A. T. Sornborger a a Department of Mathematics and Faculty of Engineering,

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This articl was downloadd by: [Univrsity of California Davis] On: 29 May 2013, At: 13:57 Publishr: Taylor & Francis Informa Ltd Rgistrd in England and Wals Rgistrd Numbr: 1072954 Rgistrd offic:

1 This articl was downloadd by: [Univrsity of California Davis] On: 29 May 2013, At: 13:57 Publishr: Taylor & Francis Informa Ltd Rgistrd in England and Wals Rgistrd Numbr: Rgistrd offic: Mortimr Hous, Mortimr Strt, London W1T 3JH, UK Intrnational Journal of Computr Mathmatics Publication dtails, including instructions for authors and subscription information: Highr-ordr oprator splitting mthods for dtrministic parabolic quations A. T. Sornborgr a a Dpartmnt of Mathmatics and Faculty of Enginring, Univrsity of Gorgia, Athns, Gorgia, Publishd onlin: 16 Jul To cit this articl: A. T. Sornborgr 2007): Highr-ordr oprator splitting mthods for dtrministic parabolic quations, Intrnational Journal of Computr Mathmatics, 84:6, To link to this articl: PLEASE SCROLL DOWN FOR ARTICLE Full trms and conditions of us: This articl may b usd for rsarch, taching, and privat study purposs. Any substantial or systmatic rproduction, rdistribution, rslling, loan, sub-licnsing, systmatic supply, or distribution in any form to anyon is xprssly forbiddn. Th publishr dos not giv any warranty xprss or implid or mak any rprsntation that th contnts will b complt or accurat or up to dat. Th accuracy of any instructions, formula, and drug doss should b indpndntly vrifid with primary sourcs. Th publishr shall not b liabl for any loss, actions, claims, procdings, dmand, or costs or damags whatsovr or howsovr causd arising dirctly or indirctly in connction with or arising out of th us of this matrial.

2 Intrnational Journal of Computr Mathmatics Vol. 84, No. 6, Jun 2007, Highr-ordr oprator splitting mthods for dtrministic parabolic quations A. T. SORNBORGER* Dpartmnt of Mathmatics and Faculty of Enginring, Univrsity of Gorgia, Athns, Gorgia Rcivd 30 August 2006; rvisd vrsion rcivd 17 January 2007; accptd 30 March 2007) Th Shng-Suzuki thorm stats that all xponntial oprator splitting mthods of ordr gratr than 2mustcontainngativtimintgration.Thrhavbnclaimsinthlitraturthathighr-ordr splitting mthods for dtrministic parabolic quations ar unstabl du to this fact. W show stability for a class of highr-ordr splitting mthods for intgrating dtrministic parabolic quations. W not that problms with backwards tim intgration will still xist for stochastic intgration mthods for which information is lost and backward timstps bcom ill-dfind. Thrfor, compltly positiv splitting mthods, such as thos dvlopd by Chin, still hav an important plac. W prsnt numrical rsults from first-, scond-, third- and fourth-ordr mthods showing that th rror bcoms incrasingly small as th ordr incrass. Kywords: Oprator splitting; Parabolic quations; Shng-Suzuki thorm; High ordr mthods; Nonrvrsibl systms AMS Subjct Classifications: 65N12; 35K45; 35K50; 65Y20 1. Introduction Oprator splitting mthods [1] ar usd in many circumstancs for intgrating ordinary and partial diffrntial quations. Particularly in th symplctic cas undr th rubrik symplctic mthods ), thy hav njoyd wid us du to thir rtntion of th phas spac charactristics of Hamiltonian systms. Advancs hav also bn mad rcntly in th application of ths mthods to quations with Lyapunov intgrals [2]. In a 1989 papr [3], Shng showd that highr-ordr ordr gratr than two) oprator-splitting mthods must contain oprators which intgrat backwards in tim. This rsult has subsquntly bn found by othrs [4 6] and is now known as th Shng Suzuki thorm. Ngativ tim intgration dos not caus difficultis for Hamiltonian systms. Howvr, for stochastic quations, information is irrcovrably lost at ach timstp and backwards intgration bcoms impossibl. Du to th Shng Suzuki thorm, ffort has bn spnt to avoid backward tim intgration in highr-ordr mthods by xplicitly including xponntials of commutators [7] in splitting mthods. In th intrmdiat * ats@ngr.uga.du Intrnational Journal of Computr Mathmatics ISSN print/issn onlin 2007 Taylor & Francis DOI: /

