page 2 2 dscretzaton mantans ths stablty under a sutable restrcton on the tme step. SSP tme dscretzaton methods were frst developed by Shu n [20] and

Transcription

1 page 1 A Survey of Strong Stablty Preservng Hgh Order Tme Dscretzatons Ch-Wang Shu Λ 1 Introducton Numercal soluton for ordnary dfferental equatons (ODEs) s an establshed research area. There are many well establshed methods, such as Runge-Kutta methods and mult-step methods, for such purposes. There are also many excellent books on ths subject, for example [1], [13] and [17]. Specal purpose ODE solvers, such as those for stff ODEs, are also well studed. See, e.g., [6]. However, the class of methods surveyed n ths artcle, the so-called strong stablty preservng (SSP) methods, s somewhat specal. These methods were desgned specfcally for solvng the ODEs comng from a sem-dscrete, spatal dscretzaton of tme dependent partal dfferental equatons (PDEs), especally hyperbolc PDEs. Typcally such ODEs are very large (the sze of the system depends on the spatal dscretzaton mesh sze). More mportantly, there are certan stablty propertesof the orgnal PDE, such as total varaton stablty or maxmum norm stablty, whch could be mantaned by certan specal spatal dscretzatons coupled wth smple frst order Euler forward tme dscretzaton, that would be desrable to mantan also for the hgh order tme dscretzatons. SSP methods are desgned to acheve such a goal. We can thus hghlght the man property of SSP tme dscretzatons: f we assume that the frst order, forward Euler tme dscretzaton of a method of lnes sem-dscrete scheme s stable under a certan norm, then a SSP hgh order tme Λ Dvson of Appled Mathematcs, Brown Unversty, Provdence, RI 02912, USA. E-mal: shu@cfm.brown.edu. Research supported by ARO grant DAAD , NSF grants DMS and ECS , NASA Langley grant NCC and AFOSR grant F

2 page 2 2 dscretzaton mantans ths stablty under a sutable restrcton on the tme step. SSP tme dscretzaton methods were frst developed by Shu n [20] and by Shu and Osher n [21], and were termed TVD (Total Varaton Dmnshng) tme dscretzatons. The termnology was adopted because the method of lnes ODE and ts Euler forward verson satsfy the total varaton dmnshng property when appled to scalar one dmensonal nonlnear hyperbolc conservaton laws. [21] contans a class of second to ffth order Runge-Kutta tme dscretzatons whch are proven SSP. [20] contans a class of frst order SSP Runge-Kutta tme dscretzatons whch have very hgh CFL numbers, as well as a class of hgh order mult-step SSP methods. Later, Gottleb and Shu [9] performed a systematc study of Runge- Kutta SSP methods, showng the optmal two stage, second order and three stage, thrd order SSP Runge-Kutta methods as well as low storage three stage, thrd order SSP Runge-Kutta methods, and provng the non-exstence of four stage, fourth order SSP Runge-Kutta methods wth non-negatve coeffcents. In [10], Gottleb, Shu and Tadmor revewed and further developed SSP Runge-Kutta and mult-step methods. The new results n [10] nclude the optmal explct SSP lnear Runge- Kutta methods, ther applcaton to the strong stablty of coercve approxmatons, a systematc study of explct SSP mult-step methods, and the study of the strong stablty preservng property of mplct Runge-Kutta and mult-step methods. More recently, Spter and Ruuth [23] found a new class of hgh (up to fourth) order SSP Runge-Kutta methods by allowng the number of stages to be larger than the order of accuracy. The same authors also proved that luck runs out n ths approach startng from ffth order: there s no SSP ffth order Runge-Kutta method wth non-negatve coeffcents [18]. Gottleb and Gottleb n [8] obtaned optmal lnear SSP Runge-Kutta methods when the number of stages s larger than the order of accuracy. They have also made an nterestng observaton to use such methods for certan specal varable coeffcent ODEs, such as those comng from spatal dscretzatons for lnear, constant coeffcent PDEs such as the Maxwell's equatons wth tme dependent boundary condtons. One mght ask the queston whether t s worthwhle and necessary to use SSP tme dscretzatons. Numercal examples shown n [9] ndcate that oscllatons and non-lnear nstablty could occur when a lnearly stable but non SSP Runge-Kutta method s appled to a TVD sem-dscrete scheme, whose forward Euler frst order tme dscretzaton can be proven stable. Thus t s at least safer to use SSP tme dscretzatons whenever possble, especally when solvng hyperbolc PDEs wth shocks. In terms of computatonal cost, we remark that most SSP methods are of the same form and have the same cost as tradtonal ODE solvers. It s true that the tme step t mght needto be smaller when SSP s proven than when lnear stablty sproven, however n many stuatons t could be taken larger n practcal calculatons wthout encounterng nstablty. SSP tme dscretzatons have been wdely used n numercal solutons of tme dependent PDEs, especally hyperbolc PDEs. ENO and WENO fnte dfference and fnte volume schemes n [21], [22], [12] and [11], and Runge-Kutta dscontnuous Galerkn fnte element methods n [4] and[5], are such examples. Other examples of applcatons nclude the weghted L 2 SSP hgher order dscretzatons of spectral methods [7], and the L 1 -SSP hgher-order dscretzaton for dscontnuous Galerkn

