Analysis Of Stochastic Numerical Schemes For The Evolution .
Available CCT.AppliedMathematicsLetters16 nalysisof StochasticNumericalSchemesfor the EvolutionEquationsof GeophysicsB. D. EWALDDepartmentof versityTX 77843-3368,U.S.A.R. TJ MAMInstituteforRawlesScientificHall, 831ComputingE. Third(ReceivedandStreet,andaccepted DecemberAbstract-wepresentand studythe stabilitya numericalscheme used in geophysics,namely,leapfrog”scheme which has been developedfor themodel. Two other schemes which might be usefulintroducedand discussed.@ 2003 omington,IN 47405-7106,UniversityU.S.A.2002)and convergence,and order of convergenceofthe stochasticversionof a deterministic“implicitapproximationof the socalledbarotropicvorticityin the contextof geophysicalapplicationsare alsoAll rights reserved.Stochasticdifferentialfluid dynamics.equations,Leapfrogscheme,1. INTRODUCTIONMuch effortof the formhas been investedin studyingnumericalschemes for stochasticdifferentialdU, u(Ut) dt b(Ut) dW,,equations(1.1)where U, E I@, a is a functionfrom Rd into itself, W is a Wiener process on Iw”, and b is afunction from I@ into !Pxm.of the expectationFor the so-called weak approximationof (1.1)) in which the approximationof functions of U is considered,extensive work is due, for example, to Talay and his collaborators,work relying on probabilisticmethods more involved than those used in this article (see, e.g., [l-3]and the references therein).The question of strong approximationof (l.l),in which the approximationof sample pathsof U is desired, has also been much studied.See, for example, the paper by Mil’shtein[4] for aof Runge-Kuttaschemes.scheme of order O(At),and that by Riimelin[5] f or an investigationThe authorsare very gratefulto C. Penlandfor bringingthese issues to their attentionand they acknowledgevery useful discussionswith her and with P. Sardeshmukh.We are also indebtedto A. Debuaschefor severalimprovementson an earlier draft, and to S. Faure, who providedthe numericalsimulationsof Section 6.This work was supportedin part by a grant from the NationalScience Foundation,NSF-DMS0074334 and bythe ResearchFund of IndianaUniversity.0893-9659/03/g- see front matterdoi: lO.l016/SO893-9659(03)00182-4@ 2003 ElsevierLtd.All rightsreserved.Typesetby A@-TEX
B. D.1224EWALDAND R. T MAMEspecially, see the text by Kloeden and Platen [6], and the companion volume by Kloeden, Platenand Schurz [7], which are a systematic investigation of numerical schemes for (1.1) in both thesense of It6 and of Stratonovich, the two stochastic calculi which are in applications by far themost useful. Their methods are analytic and are applicable to proving the convergence of a widerange of numerical schemes, and they derive a very general scheme [6, formula (12.6.2)] which,for various choices of parameters, includes stochastic analogues of such deterministic schemes asthe explicit and implicit Euler schemes, the Crank-Nicholson scheme, and the leapfrog scheme.In the geophysics community, an enormous amount of work has been spent in developing large,complex numerical models of the oceans and atmosphere. The questions therefore arise: is itpossible to add stochastic numerical noise to these already existing models in such a way that itis known to what the scheme converges (e.g., to the It6 or Stratonovich solution of some stochasticdifferential equation), to what order they may be expected to converge, etc.? While we certainlydo not answer these complex questions here, we consider a simple “implicit leapfrog” scheme fora barotropic model (supplied to us by C. Penland and P. Sardeshmukh), and demonstrate oneway of adding stochastic noise to it so that these questions can be answered for the resultingstochastic scheme (Section 3).We also examine the derivatives of a and b which occur naturally in the above schemes, andwhich can prove to be troublesome in certain applications in which these functions, especially b,are given by physical parametrizations (i.e., by “tables”) and not by analytic expressions. We consider how these derivatives can be replaced by finite differences derived from space-discretizationwhile still maintaining the existing rate of convergence (Section 4).Last, we propose a stochastic analogue for the deterministic Adams-Bashforth scheme, usingmethods similar to those of [6], as an attempt to produce alternate schemes which are higherorder in time (Section 5).2. PRELIMINARYRESULTSWe consider a stochastic differential equationdUt a(t, &) dt