828. Piecewise exact solution of nonlinear momentum conservation equation with unconditional stability for time increment

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1 88. Pecewse exact solto of olear mometm coservato eqato wth codtoal stablty for tme cremet Chaghwa Jag, Hyoseob Km, Sokhwa Cho 3, Jho Km 4 Korea Itellectal Property Offce, Daejeo, Korea, 3 Kookm Uversty, Seol, Korea 4 Yegam Uversty, Kygsa, Korea E-mal: cjag@kpo.go.kr, hkm@kookm.ac.kr, 3 shcho@kookm.ac.kr, 4 jho@y.ac.kr (Receved Jly 0; accepted 4 September 0) Abstract. Exact solto s adopted for comptato of the vscd Brgers eqato o fte dfferece grd. Ital codto ad followg compted vales of the depedet varable are assmed to be pecewsely lear betwee fxed grd pots, ad local exact solto s sed to fd the vale at the ext tme step at each grd pot. Comparsos of Pecewse Exact Solto Method (PESM), exstg pwd scheme, ad the aalytc solto show that the preset method s more accrate tha the pwd scheme. The codtoal stablty s a strog mert of ths method ad s show wth a test reslt. Keywords: vscd Brgers eqato, pecewse exact solto method, pwd scheme, codtoal stablty. Itrodcto Not oly the advecto eqato bt also the advecto-dffso eqato has bee tesvely sed to expla the flow of flds mathematcal, physcal, scetfc, ad egeerg problems [-3]. The advecto eqato becomes the advecto-dffso eqato by addg dffso terms wth dffso coeffcets. Reversely, the advecto eqato ca be thoght as a partal eqato cotag advecto pheomea oly by cttg off the dffso, or separatg the partal dfferetal operator to two, ad gettg two eqatos,.e. the advecto eqato ad the dffso eqato (fractoal step method [4, 5] or operator-splttg method [6]). The advecto eqato ofte descrbes trasport of a scalar property, e.g. cocetrato, temperatre etc. A very specfc case of the advecto eqato s the Brgers eqato, whe the depedet varable s the velocty tself [7, 8] stead of a arbtrary scalar property. The advecto eqato has bee a focs of mercal modelg de to possble mercal dffso or wggles. Calclato of the vscd Brgers eqato s cocered here. Eve thogh the Brgers eqato s olear, ad dfferet from lear advecto eqato, people have sed smlar mercal methods to solve the Brgers eqato. Fdametally, the advecto-dffso eqato: + a = ν t x x () s redced to the advecto eqato: + a = 0 t x () The vscd Brgers eqato s a specfc form of the advecto eqato, whe a=. 04

2 + = 0 t x (3) The same mercal schemes for lear advecto eqato ca also be appled to the above olear advecto eqato bt the velocty shold be represeted by a fte dfferece vale. The olear Brgers eqato also fleces the stablty problem. The vscd Brgers eqato s sefl becase t has a close lk wth the coservato laws. The vscd Brgers eqato s a example of the olear mometm coservato eqato. Soltos for the vscd Brgers eqato ca be obtaed o spatally fxed grd by sg exstg mercal methods [9, 0]. The vscd Brgers eqato ca be wrtte coservatve form as: F ( ) + = 0 t x (4) F( ) = (5) where x s the spatal axs, t s the tme, ad s the velocty the x axs. The velocty feld, ( x, t ), at the tal pot ( t = 0 ), s ( x, 0), ad s called f ( x ). The aalytc solto of Eqato (4) for the gve tal codto s: ( x, t) = ( x,0) = f ( x ) (6) L where L s the velocty for Lagragea posto: 0 x= x + t = x + f ( x ) t (7) 0 L 0 0 ( x, t) = ( x + f ( x ) t, t) = ( x, t) = ( x,0) = f ( x ) (8) 0 0 L 0 L 0 0 The problem s that t s ot always possble to get the solto a explct form for a arbtrary fcto, f. Oe of freqetly sed mercal schemes to solve the advecto eqato s the pwd scheme [9]: + + = ( or ) t + (9) where s a represetatve velocty at mber, whch cold be, +,, ( + ) /, ( + ) /. If we se the dfferece eqato for the Brgers eqato the + coservatve form, the dfferece eqato becomes: t { } ( ) ( ),or ( ) ( ) + = (0) 04