3 page 3 3 method n [3]. In fact, the (sem) norm can be replaced by anyconvex functon, as the arguments of SSP are based on convex decompostons of hgh-order methods n terms of the frst-order Euler method. An example of ths s the cell entropy stablty property of hgh order schemes studed n [16] and [15]. In ths artcle we survey the current status of the development of SSP hgh order tme dscretzatons. We provde a bref ntroducton wth a smple example for the general framework of the method n secton 2, and present the SSP Runge- Kutta methods n secton 3 and the SSP mult-step methods n secton 4. We gve some concludng remarks n secton 5. 2 General framework of SSP methods We are nterested n solvng the followng method of lnes ODE d u(t) =L(u(t);t) (1) dt resultng from a spatal dscretzaton to a tme dependent partal dfferental equaton. Here u = u(t) s a (usually very long) vector and L(u; t) depends on u ether lnearly or non-lnearly. In many applcatons L(u; t) = L(u) whch does not explctly depend on t. We would also consder an adjont problem d dt u(t) = ~ L(u(t);t); (2) whch s closely related to (1). In fact, both L and ~ L approxmate the same spatal dervatves n the orgnal PDE. The dfference s n ther assocated stablty property when the ODEs (1) and (2) are dscretzed n tme. Throughout ths we assume that the frst order Euler forward tme dscretzaton to (1): u n+1 = u n + tl(u n ;t n ); (3) where u n s an approxmaton to u(t n ), as well as the frst order Euler backward tme dscretzaton to (2): are stable under a certan (sem) norm wth a sutable tme step restrcton u n+1 = u n t~ L(u n ;t n ); (4) jju n+1 jj» jju n jj (5) t» t 0 : (6) Let us gve a very smple example to llustrate ths assumpton. Assume we are solvng the non-lnear scalar one dmensonal conservaton law v t = f(v) x

4 page 4 4 where f 0 (v) 0, wth perodc boundary condtons and a unform spatal mesh x 1 <x 2 < <x N. The followng frst order upwnd scheme f(v n vj n+1 = vj n j+1 ) f(vj n) + t ; where v n j s an approxmaton to v(x j ;t n ), as well as the adjont scheme v n+1 j = v n j t f(v n j ) f(v n j 1 ) ; are both total varaton dmnshng (TVD). That s, f we defne u =(v 1 ;v 2 ; ;v N ) T ; (7) f(v n L(u) = 2 ) f(v n 1 ) ; f(vn 3 ) f(v n 2 ) ; ; f(vn 1 ) f(vn n ) T ; ~L(u) = f(v n 1 ) f(v n N ) and the total varaton sem-norm by ; f(vn 2 ) f(v n 1 ) ; ; f(vn N ) f(vn ) T N 1 ; jjujj = NX j=1 jv j+1 v j j for any u defned n (7) and satsfyng the perodcty condton v N+1 = v 1, then we have the stablty property (5) under the tme step restrcton (6) wth t 0 =maxjf 0 (v)j; v for both (3) and (4). Hgher order TVD spatal dscretzatons can also be desgned to satsfy these propertes. Examples would nclude the Runge-Kutta dscontnuous Galerkn methods n [4]. Wth ths assumpton, we would lke to fnd SSP tme dscretzaton methods to (1), sometmes wth the help of (2), that are hgher order accurate n tme, yet stll mantan the same stablty condton (5), perhaps wth a dfferent restrcton on the tme step t than that n (6): t» c t 0 ; (8) where c s called the CFL coeffcent of the SSP method. The objectve s to fnd such methods wth smple format, low computatonal cost and less restrcton on the tme step t (larger CFL coeffcent c). We wll dscuss SSP Runge-Kutta methods n the next secton and SSP mult-step methods n secton 4. We remark that the strong stablty assumpton for the forward Euler step n (5) can be relaxed to the more general stablty assumpton jju n+1 jj» (1 + O( t))jju n jj:

