EQUADIFF 6. Erich Martensen The ROTHE method for nonlinear hyperbolic problems. Terms of use:

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1 EQUADIFF 6 Erich Martensen The ROTHE methd fr nnlinear hyperblic prblems In: Jarmír Vsmanský and Milš Zlámal (eds.): Equadiff 6, Prceedings f the Internatinal Cnference n Differential Equatins and Their Applicatins held in Brn, Czechslvakia, Aug , J. E. Purkyně University, Department f Mathematics, Brn, pp. [387] Persistent URL: Terms f use: Masaryk University, 1986 Institute f Mathematics f the Academy f Sciences f the Czech Republic prvides access t digitized dcuments strictly fr persnal use. Each cpy f any part f this dcument must cntain these Terms f use. This paper has been digitized, ptimized fr electrnic delivery and stamped with digital signature within the prject DML-CZ: The Czech Digital Mathematics Library

2 THE ROTHE METHOD FOR NONLINEAR HYPERBOLIC PROBLEMS E. MARTENSEN Mathematinches Iustitut II, Lhiirersitdt 7500 Karlsruhe I, West Germany Karlsruhe The ROTHE methd r the hrizntal methd f lines, if it is applied t parablic as well as t hyperblic evlutin prblems, reduces these prblems t a sequence f elliptic prblems. That frm a frmer pint f view, such an apprach has appeared mre natural f parablic in the case than f hyperblic prblems, may serve <is an explanatin fr the cnsiderable delay f time in studying the methd fr bth classes f prblems. S after ROTHE [11] has intrduced his methd in the early thirties f ur century, numerus parablic differential equatin prblems, linear as well as nnlinear nes, have been treated by it successfully; the names f LADYSHENSKAJA, REKTORYS, NEC'AS, and KACTJR may stand here fr many thers (references, fr instance, may be seen frm the bk f REKTORYS [10]). On the ther hand, effrts fr appjymq the ROTHE methd t hyperblic prblems firstly have been started the last decade. Results have been qiven mainly fr certain durinq linear prblems f mathematical physics, s as fr the wave equatin [1,2,5], the cntinuity equatin [3], and the MAXWELL equatins [4]; recently vibrating string prblem with discntinuus data has been the cmpletely slved by the ROTHE methd [6]. Further linear hyperblic prblems have been investigated by REKTORYS [10]. With regard t nnlinear prblems, hwever, ne is standing at the very beginning. First f MUNZ [8,9] cncerning hyperblic results the quasilinear scalar cnservatin equatin, especially have shwn the ROTHE methd as a suitable tl fr apprximatin f shcks and rarefactin waves. BURGERS In the fllwing we shall cnsider the CAUCHY prblem fr the equatin u +,+-l(u 2 )-0, (x,t)61rx(0, ), (1) t 2 x ' ' where the initial values u( x,0) = u (x), x e IR, (2) ' ' are assumed t be piecewise cntinuus with at mst a finite number f discntinuities and existing limits fr x - +». The ROTHE methd fr a fixed chsen time step length h > 0 leads t the rdinary equatin differential