3 the ( + ) / or ( + ) / correspods to the represetatve velocty. + Smlar the above pwd scheme, the Lax-Wedroff scheme [3], the modfed Lax- Wedroff scheme by Roe [5], TVD [6], the box scheme [7], or the Leap-frog scheme [8] adopt dfferet mercal schemes whch are sefl for the lear advecto eqato, bt ot for the Brgers eqato a sese that the selecto of the represetatve velocty vale prodces small or large mercal errors. There have bee trals to make se of the exact solto for fxed grd system. A Lagragea method, MAC, trodces movg partcles whch coserve physcal propertes, bt reqres cmbersome treatmet of the desty of the partcles a fxed grd space, whch s a defect of the method []. Here we propose a ew method to make se of the exact solto of the vscd Brger s eqato. The shock waves ca be geerated whle solvg the Brgers eqato depedg o codtos [4]. Pecewse Exact Solto Method If the depedet varable s tally dstrbted learly space the we fd a exact solto whch s: ( x, t ) = dx+ e () 0 x+ c = () t where c s a coeffcet, c= e / d. Eqato () satsfes the vscd Brgers eqato (4). A terestg featre of ths solto s that the tal tme, t 0, caot be arbtrarly provded, bt a fxed vale, / d, shold be assged. If the startg tme s egatve, ad tme proceeds, the we ecoter a sglarty at t = 0 whch meas a shock, as tme. If two bodary vale sets of x ad are gve, Eqato () ca be sed for both acceleratg ad deceleratg phases by sg ether postve or egatve doma for x or t. Gve the bodary vales at two adjacet grd pots, we ca fd the solto at ew tme step o a grd pot by sg above the exact solto. For postve, spatally creasg velocty the postve x axs, for stace, Eqato () ca be expressed a mercal form as: x + c x x+ c x + c ; + = = ; = (3) t t t + t where x s the spatal cremet, t s the temporal cremet, the sbscrpt or s the mber of grd pot, ad the sperscrpt or + s the mber of tme step. Droppg ot x ad t, the exact velocty vale at ode at the ew tme step becomes (Fgre ): + = ( ) t+ (4) 043

4 Fg.. Pecewse exact solto for postve creasg velocty dstrbto If the frot sde velocty tmes temporal cremet ( m m t ) exceeds m tmes the spatal cremet ( x ), see the segmet of A ( x, ) ad B( x, ) Fgre, we obta the followg exact solto: { } ( m ) m, + = + m m ( ) t+ m m m for m t < ( m+ ) x, m=,, 3, (5) Eqato (5) also satsfes Eqato (4) for m= 0, ths ts se ca be exteded for m= 0,,,. m m Fg.. Pecewse exact solto for large temporal cremet ( m ) For postve, spatally decreasg velocty dstrbto the x axs, the velocty at grd pot at ew tme step ca be calclated from the same eqato, Eqato (4), see Fgre 3. Ths solto has smlarty to exstg pwd schemes whch pstream vales at prevos tme step are sed for calclato of the veloctes at the ew tme step. For egatve veloctes (Fgre 4), the followg eqato s sed: + = ( ) t+ + (6) 044

5 Fg. 3. Pecewse exact solto for postve decreasg velocty dstrbto Fg. 4. Pecewse exact solto for egatve creasg velocty dstrbto Whe the two eghborg veloctes, ad, are the opposte drecto (Fgres 5 ad 6), oe case s that velocty creases the x axs: + = t+ + = t+ (7) (8) If the two adjacet velocty vales have opposte sgs, bt velocty decreases the x axs, the Eqatos (4) ad (6) are sed. The preset method remas vald less the spatal velocty gradet becomes fty. The fte velocty gradet correspods to the sglarty of the fcto F, ad coseqetly solto caot be obtaed mercally from the sglarty. Physcal smlarty exst e.g. shock waves, wave breakg, or hydralc jmp. 3. Reslts ad dscsso The preset method s sg pecewse exact solto for each segmet of the gve arbtrary dstrbto of velocty, ad shortly called as PESM afterwards. If the gve fcto s a lear fcto a wde comptato doma, the solto remas as a lear fcto as tme 045