5 page 5 5 Ths general stablty property wll also be preserved by the hgh order SSP tme dscretzatons. The total varaton bounded (TVB) methods dscussed n [19] and [4] belong to ths category. However, f the forward Euler operator s not stable, the framework of SSP cannot be used to determne whether a hgh order tme dscretzaton s stable. 3 SSP Runge-Kutta methods In [21], a general m stage Runge-Kutta method for (1) s wrtten n the form: u (0) = u n ; u () = u n+1 = u (m) X 1 ff ;k u (k) + tf ;k L(u (k) ;t n + d k t) ; =1;:::;m (9) where d k are related to ff ;k and f ;k by d 0 =0; d = X 1 (ff ;k d k + f ;k ); =1;:::;m 1: Thus we do not need to dscuss the choce of d k separately. Notce that n most ODE lterature, e.g. [1], a Runge-Kutta method s wrtten n the form of a Butcher array. Every Runge-Kutta method n the form of (9) can be easly converted n a unque way nto a Butcher array, e.g.,[21]. A Runge-Kutta method wrtten n a Butcher array can also be rewrtten nto the form (9), however ths converson s n general not unque. Ths non-unqueness n the representaton (9) s explored n the lterature to seek the largest provable tme steps (8) for SSP. We always need and requre that ff ;k 0 n (9). If ths s volated no SSP methods are possble. If all the f ;k 's n (9) are also nonnegatve, f ;k 0, we have the followng smple lemma whch s the backbone of SSP Runge-Kutta methods: Lemma 1. [21] If the forward Euler method (3) s stable n the sense of (5) under the tme step restrcton (6), then the Runge-Kutta method (9) wth ff ;k 0 and f ;k 0 s SSP,.e. ts soluton also satsfes the same stablty (5), under the tme step restrcton (8) wth the CFL coeffcent c =mn ;k ff ;k f ;k : (10) Proof: Snce the forward Euler method (3) s stable n the sense of (5) under the tme step restrcton (6), we have jju (k) + f ;k ff ;k tl(u (k) ;t n + d k t)jj» jju (k) jj

6 page 6 6 f the tme step restrcton (8) s satsfed wth c gven by (10). Also notce that, by consstency, We now use nducton to prove X 1 ff ;k =1: jju (k) jj» jju n jj (11) for k =0; 1;:::;m. Clearly (11) s vald for k =0. Assumng that t s vald for all k» 1, we then obtan jju () jj»»» X 1 X 1 X 1 The proof s thus completed. ff ;k jju (k) + f ;k ff ;k tl(u (k) ;t n + d k t)jj ff ;k jju (k) jj ff ;k jju n jj = jju n jj: Notce that the proof demonstrates that all the ntermedate stages u () are also stable under the same norm. Ths s mportant for practcal calculatons, as nstablty n the mddle stages could lead to unphyscal artfacts such as negatve densty and pressure n gas dynamcs calculatons, thus preventng one from even fnshng the Runge-Kutta step even though the complete step may be stable. The most popular and successful SSP methods are those covered by Lemma 1. We wll gve some commonly used examples later. If some of the f ;k 's must be negatve, we need the help of the adjont operator ~L n (2). Remember that the Euler backward tme dscretzaton to (2), gven by (4), s assumed to be stable n the sense of (5). Wth a smlar proof to that of Lemma 1, we then have the followng lemma for ths general case: Lemma 2. [21] If the forward Euler method (3) and the backward Euler method (4) are both stable n the sense of (5) under the tme step restrcton (6), then the Runge-Kutta method (9) wth ff ;k 0, and wth f ;k L replaced by f ;k ~ L whenever f ;k s negatve, s SSP,.e. ts soluton also satsfes the same stablty (5), under the tme step restrcton (8) wth the CFL coeffcent c =mn ;k ff ;k jf ;k j : (12)