b(t, ut) dW,,(2.1)for U ( 1,. . . , Ed) E Rd, where a : lR x lRd --t ll@, b : Iw x lRd - Rdxm, and W is a Wienerprocess in iw” adapted to a filtration {&}t c.We then have the ItS formula, which states that, if F : lR x ll@ --t Iw’, then Ft F(t, Ut)satisfies the stochastic differential equationhere we use the Einstein convention for repeated indices.We use the following notations from [6]. A multi-index is a row vector a (ji, j2, . . . , je) (eachji E {O,l, . . ) m}) of length e e(a) E (0, 1, . . }. We define n(o) to be the number of entriesof a which are 0. For adapted, right-continuous functions f, and stopping times p, r such that0 5 p 5 r 5 T a.s., we define (where (Y- is (I with its final component removed)Ia[f(.)],,7 if C(o) 0,f CT)7if C(a)2 1, AT( ) 0,s; -[f( lWds,s,’ la-[f(.)]p,sdW?(a’),if e(a)2 1, jqa) # 0.(2.3)We also define the operatorsd -bkJblJ1a2Lo -d ak--.atauk 2auk @ul’api @i-.-mauk’(2.4)
Analysisand, if f E Ch(R x Rd,R),whereof StochasticNumericalh 2 e(o) n(o),fa fjl f{OL’1225Schemeswe setif e(o) 0,(2.5)if e(o)2 1.Here ---a! is cr with its first componentremoved.We note that if f (t, u) E U, then fco, a, fcj) bj, etc. In what follows, unless explicitlystated otherwise,we will assume that f is this identity function.A set, A, of multi-indicesis said to be a hierarchicalset if A # 0, supcreA (o) 00, and-CX E A whenever o E d-(v).We then define the remainderset B(d) of A by a(d) (CX ] (Y Aand --Q: E A}.We can now provide a stochastic Taylor expansionfor U satisfying(2.1): iff : lRf x lRd R, then, provided the derivatives and integrals exist,f (7,UT) c LY[fcxhUP)lP,T cLISAwhere A is some hierarchicalNow, for y 0.5,1.0,1.5,.Lc[fa(.,U.)lP,7,set. . , we set27 or f!(o) n(o)We call the stochasticTaylorexpansion3. A STOCHASTICThe barotropicNationalOceanic(2.6)&B(A)withA d,“IMPLICITvorticitymodeland Atmospheric y the expansionto orderLEAPFROG”(2.7).y.SCHEMEsuppliedto us by C. Penlandand P. Sardeshmukhof theAdministrationin Boulder,Colorado(see [8]), takes the forma - -vat. (WC) s - 7-t - KV41 ,(3.1)where C V211, f E f and v R x 04.Here, C is the total vorticity,u is the velocityvector, f is the Coriolis term, S is a (deterministic)forcing, T and K are constants, and is thelocal vorticity.The numericalscheme they providedfor this uses spherical harmonics,and, writingF for-V . (UC), the equation becomes(3.2)Thenthe scheme has two steps. First,tr(tfollowedby an implicita leapfrogstep: At) ,“(t - At) 2At[F,“(t) S;(t)],(3.3)step:i;(t‘,“(t At)If we simplify notationand writean “implicitleapfrog”scheme 1 2At At)[r K [(n(n 1))/a2]“]al for F S and as for -r[(3.4)’- KV [,E(t At) Y(t - At) 2Atai(t,Y(t)),Y(t At) F(t At) 2Atas(t At, Y(t At)),we see thatthis is just(3.5)
B. D. EWALDAND R. TBMAM1226for the equationduct) [al@, U(t))(3.6) z(GU@))l&.Therefore, we consider a stochastic differential equation of the formdU, (q(t,Ut) ua(t, Ut)) dt b(t, Ut) dW,.Note that we have simply added a general diffusion term to the deterministiction (3.6).We will consider the schemeFn 2 Y,K 2 2al(t, l,Y, l)At M,(Y,)(3.7)differential equa- Mn l(Yn d, 5L 2 ‘Jaz(t, z,Y, z)At,(3.8)whereM,,(Y) b(tn, y)AwnTHEOREM bb’(tn, Y) (I,I),,.W-03.1. Suppose that the coefficient functions fa satisfyIf&,x)- fa(t,Y)II Klx - YI,(3.10)fa E %,(3.11)for all Q E dl.0, t E [O,T], and z,y E I@;fern E es2for all cxE Al.0UB(dl.0);andand(3.12)Ifa(t,xc)l I KC1 I4Lfor all CYE Al.0 U f3(dl.o), t E [O,T], and x E Rd. Choose At 5 1 and set N T/At, anddefine t, nAt for n 1, . . . , N. Suppose that some appropriate numerical scheme is used togenerate Yl such that IE[IUt, - Yl12 ( Fol1i2 5 CAt. Then,l/2E o;wJNI% - K12I ro--4.SPACE1 CAt.(3.13)DISCRETIZATIONIt sometimes happens in applications that the functions a and b may only be known empirically(i.e., in tables) rather than analytically. In such cases, analytic derivatives of these functions canbe difficult to obtain. It is, therefore, useful to replace these derivatives by discrete approximations. As a first example, consider this modification of Mil’shtein’s schemewhere ee is the vector (0,. . ,0, l,O, . . . , 0) with 1 in the eth position, and we have chosen Ax 0.We have also suppressed the dependence of a and b on time to simplify notation.We then have the following theorem.THEOREM 4.1. Suppose that a and b have the regularityconverge to the solution U to order At. Then,required for Mil’shtein’sschemeto(4.2)Note that if we want to maintain the order of convergence of Mil’shtein’s scheme, we need thatAx O(At1i2).
