NUMERICAL SIMULATION OF MHD-PROBLEMS ON THE BASIS OF VARIATIONAL APPROACH V.M. G o lo v izn in, A.A. Sam arskii, A.P. Favor s k i i, T.K. K orshia In s t it u t e o f A p p lie d M athem atics,academy o f Sciences Moscow, USSR 1. Many o f modern s c i e n t i f i c in v e s tig a tio n s d e a l w ith s o lu tio n o f m u lti-d im en sion al MHD-problems. For th ese problems the f i n i t e - - d iffe r e n c e methods a re o f th e most u n iv e rs a l and e f f e c t iv e use* The d iffe r e n c e scheme may by con sidered as the d is c r e te model.of a medium. O bviously th e scheme o f b e tte r q u a lity r e s u lts In b e t t e r sim u lation o f th e medium p r o p e r tie s. For example th e divergen ce schemes ^1^ p ro v id e an adequate accuracy o f th e discontinuous flo w com putations. The com p letely c o n serv a tiv e schemes f 2 lm a y be used s u c c e s s fu lly even on crude g r id s. The d i f f e r e n t i a l equations o f n o n d is s ip a tiv e MHD and t h e ir most im portant p ro p e r tie s w ith con servation laws in clu d ed may be obtained brom v a r ia t io n a l p r in c ip le s s im ila r to those o f H am ilton-o strograd- s k ii in c la s s ic a l mechanical systems t 3, 4 l. The com p letely conserv a tiv e f i n i t e - d iffe r e n c e schemes can be obtained through th e analogous approach. 2. As an example we con sid er the liq u id volume I? o f an id e a ly condu ctive a d ia b a tic plasma moving in space. L e t -Q 3 be th a domain th a t corresponds to Q- in Lagrangian coord in a tes J7 the C a rte- z ia n coord in a tes o f p a r t ic le s o f the medium. The Jacobian ФСХ) У О ).off JSj^ :_Q.3 # The Lagrangian o f liq u id volume Q i s Here is th e v e lo c it y v e c to r, the v e c tb r o f fr o z e n - in m agnetic f i e l d. The law o f mass con serva tion is d ) The frozen-in condition fo r magnetic f i e l d yields the equations
249 Н X + ц й а з у + ц Ш Ю + и Ш Ю + п а М щ ) 4 > с / ф 4 c j *, p н Ф с л р и fa &г > н* Ф с ^ ) Фсг,х? и Фсх. я) Ф С [М ) * < ) In th is paper we con sid er th at a l l flo w param eters depend' on on ly twt space coord in ates 3C and У i. e. ol and R e s p e c t iv e ly, in Lagran* gian coord in a tes. In t h is case we can accept J 2, - H i { 0 4 * АЪ a. - г, ш = n e j 0 Equation (2 ) con verts in to The r e la t io n? $ ) «& C d, j O л и * 2 2 л + Q X & +0X5, 0 (? ) fo llo w s from ( 5). The tim e d iffe r e n t ia t io n o f ( 5 *) g iv e s the equation o f induct io n o f magnetic f i e l d where 3 г Ф С -, Я > ф С у.; 0 0 l - + t фы.р 7 7M v By v a ryin g th e fu n c tio n a l o f a c tio n S ~ and tak in g into account a d d itio n a l c o n s tra in ts ( 2 ' )» ( 3 ' )» and a d ia b a tic ity ^ con d itio n! - 5 Aj ft)
and by s e ttin g the f i r s t v a r ia tio n S^S E u ler equations equal to z e ro, we ob tain the P } dm =-V7ft (p.