7 page 7 7 SSP methods covered by Lemma 2 wth negatve f ;k 's are less popular n applcatons, partly because people do not want to deal wth the unfrendly ~ L (although ts constructon for PDEs s n general very easy and smlar to that for L, see for the smple example gven n the prevous secton), and partly because of the extra computatonal and storage costs assocated wth the addtonal operator ~ L. We now gve examples of some of the most useful SSP Runge-Kutta methods. 3.1 Lnear SSP Runge-Kutta methods The stuaton s dramatcally smplfed f L(u; t) = Lu s lnear wth constant coeffcents. The optmal m stage, m-th order SSP Runge-Kutta method for such lnear constant coeffcent case s gven by [10]: u () = u ( 1) + tlu ( 1) ; =1;:::;m 1 (13) u (m) = where ff 1;0 =1and m 2 X ff m;k u (k) + ff m;m 1 u (m 1) + tlu (m 1) ; ff m;k = 1 k ff m 1;k 1; k =1;:::;m 2 (14) ff m;m 1 = 1 m! ; ff m;0 =1 m 1 X k=1 ff m;k : Ths class of Runge-Kutta methods s SSP wth the CFL coeffcent c = 1 n (8). Notce that, for ths lnear case, there exsts only one m stage, m-th order Runge-Kutta method. So f we gnore the possble dfferences n the mddle stages u () for 1»» m 1, we are n effect clamng that ths unque m stage, m-th order Runge-Kutta method s SSP. Such methods are very useful when one solves a constant coeffcent PDE such as the Maxwell's equaton usng a lnear method, such as the dscontnuous Galerkn method wthout usng nonlnear lmters. Another applcaton s to the lnear and coercve approxmatons for parabolc type problems, see [10] and [14]. If one relaxes the condton on the number of stages so that t can be bgger than the order of accuracy, then the avalable Runge-Kutta method s no longer unque. For example, t s possble to get a (m + 1) stage, m-th order SSP Runge- Kutta method wth CFL coeffcent c = 2 n (8). See [8] for detals. Fnally, we quote a very nterestng result of [8], whch clams that one can use ths class of SSP Runge-Kutta methods on certan tme dependent lnear problems of the form d u(t) =Lu(t)+f(t) (15) dt whch could arse n, e.g. a dscretzaton of a lnear, constant coeffcent PDE such as the Maxwell's equaton wth a tme dependent boundary condton. The trck

8 page 8 8 s to approxmate f(t) by a polynomal n t, for example usng the Legendre or Chebyshev seres approxmatons, or by any expanson f(t) = KX a k ff k (t) where d ff(t) =Bff(t) dt for some constant matrx B. Here ff(t) =(ff 0 (t); ;ff K (t)) T : Trgonometrc expansons and exponental expansons also belong to ths class. Clearly, f we denote then equaton (15) becomes where w =(u; ff) T ; A = dag(a 0 ;a 1 ;:::;a K ) d w(t) =Cw(t) dt L A C = 0 B s a constant matrx. Hence SSP methods gven n ths subsecton would apply. 3.2 Second order nonlnear SSP Runge-Kutta methods The optmal second order, two stage nonlnear SSP Runge-Kutta method s gven by [21], [9]: u (1) = u n + tl(u n ;t n ) (16) u n+1 = 1 2 un u(1) tl(u(1) ;t n + t) wth a CFL coeffcent c = 1 n (8). Ths s just the classcal Heun method or modfed Euler method. Ths method s clearly no more expensve or complcated than any other second order Runge-Kutta methods, wth ts added assurance of SSP wth a healthy CFL coeffcent c = 1. Thus t should be used whenever second order accuracy n tme s desred and non-lnear stablty s a concern. If one relaxes the condton on the number of stages so that t can be bgger than the order of accuracy, then SSP methods can be found to have larger CFL coeffcents. For example, t s possble to get a three stage, second order SSP Runge-Kutta method wth CFL coeffcent c = 2 n (8), or a four stage, second order SSP Runge-Kutta method wth CFL coeffcent c = 3 n (8). See [23] for detals.

