .tn. ['z�, follows from convexity of the curve ' A_, ------.r,n,1-- z:. '%2 - -;:: M., "X-1 -x.-1 Zz -

follows conea vi ty of Zz from the curve "'2 - z., we conclude tha t

30 Now we note that : [(£ + �').r:z- ; [: tudJ:r :[v-(z) dz are the liJ which in co p ting by our scheme we assign to mean value s E, m u I interval and denote by £"""' the "point" 2 dur�g the time 7: i ' � ,_,.f In thi s way we arrive at the ine quality

Applying the rea oning analogous to that used in 3 in . the proof of the inte gral s the i conservation laws we can show , ueing fact that at each me sh po nt the inequality of the type holds, that as the mesh size tends e ro the (9) to z limit aa fies for eve ry closed contour the in gral eq li aolution tia te in ua ty

(condition guarante eing uniqueness).

Using simila r arguments we can show that for the system (la) the corresponding iq e s condition is satisfied. alao, un uen s

S 5. Computation of Euler coordinates

Usually after solving the system (1)

one still has to solve the equation for the Euler coordinates of the ga s pa rticles

= (}r u. at

We propose to at intege r points by the formula dete rmine ..t

31 It of inte rest to note that from the prec eding it follows is formula and (4) if initially tha t

(10) , .. hB fm�t - 1', )' r"' � 1

then this e quation will be satisfied for all late r times. It means that the

volume of the ga s laye r can be determined from the knowledge of its boundarie s.

Equation (1 can be u sed to dete rmine place of he second 0) 'V' in t of.formulae (4). When computations are carried out on an electronic computer it is more convenient to use formula (10) because decrease s the number of it quantities to be stored at each step.

Some results of numerical compu tations g 6 .

Application we list the results of computation of a s teady ­ In U state travelling carri d out using and (8 ). It is shock wave e formulae (4) seen from the curves that the computation i begun from t p function which if s a s e satisfies the shock conditions at the point , then afte r several ime teps 0 t s the profile of each quantity settles down to a steady-state shape which propagates time with the velocity equal to the velocity of the shock wave with the same in jumps in pressures and velocities. Only near the point there remains 0 a

bump in the curve of • This is explained by the fact that in the process of

11-' r reaching the steady s ta the "erred" in en tropy, which is c on erved te scheme s in the smooth region behind the front of the wave . the smooth region the In scheme is sufficiently ac c urate to show this conservation of entropy. After

the state is re a e pre ssure behind the front of the wave equalizes steady ch d the and is not c o r re c t th ere this leads to ppea rance of the bump since ;nr¥ the a on the curve of � . Analogou s entropy traces remain also afte r the computation of othe r unsteady e es for example process a shock pr oc ss , the of formation of wave du ing the impact of a moving gas onto a rigid wall (see pplication r A III). entropy tra c e s usually cove r twoor me sh points and the refore do These three not influence th e results of computation for a sufficiently small me sh size.

32 A certain effect obtained in the computation of contact § 7. di acontinuitie s

All considera tiona which we adduced in arriving at our scheme � were obtained by conside ring the case of constant x-steps and with the

assumption that the entire computational process occurs in an infinite gas;

howeve r, the numerical scheme obtained ha s such a clear physical meaning

that it difficult to resist the de sire to apply it also at the bounda ries between is twome dia -- contact discontinuities. For this it is sufficient to include the

contact discontinuity among the number of inte ge r points and in computing a.. and at this point use for constants characteri zing the gaS' located h a. , to the left of the separation line and for constants referring to the gas b , located to the right.

The results of our computations show that the application of

the scheme so constructed on the contact discontinuity is allowable, but, as is not difficult to verify, it leads to a decrease in accuracy.

this pa ragraph we wish to describe one effect which is a In consequence the decrease accuracy and which wa s ob served during an of in analysis of computations near contact di scontinuitie s. This effect appeared

the computation of smooth solutions , it bears no relation to the shock wave s in and therefore it is natural to attempt to explain it starting with the assumption

tha t our system of equations can be-approximated by a linear system. Com ­

putations based on such a linearized system of equations yielded the magnitude

of the effect, which agreed with the one observed in computation of gas

dynamical problems.

Suppose processes certain ga s are de scribed by the system in a

tJu. 1"1 B ap - at + a� v, (11}

f) 8 au = 0. 9t?I"'- �%

33 The equation of state of a. gas in of small variations preuure case in admits the following linearized representation:

Using thi s sta te the system can be (11) rewritten e quation of

-+f}p .:1 -t)U .o. i! { () � contact discontinuity be at i. e. , let the coefficients be different Le t the y..aQ • 0 than for . Set for X > -:t <0

1: Af lot' > 0 I A { % <0, A_ for z

fer � > o. 8 = { B.,. B_ for "Z < o, The system of equations

fo l' � > 0

1JI.I. + 8 .!.E. J: 0, at - or

fo r x < o

34 with the continuity condition on at the U. and ;IJ Z • 0 admits following solution

-ot!. -fit + ¥, (/.,-A� 1;

rof' X. > o,

(1 Z)

for -x, < 0,

satisfies following initial conditions: which at t • 0 the

lof' -:t > o,

(13)

for % < 0.