6 proceeds. I order to exame ts accracy PESM s appled to a case of a lear tal codto for postve x: ( x,0) = x, x=.0, 0 x 00, t=.0, 0< t 5. Fg. 5. Pecewse exact solto for flatteg veloctes of opposte sgs Fg. 6. Pecewse exact solto for steepeg veloctes of opposte sgs 046 Fg. 7. Test reslts of PESM ad Godov for case of lear velocty dstrbto

7 Fg. 8. Zoomed plot of Fgre 6 for t =0.0 Fg. 9. Zoomed plot of Fgre 6 for t =5.0 The reslts of the preset method cocde wth the aalytc solto throghot the tme as expected, becase the preset method ses pecewse exact solto, ad the gve fcto has a sgle gradet wth the whole doma. The reslts were also compared wth the reslts of a exstg scheme [9]. Godov s scheme s a frst-order pwd scheme. The pwd scheme s ot free from mercal error at each tme step, ad accmlates larger total error as tme proceeds, see Fgs. 7, 8, ad 9. Next, the PESM was appled to a expaso wave composed of three lear fctos, costat (left velocty), straght coecto le, ad costat (rght velocty): l ( x,0) = = 0.5, x< 5 l 3 (,0) = ( ) ( 6) + = l r 4 +, 4 5 x 7 ( x,0) = =.0, x> 7 r x=, t=, 0 t 6. 3 Reslts for both the PESM ad Godov s scheme show smlar behavor a large scale plot, Fgre 0. The compted cetral posto of the wave wth both the PESM ad Godov s 047 r

8 scheme are exact. However Godov s scheme becomes farther from the aalytc solto as tme proceeds. Nmercal dffso of the PESM s slghtly smaller tha that of Godov s scheme, see Fgres 0(a) ad 0(b). Smlarly, the PESM ad Godov s scheme were appled to a shock wave case: ( x,0) = =.0, x< 5 l 3 (,0) = ( ) (6 ) + = l r 4 +, 4 5 x 7 ( x,0) = = 0.5, x> 7 r x=, t=, 0 t 6. 3 The PESM aga prodces smlar shape of comptato reslts to Godov s frst order pwd scheme a large scale plot (Fgre ). Nmercal dffso of the PESM s slghtly smaller tha Godov s scheme, see Fgres (a) ad (b). The PESM shows more accrate solto for both spreadg ad shock waves, whch dstgshes tself from the exstg frst order pwd schemes. Fg. 0. Calclated reslts of expaso wave by PESM ad Godov Itally smooth dstrbto of the velocty space ca be approxmated by pecewse lear fctos ad relatvely smooth varato of ther gradets. The PESM was appled to the velocty dstrbto of a bell-shaped fcto wth a rogh resolto: x ( x,0) = exp, < x< 60 x= 0.0, t= 0.0 or 0.03, 0 t

9 Fg.. Calclated reslts of shock wave by PESM ad Godov At t= 0.09, the compted velocty feld for the tme cremet of 0.0 clded maxmm total error of 0.5 arod the bell crest, whch s smlar to the maxmm total error of 0.3 Godov s scheme, see Fgre. Now, a creased tme cremet, t= 0.03, was appled to the same codto, whch does ot satsfy the CFL codto. The compted reslts at tme of 0.09 by sg the PESM are stable, whle those by sg Godov s scheme show the symptom of stablty, see Fgre 3. Ths demostrates the codtoal stablty of the PESM wth respect to the tme cremet, less the problem actally reaches sglarty. Fg.. Comparso wth the velocty dstrbto at tme of 0.09 for t= 0.0 It cold be sad that the PESM gves more accrate soltos tha Godov s frst order pwd scheme becase of ts exactess. Therefore, replacg the frst-order fte dfferece 049