+ii5\ + 2. + - iijid. (K) * v di +gfr ) +^ы 4^ <2>/3 4 p w From (4 ) we ob tain th e equation f o r in te r n a l s p e c ific energy p <k dj? p Г ^ фсхдлл)*! ^ * u a t 1 L t>coi(sp3 \ 3* We assume th a t S lu C d ijо is a square. L et us in t r o duce the resta n gu la r d iffe r e n c e g r id w ith the in t e r v a ls f^ and b } CO b ein g the set o f i t s c e l l s and CO the s e t o f i t s nodes* L e t us approxim ate th e Lagrangianby the d is c r e te expression ^-ZSgBjCZ I f.r,) <0 where Ш 4С, j ) = { O. j ), C W, j ), C i+ l,'p 4 ), and Si j i s the volume o f th e Lagrangian c e l l.. The d is c r e te analbgs to the law o f mass con servation and fr o z e n - -in c o n d itio n are U*S~ $ ^? t = $ = oms.i ) U J jc i,^ The rep eated index denotes the summation. The valu e JCic_ 4T,A>'ij (]4 Д ^ 'bb;e m agnetic f lu x across d iagon als o f the c e l l f t j j ) in the d ir e c tio n to the k -th node. According to (4 ) th e en ergy change law i s Ф). 11 ') After the time differentiation frozen-in condition (3 **) yields the magnetic induction equation
251 Varying th e a c tio n fu n c tio n a l w ith c o n s tra in ts (2 ) - ( 4 ) resu lts i n th e MHD-equations <P) M S + ьсшяа,\) Here Ш2С1,р- Ч1 V ^ 4 H >!) Schemes C5 ) are v a lid in the in n er nodes o f th e g r id and accurate to the second-order approxim ation in j^ and ll^. Simple g e n e r a liz a tio n oi th e a c tio n fu n c tio n a l extends (5 * ) in to th e boundary nodes. D iffe r e n t ia l- d iffe r e n c e schemes (5*) a re com pletely con servative M - 4. The d iffe r e n c e scheme is deduced from equations (5 * ) through re p la c in g th e tim e d e r iv a t iv e by the f in it e - d iffe r e n c e r e la t io n s. By tim e cen terin g the r ig h t hand s id e terms the d iffe r e n c e schemes are made com p letely c o n s e rv a tiv e. 5* The lin e a r iz a t io n o f equations ( 2 " ) - ( 5 * ) produces th e d i f f e r e n t ia l- d iffe r e n c e schemes fo r, MHD-acoustics. The v a r ia t io n a l approach p ro vid es f o r a space op era to r o f th is schemes to be s e lf-a d jjo in t and p o s it iv e м. The scheme s t a b i l i t y depends on the im p lic itn e s s o f the scheme. The d iffe r e n c e equations a re s o lv e d by the Newton method. To c a lc u la te th e discontinuous flo w s a r t i f i c i a l v is c o s it y is used M * 6. The v a r ia t io n a l approach has been used as a. b a sis f o r numer i c a l algorithm s and techniques employed f o r s o lv in g a number o f model and a p p lie d problems 5» 6» 7 ^ R E F E R E N C E S I. Sam arskii, A,A. and Popov, Yu.P. D iffe re n c e schemes o f gasdyna-
252 m ics, Moscow, "Пайка", 1975» 2. S e lig e r, R.L. and Whitham, G.B. V a r ia tio n a l p r in c ip le s in continuum mechanics, P ro c. R oy.s oc., AJ05, 1-25 (1 9 6 8 ). 5. Goloviznin, V.M., Samarskii, A,A., and Favorskii, A.P. Variational approach to constructing the fin ite-d iffe re n c e mathematical models in hydrodynamics, Dokl.Akad.Hauk SSSR, 1977» v.255» H6. 4. Goloviznin, V.M., Korshia, T.K., Samarskii, A.A,, and Favorskii A.P. On variation-difference schemes of MED, Preprint IPM, Akad. Hauk SSSR, H 57, 1977* 5* G a s ilo v, V.A., G o lo v izn in, V.M., T ish k in, V.F., and F a v o rs k ii, A.P., numerical solution of the model problem on Rayleigh-Taylor in s ta b ility evolution, Preprint IPM, Akad.Пайк SSSR, N 119» 1977* 6. Volkova, R.A., Goloviznin, V.M., Samarskii, A.A., TJlinitch, F.R., and Favorskii, A.P., Calculation of magnetic cumulation. 7.. V o lo s e v itc h, P.P., G am alii, E.G., G ulin, A.V., Rozanov, V.V., Sam arskii, A.A., Dyurina, П.П., and F a v o r s k ii, A. P.,!Pwo-dimen- s io n a l e f fe c t s during th e la s e r compression o f s p h e ric a l ta r g e ts, Pis»m a v JETF, v.2 4, N 5» 285-286 (1 9 76).