3 888 A. T. Sornborgr cas of dtrministic parabolic quations, whr diffusiv ffcts dstroy tim-rvrsal symmtry yt thr is a dtrministic non-stochastic) quation dscribing th volution, thr hav bn claims that th backward intgration stps of highr-ordr mthods caus instability [3, 7]. In this papr, w prsnt a stability analysis for a class of xponntial oprator splitting mthods and a countr-xampl dmonstrating that, although it is tru that highr-ordr splitting mthods all must hav ngativ tim volution oprators, nvrthlss, thy ar not all unstabl for th intgration of linar parabolic quations. W prsnt simulation rsults for third- and fourth-ordr oprator splittings which ar stabl and dmonstrat incrasd accuracy rlativ to lowr-ordr splittings. First, w giv a brif rcapitulation of how our intgration mthods wr drivd [6]. Thn, w show that thy ar stabl. Finally, w intgrat a discrtizd vrsion of th linar quation φ t = xφ) x in on dimnsion. W can rwrit this quation as + 2 φ x 2 1) φ = Dφ 2) whr D is th discrtization to fourth ordr in x of th oprator x x + x 2. Intgrating for a tim t, w hav th solution φt + t) = td φt). 3) Hr, D = D 1 + D 2 whr D 1 is th oprator x x discrtizd to fourth-ordr in x and D 2 is th oprator x 2 also discrtizd to fourth-ordr. W intgrat quation 1) in tim using a fourth-ordr Rung Kutta schm and tak th rsults of this unsplit schm as th solution to compar th split schms against. Th particular quation usd hr dos not rquir a splitting, xact solutions ar known. W hav usd it bcaus th two oprators do not commut, and th continuous problm has an xact solution to chck our mthods against. Splitting mthods ar particularly valuabl for quations whr an analytical solution is not availabl for a sum of oprators, but analytical prfrably quickly calculabl) solutions ar known for ach oprator in th sum. W not that th splitting mthods usd hr can b usd for arbitrary sums of oprators, i.. D 1 + D 2 + +D N, whr th D i ar all oprators, non of which nd commut with th othrs s [6]). In particular, ths mthods may b usd for linar parabolic quations, whr xactly solvabl diffusiv trms as wll as othr asily intgrabl trms can b split. 2. Oprator splitting mthods Our mthods wr drivd in th following mannr [6]. W want to xprss th oprator N ) xp n=1 A n as a product of individual factors xpa n ) s. Thr will b many possibl N ) combinations that approximat xp n=1 A n to a givn ordr. W choos to invstigat

4 Highr-ordr oprator splitting mthods for dtrministic parabolic quations 889 combinations of fundamntal units of th form aa 1 aa 2... aa N ) α 4) with paramtrs a and α allowing for transposs of th ntir product as wll as raising all th xponntials in th fundamntal unit to th sam powr. Choosing to sarch for approximations combining fundamntal units of this form has th bnfit that all approximations will b valid for an arbitrary numbr of non-commuting trms N. By itrating th Campbll Bakr Hausdorff formula, w find an xprssion for th fundamntal unit 4) in trms of a singl xponntial aa 1 aa 2... aa ) α N = xp αa p B p N 5) that dfins th B p N in trms of th A n. Hr, p is an xponnt on a and a labl on th matrics B p N. W tak α =±1. Combining i = 1,...,I fundamntal units with paramtrs a i and α i and using th Campbll Bakr Hausdorff formula, w find p=1 xp α 1 a p 1 Bp N xp α I a p I Bp N = xp p=1 p=1 X σ X I BX N Hr, th BN X ar commutators of th Bp Ns. X rprsnts a labl pq rs, whr B pq rs N ). 6) B pq rs N [ B p N, [ B q N,...,[ B r N,Bs N] ]]. 7) is of ordr p + q + +r + s. Up to 5th ordr, w can tak X {1; 2; 3, 12; 4, 13, 112; 5, 14, 23, 113, 221, 1112}. 8) Th σ X I ar dfind in trms of α i and a i by 6). Aftr som calculation, w obtain th following quations for th σ X I. σ p I = I α i a p i 9) for p = 1,...,5, for pq = 12, 13, 14, 23, σ ppq I σ pq I = 1 2 σ p I σ q I I = 1 2 σ p I σ pq I 1 6 σ p I )2 σ q I a q p [ p i σ i )2 σ p i 1 )2] 10) I a q p [ p i σ i )3 σ p i 1 )3] 11)