9 page Thrd order nonlnear SSP Runge-Kutta methods The optmal thrd order, three stage nonlnear SSP Runge-Kutta method s gven by [21], [9]: u (1) = u n + tl(u n ;t n ) u (2) = 3 4 un u(1) tl(u(1) ;t n + t) (17) u n+1 = 1 3 un u(2) tl(u(2) ;t n t); wth a CFL coeffcent c = 1 n (8). Thssby far the most wdely used SSP method n the lterature. It s clearly no more expensve or complcated than any other thrd order Runge-Kutta methods, wth ts added assurance of SSP wth a healthy CFL coeffcent c =1. Thus t should be used whenever thrd order accuracy n tme s desred and non-lnear stablty s a concern. If one relaxes the condton on the number of stages so that t can be bgger than three, the order of accuracy, thenssp methods can be found to have larger CFL coeffcents. For example, we quote the followng four stage, thrd order SSP Runge-Kutta method from [23] whch has a CFL coeffcent c = 2 n (8): u (1) = u n tl(un ;t n ) u (2) = u (1) tl(u(1) ;t n 1 + (18) 2 t) u (3) = 2 3 un u(2) tl(u(2) ;t n + t) u n+1 = u (3) tl(u(3) ;t n t): 3.4 Thrd order nonlnear SSP Runge-Kutta methods wth low storage The general low-storage Runge-Kutta schemes can be wrtten n the form [24], [2]: u (0) = u n ; du (0) =0; du () = A du ( 1) + tl(u ( 1) ;t n + d 1 t); =1;:::;m; (19) u () = u ( 1) + B du () ; u n+1 = u (m) : =1;:::;m; Only u and du must be stored, resultng n two storage unts for each varable. Followng Carpenter and Kennedy [2], the best SSP thrd order method found by numercal search n [9] s gven by the system, wth the free parameter b chosen as b = 0:924574: z 1 = p 36b 4 +36b 3 135b 2 +84b 12

10 page z 2 =2b 2 + b 2 z 3 =12b 4 18b 3 +18b 2 11b +2 z 4 =36b 4 36b 3 +13b 2 8b +4 z 5 =69b 3 62b 2 +28b 8 z 6 =34b 4 46b 3 +34b 2 13b +2 d 0 =0 A 1 =0 B 1 = b d 1 = B 1 z 1 (6b 4b +1)+3z 3 A 2 = (2b +1)z 1 3(b +2)(2b 1) 2 B 2 = 12b(b 1)(3z 2 z 1 ) (3z 2 z 1 ) 2 144b(3b 2)(b 1) 2 d 2 = B 1 + B 2 + B 2 A 2 A 3 = z 1z (2b 1)b 5 3(2b 1)z 5 24z 1 b(b 1) 4 +72bz 6 +72b 6 (2b 13) B 3 = 24(3b 2)(b 1) 2 (3z 2 z 1 ) 2 12b(b 1)(3z 2 z 1 ) wth a CFL coeffcent c = 0:32 n (8). Ths s of course less optmal than (17) n terms of the CFL coeffcent, but the low storage form s useful for large scale calculatons. Ths method can be coded up usng only two arrays, one for u and the other for du. Thus ths method s a favorte when storage s the paramount consderaton, such as for large scale three dmensonal calculatons usng small computers. 3.5 Fourth order nonlnear SSP Runge-Kutta methods It s proven n [9] that all four stage, fourth order SSP Runge-Kutta scheme (9) wth a nonzero CFL coeffcent c n (8) must have at least one negatve f ;k. Such schemes are very ugly lookng [21], [9] and they have never ganed any popularty n the applcatons. So we wll not lst them here. Interested readers can fnd them n [21] and [9]. If one relaxes the condton on the number of stages so that t can be bgger than the order of accuracy, then SSP methods can be found to have non-negatve ff ;k and f ;k and postve CFL coeffcent c n (8), see [23]. For example, the followng fve stage, fourth order Runge-Kutta method [23] s SSP wth a CFL coeffcent c =1:508 n (8): u (1) = u n +0: tl(u n ;t n ) u (2) =0: u n +0: u (1) +0: tl(u (1) ;t n +0: t) u (3) =0: u n +0: u (2)