10 part may exstg schemes wth the preset method cold be recommeded here. For example, the exstg form of Fromm s scheme [] s to be sed to solve the vscd Brgers eqato, the: + t t t = + 4 a a a (9) cold be modfed to: + t t = + ( ) t x 4 x x x x + a a (0) where a s a velocty or characterstc speed the x-drecto, ad a dscrete vale shold be assged f t s sed for the Brgers eqato. Fg. 3. Comparso wth the velocty dstrbto at tme of 0.09 for t= 0.03 Ad the exstg Lax-Wedroff scheme [3]: + t t t = + a a a () cold be modfed to: + t t = + ( ) t x x x x x + a a () Usg Eqatos (0) ad () may reprodce more accrate speed of the wave phase tha Eqato (9) ad (), as was demostrated Fgres 8 ad 9, eve throgh these replacemets are ot the major terest of ths paper. 050

11 Coclsos Accracy s oe of the most mportat objects drg solvg the mometm coservato eqato. Most exstg mercal methods have bee developed for the lear advecto eqato. The PESM s a farly smple method, ad stll prodces more accrate soltos for the vscd Brgers eqato tha Godov s frst order pwd scheme whch s adeqate for the lear advecto eqato. The PESM s aga a codtoally stable method for solvg the vscd Brgers eqato. If the tal codto s descrbed by cotos pecewse lear fctos, the solto ca be obtaed wth arbtrary tme cremet less ay shock wave develops. The preset PESM cold be sed effectvely to solve the mometm coservato eqato, especally whe t s combed to fractoal-step method. Ackowledgemets Ths research was a part of the Project ttled Mare ad Evrometal Predcto System (MEPS) fded by the Mstry of Lad, Trasport ad Martme Affars, Korea 0 ad spported by Kookm Uversty as a Uversty Grat 0. Refereces [] Morto K. W., Mayers D. F. Nmercal Solto of Partal Dfferetal Eqatos. Cambrdge Uversty Press, Cambrdge, UK, 005, p [] Ibragmov N. H. Hadbook of Le Grops to Aalyss of Dfferetal Eqatos. Vol., CRC Press, Boca Rato, USA, 994. [3] Polya A. D., Zatsev V. F. Hadbook of Nolear Partal Dfferetal Eqatos. Chapma &Hall/CRC, Boca Rato, USA, 004. [4] Perot J. B. A aalyss of the fractoal step method. J. Compt. Phys., Vol. 08, 993, p [5] Armfled S., Street R. Modfed fractoal-step methods for the Naver-Stokes eqatos. ANZIAM J., Vol. 45, No. E, 004, p. C364-C377. [6] Karlse K. H., Rsebro N. H. A operator splttg method for olear covecto-dffso eqato. Nmer. Math., Vol. 77, No. 3, 997, p [7] Brgers J. M. The Nolear Dffso Eqato. Redel, Dordreht, Netherlads, 974. [8] Brgers J. M. A mathematcal model llstratg the theory of trblece. Adv. Appl. Math., Vol., 948, p [9] Godov S. K. A dfferece scheme for mercal solto of dscotos solto of hydrodyamc eqatos. Math. Sbork, Vol. 47, 959, p [0] Va Leer B. Mootcty ad coservato combed a secod-order scheme. J. Compt. Phys., Vol. 4, 974, p [] Welch J. E., Harlow F. H., Shao J. P., Daly B. J. The MAC Method. Los Alamos Scetfc Lab. Rept. LA-345, Los Alamos, New Mexco, USA, 966. [] Fromm J. E. A method for redcg dsperso covectve dfferece schemes. J. Compt. Phys., Vol. 3, 968, p [3] Lax P. D., Wedroff B. Systems of coservato laws. Comm. Pre Appl. Math., Vol. 3, No., 960, p [4] Sarra S. A. The method of characterstcs wth applcatos to coservato laws. J. of Ole Math. ad Its Appl., Vol. 3, 003. [5] Roe P. L. Approxmate Rema solvers, parameter vectors, ad dfferece schemes. J. of Compt. Phys., Vol. 43, 98, p [6] Harte A. O a class of hgh resolto total-varato-stable fte dfferece schemes. SIAM J. Nmer. Aal., Vol., 984, p. -3. [7] Thomee V. A stable dfferece scheme for the mxed bodary vale problem for a hyperbolc, frstorder system two dmesos. J. Soc. Idst. Appl. Math., Vol. 0, 96, p [8] Flather R. A., Heaps N. S. Tdal comptatos for Morecamble Bay. Geo-Phys. J. R. Astr. Soc., Vol. 4, 975, p