5 890 A. T. Sornborgr for ppq = 112, 113, 221, and σ 1112 I = 1 2 σ 1 I σ 112 I 1 3 σ 1 I )2 σ 12 I 1 24 σ 1 I )3 σ 2 I I a i [ σ 1 i ) 4 σ 1 i 1 )4]. 12) For approximations to xp N n=1 A n), all σi X must b zro, xcpt for σi 1, which should b gratr than zro. Thr ar no non-trivial solutions to quation 9) with p = 3 whn th product α i a i is positiv for all i. Thrfor, for third-ordr mthods, α i a i must b ngativ for at last on i. Thisimplisthatthrmustbatlastonopratorwithngativtimvolutioninoprator splitting mthods of ordr 3. Similarly, with p = 3 and p = 4 in quation 9), it can b provd that fourth-ordr mthods must hav at last two oprators with ngativ tim volution [6]. Th oprator splitting mthods givn blow wr obtaind by solving 9 12) for approximations with intgral cofficints to th oprator xp N ) n=1 A n up to first-, scond-, thirdand fourth-ordr. W us th notation t) to rprsnt and t) T to rprsnt So, for xampl, our first-ordr mthod is rprsntd by and our scond-ordr mthod is rprsntd by td 1 td 2 ) 13) td 2 td 1 ). 14) td 1 td 2 ) 15) t) 16) td 1 td 2 ) td 2 td 1 ) 17) t) t) T. 18) Our third-ordr mthod is t) T t) t) t) t) T 2 t) T t) t) t) 19) whr th radr will not th oprator with ngativ tim volution, and our fourth-ordr mthod is t) T t) t) T 2 t) t) T t) T t) T t) T t) t) T t) t) t) t) 2 t) T t) t) T t). 20) This fourth-ordr mthod has two ngativ tim volution oprators.

6 Highr-ordr oprator splitting mthods for dtrministic parabolic quations Stability THEOREM 3.1 All mthods of th form 6) ar stabl for th intgration of dtrministic parabolic quations if th fundamntal unit is stabl. Proof W prov th thorm for matrix oprators. W start by assuming that th fundamntal unit, a first-ordr oprator splitting mthod, is stabl. This can b chckd with standard tchniqus. A stabl first-ordr mthod will b non-incrasing aa 1 aa2 aa ) α N = aα 1, 21) whr N d n=1 k=1 λk n 0, d is th dimnsion of th matrics {A n}, λ k n is th kth ignvalu of th nth matrix and w hav usd th fact that M T = M for a matrix, M. To dtrmin stability for th full mthod, w chck that it is also non-incrasing I a ia 1 a ia2 a ia N ) α i = I a iα i = σ 1 I 1, 22) but this will always b tru sinc σ I 1 > 0 for our mthods. For th third-ordr mthod usd hr, σ I 1 = 6, and for th fourth-ordr mthod, σ I 1 = 12. From th abov thorm, w s that, although som intgrations in th oprator splitting mthod ar backwards in tim, th ovrall intgration has an amplification factor that is lss than on. Possibl instabilitis arising from th ngativ tim intgration ar canclld and th highr-ordr mthods ar stabl. Blow, w dmonstrat this fact numrically and show that our mthods ar also mor accurat as th ordr incrass. 4. Numrical rsults W intgratd th split oprators 16), 18), 19), 20) using fourth-ordr spatial finit diffrncs and a fourth-ordr Rung Kutta algorithm [8]. As statd abov, w compard th volution as intgratd by th split spatially discrtizd) oprators to th unsplit spatially discrtizd) oprator. As initial conditions, w took φx, t 0 ) = t 0 x 1/2x2 with t 0 = 0. W discrtizd spac with x = 0.05, t = x 2 /60 with 500 gridpoints. Ths paramtrs gav us a stabl first-ordr split oprator intgration. At th boundary w took drivativs to b constant. W chckd our spac and tim discrtization by vrifying that th discrt solution intgratd using th unsplit oprator was a good approximation to th solution of th continuous quation i.. φx, t) = t x 1/2x2 ). Th solution was intgratd for 100, 000 tim stps. In figur 1a), w plot th logarithm of th spatially intgratd rror E = φ unsplit x) φ split x) dx 23) for ach mthod vrsus th logarithm of tim countd in units of th numbr of timstps). Th first-ordr spatially intgratd log-rror is upprmost, with th scond-ordr log-rror bnath it, and so on down to th fourth-ordr log-rror at th bottom. Not that th fourthordr rror nvr gts much largr than th numrical roundoff accuracy ) of th