3 388 u + -^ (u 2 ) ' = U Q xglr (3) which fr qiven u (x), x 6 ]R, has t be slved successively accrdinq t the next time step. On the slutins u(x), x G ]R, f (3), there are impsed piecewise cntinuity with at mst a finite number f discntinuities and existinq limits fr x - + ; furthermre, the square [u(x)] 2. x IR, is asked as a piecewise cntinuusly differentiable functin (as a cnsequence f the freqing, fr the derivative there may ccur at mst a finite number f discntinuities). Withut mentining in detail, the fllwing assertins will cncern t slutins f (3) having at least these prperties. Therem 1 (Behaviur at the infinity). Fr a slutin u(x), xg E, f (3) it hlds lim u(x) = lim u (x ) (4 ) X-* + x-+ Prf fllws immediately frm (3) in cnnectin with the secnd L'HOSPITAL rule: n.. h[u(x) ] 2.. (h d r,,,,\,. 0 = l1 2 - = l l m \2 d^ [ u ( x ) ] J = ll T U (x) - lim u(x) J X-+c X- + x-» ^ X-* + Remark. The prf f Therem 1 makes nly use f the cnservatin prperty f the underlying partial differential equatin (1). Thus the accrdance f the limits (4) will be btained analgusly fr ther hyperblic prblems when they are given in cnservatin frm. Fr instance, this hlds fr the EULER equatins. Therem 2 (Glbal uniqueness). There exists at mst ne cntinuus slutin u(x), x G ]R, f (3). Prf. Assuminq that there exist tw different cntinuus slutins u( x ). v ( x ). xge, s the cntinuus functin w(x):=u(x)-v(x), x 6 E ; des nt vanish everywhere. Nte that because f Therem 1 it hlds lim w(x) = lim u(x) - lim v(x) = 0 (5) X-+ x + x + Let nw x Gl be a pint with w(x )4= 0. If w(x) has at least ne zer ^ 7 in the pen interval (-.x ), then fr cntinuity there exists a maxi mum zer in this interval and we dente it by a< x ; if, hwever, there i ' ' ' are n zeres in (-»,' x ),' we put ^ a = -«. Analqusly vi let b>x dente 0 the minimum zer fr w(x) in (x») r stand fr, respectively. Tgether with (5) we get lim w(x) = lim w(x) = 0. (6) x-*a x-b Observing the cntinuity and piecewise cntinuus differentiability

4 ЗS f [ u( x ) ] 2, [ v ( x ) ] 2, x e 1R, it fllws frm H) and (b) by imprper integratin that Гw(x)dx= {u(x)-v(x)} dx A[ u (x) ] 2 - A[ v(x) ] 2 \> dx d x d x = -^ [LU(X) ] 2 - [ v(x) ],b ^ w(x)(u(x)+v(x)) This is a cntradictin t w ( x )-f 0, xf= ( a, b ) Therem 3 (Lcal unigueness ). Let ( a, b ) :=JR be an arbitrary finite r infinite pen interval and let the abve rdinary differential equatin prblem be frmulated analgusly fr (a,b) instead f IR. Let further u ( x ), x G ( a, b ), be a psitive cntinuus (nega_iy?_^2 n la n^9 u 5) slutin f (3) which has a psitive_limit fr x-a (negatiye imit fr x-b). Then there des nt exist anther cntinuus slutin f (3) with the same limit fr x-a (x-b). Prf nly fr the first case. Assume that there exists a cntinuus slutin v(x), xg (a,b), different the same limit fr x-a. Then the difference w(x) frms a cntinuus functin lim w(x) x a satisfying frm u(x), xg (a,b), but with :=u(x)-v(x),xg(a,b) 0 (7) Next we are able t find a pint x G (a,b) with prperties w(x ) 4= 0, u(x)+v(x)>0. (8) T ' -= Indeed, if u(x) + ( v ( x ), xg (a.b), has n zeres, frm cntinuity and lim {u(x)+v(x)} = 2 lim u(x) > 0 x-a x-a it fllws that u(x)+v(x)>0, x G (a,b), and s it is trivial t find x G(a,b) satisfying ^ (8): if, hwever, u(x)+v(x), ' ' ' xg(a,b), has a zer x G (a,b), s this zer immediately J fulfills the secnd cnditin in (8) and the first cnditin fllws frm v(x ) =-u(x ) as w(x ) = u(x ) v(x)=2u(x)>0 Nw we dente by a* < x the maximum zer fr w(x) in the pen interval (a,x ) if there exists a zer at all, therwise we put a* = a. S in any case when bserving (7), we get lim w(x) = 0 x-a* Then by imprper integratin, it fllws frm (3) and (9) that (9)