We shall inve s tigate what the solution of the diffe rence

i s the same initial data. It is more inte resting because any equations for smooth solution of our system near the point =t0 can be X=0 t represented in the form

35 .;(),. :t > o,

fort -x < 0.

The refo re the behavior of the difference solution near will z. -o characterize the behavior near the contactdis continuity of the quantities obtained as the result of numerican computation of any smooth solution of our system .

We begin by giving the explicit expressions for the diffe rence scheme for the present case (we have explained at the beginning of this paragraph how to obtain these formulae ). We will as surne that the step h equal to the difference of coord nates of the two neighboring integer X i ints, may be diffe rent in the right and left of po 1:: = 0 regions to and fo . N mely, for Z > 0 h =h+ The computation � r � < 0 h =11- formulae are:

36 l�r1 - i 1 Pm +--J. - pr; ______. ----=---- - . , Pm = 2 i B; 2

fof' m > 0,

2

I LJ t �- + -1'1 r r - m - z. m + 2 ______, -- P,= - -· 2. 8_ 2

lof' m < 0,

rm -- � -z.I- b f rm + �

( 14)

, -· f?'E·- + !'.!.& + of 8+ P. = &+ , 0 ·-& If - -t {fT {f-8_

fot m )J 0,

T: · B_ u. m+{ ,. In m ' tL 1 - 'm fl - P.. } m + .� --;;:- ( lo,.m < o .. fJm + - i - • pj,�� :�- (tJm+l t!, }

37 For our difference e quations one can find a solution which,

is a linear function of nd in each of the region s tiJce (1 Z), :t. a t %' > 0 nd has in the se regions g adients identical with (1 Z). Namely , t < 0 a r Jl'ld turns out tha t such s,olution will be jt

lo r � > 0,

(15 )

fX. -;- -I • -Bh - t.L = - X - ..8-t d', A_ fA _ + 2 8_ z < lo r 0 . .,.B t I o�,h_ . f n fJ = -� -a,+ -· -;=::===- r7 B- Z fA - f3_

We leave to the reader the completely elementary verification of this fact.

( f To compute by the se formulae tL 111 r i" and for an a r i tra ry e. '/m.,.l b , g. i s necessary to let me step it hz:, nth, ti t- ;r., = (m "f i)h we begin the computation by our scheme from the initial lf conditions (13) then experimental computations show that near the contact di scontinuity the solution of the diffe rence equations (15 ) tends to a steady state with certain and o e the process . 0 e btain d in we compute the and by o m l e (15) 1i valu� s of u. I' f r u a at

'X we will see tha t the se quantities assume at this point diffe rent values = 0 to the left and right; diffe rences between them are:

38 solving the diffe rence equations the values of u.. and � are . d I when

, • • only a t half ·integer po ints • �ted ( - ! h. , - � h-, f h+, f h+ , ) . tJII"P · · refore on the c ontact d'11co ntmu1ty we d o not f'1 n d any rea d'uco nt' mu1h ..e e the 1 ur and velocity. howeve r, the pressure and velocity are linearly e u �p r e IC, a lated to the poin obtain exactly he values compu d f' po = te at ,_cr \ � 0 we t 0 by formulae ,_.. ( 15). From what we have said so far it follows that the values of

ressures and velocities extrapolated from the right and left will in general if e r d. f , on the contact discontinuity and the diffe rences b tween them will be dete rmined e from the fo rmulae

(16)

This disagreement of velocities and pressure s on the contact

discontinuity is e s e c iall noticeable on the graphs of and and p y "- fJ obviously characte rize s the inaccuracy of our scheme. Indeed our scheme , if were exact fo r the linear functions it would compute solution {12.) exactly and we would not observe any discrepancies in ct. and p .

. . � · In or d e r to co unteract t is effect we should as one can see h from (16) choose the steps s uc that as closely as poss ble h h i

We have already explained that h /i A /3 represents the la r ge s t allowable time step which doe s not violate the stability of the difference s c he m e Thus we a tte m t to choose the space steps in such a way . should p that la e s steps requirement are if po ssible th e rg t time consis tent with the stability equal or approximately for the gases on both sides of the contact equal discontinuity .

39 is not po ssible to satisfy this condition exactly for nonlinear It problems of gas dynamics because the speed o wh c h determines the time' f sound i * step at different stages of the problem . is diffe rent in computations we succeeded in cho s ing the steps in If o different regions in suc h a way that the above condition was not strongly

violated, the effect betng studied on the interior boundaries was almost absent

which s ign fi d an in rea s e in a curac The gre ate r accuracy in thoseca ses i e c c y . was also noted by c omparis on with the method of characteristics.