7 892 A. T. Sornborgr Figur 1. Panl a), Th logarithm of th spatially intgratd rror is plottd as a function of th logarithm of tim. Th rror for th first-ordr oprator splitting mthod is topmost, followd succssivly by th scond-, third- and fourth-ordr rrors. Th fourth-ordr rror is of ordr th numrical roundoff accuracy of th computr. Panl b), Th unsplit, xact solution is plottd as a function of th logarithm of tim from 0 to 100, 000 timstps. Panls c) f), Th spatial distribution of th logarithm of th absolut valu of th rror for first- through fourth-ordr oprator splitting mthods as a function of th logarithm of tim. computr. In figur 1b), w plot th xact, unsplit solution in spac as a function of log-tim. In figur 1c) f), w plot th spatial distribution of th log-rror as a function of timstp for ach of th mthods from first-ordr c) to fourth-ordr f). Th fourth-ordr log-rror looks lss structurd than th first- through third-ordr log-rrors bcaus it is at th dg of th numrical roundoff accuracy of th computr. By th nd of th simulation, all rrors ar of ordr th numrical roundoff accuracy sinc th solution rlaxs to zro along th ntir grid as sn in figur 1b)). For th class of mthods discussd hr, w xpct th nth-ordr rror pr timstp from th splitting schm E n R n+1 t) n+1 s Appndix 3 in [6]), whr R n+1 is a cofficint proportional to an n + 1)st-ordr commutator. For our quation, R 1 [D 1,D 2 ]= 2 2 x. Rmmbr that for our simulation t = x/60, giving a first-ordr rror E , and th rrors of highr-ordr mthods li proportionally bnath th first-ordr rror. In th figur, w s that th rrors ar as xpctd in that thy li proportionally bnath ach othr in log units) and thir spatial support is whr x is rlativly larg s figur 1b)). 5. Conclusions W find that th class of oprator splitting mthods givn by th factoring 6) ar stabl for dtrministic parabolic quations providd th fundamntal unit 4) is stabl. Thr is

8 Highr-ordr oprator splitting mthods for dtrministic parabolic quations 893 no instability inducd by ngativ tim-volution in th splitting. W thn dmonstrat that 1) th rrors ar spatially distributd in th way that would b xpctd givn th form of th commutator of th trms in th dtrministic parabolic quation that w chos to intgrat as an xampl, and 2) this class of mthods is as accurat as w would xpct it to b, givn th valus of th rror calculatd for a givn splitting, for dtrministic parabolic quations. Rfrncs [1] Strang, G., 1968, On th construction and comparison of diffrnc schms. SIAM Journal of Numrical Analysis, 5, [2] McLachlan, R., Quispl, G. and Robidoux, N., 1998, Unifid approach to Hamiltonian systms, Poisson systms, gradint systms, and systms with Lyapunov functions or first intgrals. Physical Rvu Lttrs, 81, [3] Shng, Q., 1989, Solving linar partial diffrntial quations by xponntial splitting. IMA Journal of Numrical Analysis, 9, [4] Suzuki, M., 1991, Gnral thory of fractal path intgrals with applications to many-body thoris and statistical physics. Journal of Mathmatical Physics, 32, [5] Goldman, D. and Kapr, T., 1996, Nth-ordr oprator splitting schms and nonrvrsibl systms. SIAM Journal of Numrical Analysis, 33, [6] Sornborgr, A. and Stwart, E., 1999, Highr-ordr mthods for simulations on quantum computrs. Physical Rvu A, 60, [7] Chin, C.A., 2004, Quantum statistical calculations and symplctic corrctor algorithms. Physical Rvu E, 69, [8] Prss, W.H., Tukolsky, S.A., Vttrling, W.T. and Flannry, B.P., 1994, Numrical Rcips in FORTRAN, Th Art of Scintific Computing, 2nd dn Cambridg: Cambridg Univrsity Prss).

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