5 390 [ w(x)dx = [ {u(x)-v(x)}dx = -j [" A[ u(x) ]- _^L[ v(x) ] 2 j dx a * a * a * = ----w(x)(u(x)+v(x)) 2 = ---- w ( x ) ( u ( x )+v(x )) L J a * 2 here because f (8), we have the cntadictin, that the left hand side has the sign f w(x ) ^ 0 whilst the right hand side either has the ppsite sign r vanishes. Remark 1. Therem 3 gives a hint hw t prceed fr slving the differential equatin (3) uniquely. S if starting at sme pint with a psitive r negative initial value, ne has t integrate t the right r t the left, respectively. On the ther hand, the sign f the exact slutin analgusly indicates the directin f the characteristics. S it turns ut that lcal uniqueness fr the ROTHE slutin is assured by integrating int the directin f characteristics. Remark 2. As it can be seen frm the example u ( x ) = 1, x G 1R, the sign cnditin in Therem u(x) = 1, x G 1R, is the nly ne f 3 plays a significant rle. S the slutin (3) with limit 1 fr x -<», but there exist an infinite number f further slutins with limit 1 fr x - ; indeed, with an arbitrary real cnstant C, such a slutin u(x), x6 3R, may be btained as the inverse f the mntnusly functin x(u)=-h{u+ln(u-1)}+c, u G ( 1, ) decreasing We shall make use f the freging therems when discussing the fllwing fur examples. Examle 1 (MUNZ [8]). If u (x), xg 1, is the step functin with ' value 2 fr negative r 1 fr psitive x, respectively, the exact slutin u(x,t),(x,t)g3rx[0,<»), f the evlutin prblem (1) and (2) is given as a shck wave at x =--t with value 2 left r 1 right f the shck, respectively. Assume that fr an arbitrary time step a ROTHE slutin exists which, fr cnvenience, will be dented by u (x),x G ]R; 1 besides the general prperties mentined abve let this slutin be mntnusly nnincreasing with lwer bund 1, let it have the value 2 fr xg (-» f0), and let it be cntinuus fr xg (0,«). Nte that fr such slutin the limits fr x + exist and that everything hlds fr the given initial functin. The next ROTHE step u(x), x G 3R, then may be cmputed frm (3) as a cntinuus slutin with value 2 fr xg (-«,0 ) ; fr xg[0, ) the slutin fllws by means f the initial cnditin u(0) = 2 in cnnectin with the lwer functin u (x) and the upper

6 :TИ functin 2. This especially yields 1 < u ( x ). < u ( x ), x G ( 0, «) (10) = = 4 Frm the differential equatin (3) tqether with i\0) it t>lk>ws that u'(x) <0, x G ( 0, ) ; s u(x), x G 1R, is mntnusly ru>iiiiinr,ir. mq,md because f (10), it has the lwer bund 1. Thiem '?,is we 1 1.is the first case f Therem 3 say that there JS n fur tine emit lnmni;; slutin, s the next ROTHE step is well-defined. Finally by inductin, all ROTHE slutins are uniquely determined. Because f Therem 1, f <-> r every ROTHE slutin the limit 2 fr x - -<*> r 1 fi x - is btained, respectively. Example 2 (MUNZ [8]). Here u (x), x G IR, is ens I d rd as a step E: ' functin with value 1 fr neqative r 2 fr psitive x, respectively. The exact slutin is a rarefactin wave with values fr t «' x ' 2t 0 < t < and value 1 left r 2 right f the wave, respect tvly. As it turns ut quite similarely t Example 1, the ROTHE methd again e,in be carried ut uniquely. Example 3 (MARTENSEN [7]). The initial values u (x), xp 1R, aregiven as -1 fr negative r 1 fr psitive x, respectively. The exact slutin is a rarefactin wave with values fr -t < x < t, 0 < t < and value -1 left r 1 right f the wave, respectively. Evidently Therem 3 is nt applicable with respect t bth the infinities, rf beginning with the first time step, the ROTHE slutins u(x), xg TR, are further asked t be cntinuus, mntnusly increasing, and skew-symmetric with respect t the rigin, then such slutins can be successively b.y means f a fixed pint methd. Uniqueness cnstructed is nw assured by Therem 2. As a secndary result it turns ut that rill the ROTHE slutins (cntrarily t their sguares) are nt frm each sidedifferent iable at the rigin. tive r -1 fr psitive x, respectively, the exact slutin is btained as a shck wave at x =-- t with value 2 left r -1 right f the shck, respectively. Fr the piecewiese cntinuus ROTHE slutin u(x), x G TR, beginning with the first time step, the further suppsitin is made that the square [u(x)] 2, x G TR, remains cntinuus when passing thrugh a discntinuity; in such a way there is made use f the cnservatin prperty gverning the ROTHE differential eguatin (3). In particular, with a well-defined discntinuity x* G ( 0,«), the ROTHE slutin u ( x ), x G 3F, is btained with cnstant value 2 fr xg (-,0), as a mntnusly decreasing slutin f the differential eguatin (3) fr xg [0,x*] satisfying the initial cnditin u(0) = 2 and the free bundary Example 4 (MUNZ [9]). If u (x), x6 E, has the value 2 fr nega cnditin