11 page : tl(u (2) ;t n +0: t) u (4) =0: u n +0: u (3) (20) +0: tl(u (3) ;t n +0: t) u n+1 =0: u n +0: u (2) +0: u (3) +0: tl(u (3) ;t n +0: t) +0: u (4) +0: tl(u (4) ;t n +0: t): 4 SSP mult-step methods In [20], a general m step method for (1) s wrtten n the form: u n+1 = mx ff u n+1 + tf L(u n+1 ;t ) n+1 : (21) =1 Agan, we requre that ff 0 n (21). If ths s volated no SSP methods are possble. Smlar to Lemma 1 and Lemma 2, we have the followng smple lemmas for the mult-step methods n (21) to be SSP. Lemma 3. [20] If the forward Euler method (3) s stable n the sense of (5) under the tme step restrcton (6), then the mult-step method (21) wth ff 0 and f 0 s SSP,.e. ts soluton also satsfes the same stablty (5),under the tme step restrcton (8) wth the CFL coeffcent c =mn ff f : (22) Lemma 4. [20] If the forward Euler method (3) and the backward Euler method (4) are both stable n the sense of (5) under the tme step restrcton (6), then the mult-step method (21) wth ff 0, and wth f L replaced by f ~ L whenever f s negatve, s SSP,.e. ts soluton also satsfes the same stablty (5), under the tme step restrcton (8) wth the CFL coeffcent c = mn ff jf j : (23) The proofs of these lemmas are smlar to that for Lemmas 1 and 2, namely usng the fact P ff = 1 from consstency, and the observaton that u n+1 can be wrtten as a convex combnaton of forward Euler steps wth sutably scaled t's.

12 page As before, we would lke to have mult-step SSP methods wth non-negatve coeffcents covered by Lemma 3, f at all possble, to avod the complcaton and extra computatonal and storage costs assocate wth ~ L. It s proven n [10] that we need at least m +1 steps for an m-th order SSP method wth non-negatve coeffcents. The SSP mult-step methods seem to be less popular n applcatons than the SSP Runge-Kutta methods, perhaps because of the complcaton of the start-up of the frst few tme steps, the dffculty of adjustng tme steps n the mddle of the calculaton, and the relatvely larger storage requrements, all these beng generc problems wth mult-step methods. On the other hand, t s usually easer to satsfy the order of accuracy condtons for mult-step methods than for Runge-Kutta methods, especally for orders of accuracy hgher than four. We now gve examples of some of the SSP mult-step methods. 4.1 Second order SSP mult-step methods The optmal second order, three step SSP method wth non-negatve coeffcents s gven by [20], [10]: u n+1 = 3 4 un tl(un ;t n )+ 1 4 un 1 (24) wth a CFL coeffcent c = 1 n (8). 2 We remark that ths SSP mult-step method has the same effcency as the optmal two stage, second order Runge-Kutta method (16). Ths s because there s only one L evaluaton per tme step here, compared wth two L evaluatons n the two stage Runge-Kutta method. Of course, the storage requrement here s larger. Agan, f one ncreases the number of steps, then SSP methods can be found to have larger CFL coeffcents. For example, t s possble to fnd a four step, second order SSP method wth non-negatve coeffcents and a CFL coeffcent c = 2, 3 see [20] and [10]. Notce that unlke n the case of Runge-Kutta methods, here the computatonal cost s not ncreased much wth an ncrease of the number of steps, because the most expensve evaluaton of L s performed only once. However, storage would be ncreased wth ths ncrease of the number of steps. 4.2 Thrd order SSP mult-step methods The optmal thrd order, four step SSP method wth non-negatve coeffcents s gven by [20], [10]: u n+1 = un tl(un ;t n ) un tl(un 1 ;t n 1 ) (25) wth a CFL coeffcent c = 1 n (8). 3 We remark that ths SSP mult-step method has the same effcency as the optmal three stage, thrd order Runge-Kutta method (17). Ths s because there

13 page s only one L evaluaton per tme step here, compared wth three L evaluatons n the three stage Runge-Kutta method. Of course, the storage requrement here s larger. Agan, f one ncreases the number of steps, then SSP methods can be found to have larger CFL coeffcents. For example, t s possble to fnd a fve step, thrd order SSP method wth non-negatve coeffcents and a CFL coeffcent c = 1 2,see [20] and [10]. Because there s no sgnfcant ncrease n the computatonal cost when the number of steps s ncreased, when storage s not a consderaton, t s advantageous to use a SSP mult-step methods wth more steps and hgher CFL coeffcents. 5 Concludng remarks We have gven a bref survey of the strong stablty preservng, or SSP, hgh order tme dscretzatons for method of lnes ODEs resultng from a spatal dscretzaton of PDEs, especally the hyperbolc type PDEs. The most popular SSP methods n applcatons have been lsted.

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