The stability of our difference scheme on the contact discontinuities � 8 .

the preceeding paragraph we have given formulae �y one In which a ntact s c onti i can compute solutions to equa tions near co n ty. our di u Now we shall investigate the stabili these formulae. This investigation will be ty of carried o t on the difference scheme for the linea r system. u

fJU +8 !e. •0 , gt ()�

'()prA �u ·0 ot ��

coefficients and which a e constant in each of the regi ns ;t. > with A 8 r o 0 or (in the preceding paragraph we studied the computational phenom eno t ( 0 n described there by onsidering just such a sys tem) c .

After A. Filippov (see by stability we will mean uniformly F. (4)) continuous dependence decreasing mesh size) of solutions to the (with diffe rence equations on their right hand side s and on the initial data .

o rder to prove stability it suffices to define ol n s In for the s utio of difference equations a norm hi h the l it as the mesh size tends to w c in im zero goes ove r into a norm for the solutions of differential equation s certain such tha t

We h ave § 4 Chapter that the time step in e dete r mined in probleof ms II admissibleFor t solution of gasdynamical oc J, 8/f1d.J:.(fl ,.,.,1 b,) . h is t: / the smooth solutions considered in section sufficiently a111 , and this and small h , bm e ua the conve cted ve locity of sound. q l 40 - we here an infinitely dimensional vector n understand By 1.1. defined by the values of the solution (�;+2 , P;.,.1) to the difference equations for the time step .. nth shall assume that the time step is chosen by the stability We conditions inside each e gion we noted earlier this implies that the of r . As following inequalities are satisfied

(. I'

We introduce the notation

From (14) of the preceeding paragraph one can conclude without difficulty that the following equalities are satisfied

Bf {E{F- 8: . . �, + I! sp , - -s .,. fll � . 1B:K + ra.:

fo f' tn ( - f,

ro r h1 � o, z(A; rz-rA: + � M �� II sn+t . (.t -r.)sn 18; r:.. -ra; -· - -' - � 1 - � - E + - + + If- {F84- {fB_ + 8_ f/+ l ( 1 � , _ -2. s, � 1-r_ ) s" + f' S" for m � +� m + -z m + z

41 m conside rations out the assu ption that All be will carried with subsequent e We leave ade the entirely {A'f/ B+ > fA-/ 8_ . to the r r analogous the considerations in case B+ < B- when /A+/ fA-/ Set

lo r att m,

S� , = I fo r m � 0, mf� $ m k + -z

* lor m < 0. · sm i- ..J..z

n + l formulae From the expre ssions analogous m o�- for and S i and We them be low for follow. list fo r m> �

2 ,-;;c u r;c-- S::' -, 1 1& -18: s ( , z - + + If8_ for m< -I}

-lo r m? 0,

177 � - 2 . =(t-r)- 5" fo r m + !.&

42 Since the sum of the absolute values of coefficienta of the j and in the right side of each of the se equations e quals unity (recall that A,f' hand and are leas than then A- I )

This inequality proves stabil ity choo s e for the norm if we

Thus established stability of the difference scheme for the li ea r n system ma y serve as some kind of a justification for its application in the case of a nonlinear addi ion let us once more that in all the system . In t , state numerous computations . using our scheme , car ried out with consideration of our bound on step, the computations were always stable. the time

Appli cation I

Below {Figure graph of pressure in steady-state 2) is the the shock wave for the system

�a ap ('v) = 0 ' at -r a1:

= o,

43 ,

J.D

u

I • I I I • I

Figure l

computed with the scheme of second order accura cy

where

tip A=- -· dtr

Application II

Computa tion for the steady·state shock wave for the gas with

Numbe rs nea r curve s (F igures 5) indicate the layer )" = . number. 5/3 3 , 4,

44 ,

• ...___5� =s::e-��

I

,

'

Figure 3

------• -z � --�--��������--�I lllt•r ..f ..L..J. r tr �� 11 sr lc 'C q 1

Fi gure 4

45 ' ,,

u

Ul

Figure 5

, ·r------e��---

I

'' I I .til I

Figure 6

46 Application Ill

The im pact of an absolutely cold gas moving into ( 1 • 5/J ) a wall. At the ini tial moment we presc ribed =I, I' • rof' � > () � 0 we speciijed the boundary condition which waa taken into For :t • 0 tl•0 account in computa tions of resolutions of discontinuities at this point.

II I

Figure ,. 7

6, 7) On the curves (Figures now the right travelling shock wave forms.

20 Received March 1956

47 Refe rences:

1. Neumann, Method for the Nume rical Calculation I. R. Richtmye r: A of Hydrodynamic Shocks, Journ. Appl. Physic s, No. 21, 3 ( 1950) 2.32.:237.

P. D. Solutions and 2. Lax: Weak o£ Nonlinear Hyperbolic Equations their Numerical Computa tion , Comm. on Pure and Appl. Math,

No. 1 (1954) 159-193. vn,

3. L. D. Landau and E. Lifshits: Mechanic s of Deformable Media . M.

A. S ili f Diffe rence Equations, N. F. Filippov: On ty ? D. 4. tab A. Vol. 100, No. 6 (1955) 1045-1048 .

48