7 392 u ( x* ) = 1, and with cnstant value -1 fr x 6 (x* (») i Here Therem 3 leads t lcal uniqueness fr the left interval (_<» x*) as well as fr the right ne (x*,»); furthermre by means f Therem 3, this ROTHE slutin turns ut t be the nly ne with exactly ne discntinuity whilst a cntinuus slutin des nt exist. With regard t the cmplete ROTHE methd, the discntinuities frm a mntnusly increasing sequence. Fr the examples mentined befre numerical cmputatins have been dne by standard methds, where the results have shwn a high accuracy in cmparisn with the exact slutins [7,8,9], Recently fr such nnlinear hyperblic prblems the L -cnvergence f the ROTHE methd with respect t any cmpactum in the upper (x.t)-plane has been prved [9]. The pintwise cnvergence, hwever, remains still as an pen questin. References [1] Gerdes, W.; Martensen, E.: Das Rtheverfahren für die räumlich eindimensinale Wellengleichung. ZAMM 5J3 (1978) T367-T368 [2] Halter, E.: Das Rtheverfahren für das Anfangs-Randwertprblem der Wellengleichung im Außenraum. Dissertatin, Karlsruhe 1979 [3] Halter, E.: The cnvergence f the hrizntal line methd fr the cntinuity equatin with discntinuus data. ZAMP 35 (1984) [4] Martensen, E.: The cnvergence f the hrizntal line methd fr Maxwell's equatins. Math. Methds Appl. Sei. j_ (1979) [5] Martensen, E.: The Rthe methd fr the wave equatin in several space dimensins. Prc. Ry. Sc. Edinburgh 84A (1979) 1-18 [6] Martensen, E.: The Rthe methd fr the vibrating string cntaining cntact discntinuities. Meth. Verf. math. Phys. 26 (1983) [7] Martensen, E.: Apprximatin f a rarefactin wave by discretizatin in time. Applicatins f Mathematics in Technlgy, V. Bffi and H. Neunzert eds. Stuttgart: Teubner 1984, [8] Munz, C.-D. : Über die Gewinnung physikalisch relevanter Stßwellenlösungen mit dem Rtheverfahren. Dissertatin, Karlsruhe 1983 [9] Munz, C.-D.: Apprximate slutin f the Riemann prblem fr the Burgers equatin by the transversal methd f lines. T appear in ZAMP [10] Rektrys, K.: The Methd f Discretizatin in Time and Partial Differential Equatins. Drdrecht/Bstn/Lndn: Reidel Publishing Cmpany 1982 [11] Rthe, E.: Zweidimensinale parablische Randwertaufgeben als Grenzfall eindimensinaler Randwertaufgaben. Math.Ann. 102 (1930)