Finite difference 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 HAL Id: hal-01620642 https://hal.archives-ouvertes.fr/hal-01620642 Submitted on 25 Jul 2019 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<!ted to use, for the solution of fluid dynamics equations, difference equationa into which viscosity was introduced artificially; this has the effect of smearing out the shock wave over aeveral mesh points. Then it was proposed to proceed with the co�putations across the shock waves in the ordinary manner. 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 differential equation. Such a require­ m nt however, is not always satisfied. e , For example, consider the heat equation �!.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.