1227Analysis of Stochastic Numerical Schemes5.A STOCHASTICADAMS-BASHFORTHSCHEMEThe followingis a stochastic version of a scheme which is very effective and commonlyusedin computationalfluid dynamics.The deterministicAdams-Bashforthscheme for the ordinarydifferentialequation4’ F(4) takes the form4n-i-1 4n 13F(4,)This scheme is order At2 in the deterministiccase.We consider the followingstochastic Adams-BashforthY n 2 Y, I (t,,l,% I)(5.1)- F(&-I)].(SAB)scheme:1At - ;AtA,(t,,- ;&,Yn)Y,) B,(t,,Y,),(54in whichA,(t,z)wherethe randomintervals&(t, Lja(t,z)AFVj Lj’Lj2a(t,x)l jl,j,),are from time t, to & I, and Lob”@, z)I ,, )x) bj(t, x)AWj L%qt,Z)l jl,jZ) L a(t,x)l j,o L”Lj’bj2(t,s)l o,j,,j,) Lj’L” (t,Z)I j1,0,j2) Lj’Lj2Lj3 (t)2)I jl,jZ,j3,j4)’where the random intervals are those from time t, to tn 2 minusWe then have the followingtheorem.5.1.Supposethat(5.4) LjlLj2a(t,z)l j,,j,,0) L31Lj2 3(t,Z)l jlrj2,j3)THEOREM(5.3)the coefficientfunctionsthose from time t, to tn l.fa satisfyI.fa(t,xC)- fcY(CY)I I 0- YL(5.5)fa E 3-10,(5.6)for all (Y E d2.0, t E [O,T], and s,y E IL@;f--afor all (Y E dz.0 U I3(dz.c);andE c1 2andIfa(t,‘C)I5 KC1 14Lfor all cx E AZ.0 U B(dz.o), t E [O,T], and x E Rd. Choose Atdefine t, nAt for n 1,. . . , N. Suppose that some appropriategenerate Yr such that lE[jU,, - Yl12 I .Fo] / I CAt2. Then,(5.7)5 1 and set N T/At,andnumericalscheme is used tou2SUPO n NIU&I-K12 1306. NUMERICAL CAt2.(5.3)SIMULATIONThe object of this section is to test numericallythe accuracy of the scheme of Section 5 andcompare it to the theoreticalresult above (i.e., O(At2) accuracy) and to the accuracy of the Eulerand Mil’shteinschemes (respectively,O(At1i2)and O(At)).All the numericalresults below areconsistent with the theoreticalones.
1228B. D. EWALDAND R. T MAM----J--------- --------slope l/21, , ,-.-,-.-.-.- - -. --.-.-slope 1 -, , . .-.- -‘-’II-6.2-6I1-5.6(8) lope l/, . .-.-‘-.-.-.-.-.--slope 1, . . ,L.-.- -’, . .-’A6----------l(cP& 4-6.2-6-5.0-5.6log(time(b) StochasticFigure1. Resultsobtainedwith-5.4step)equationthe 6
Analysisof StochasticNumericalSchemes1229We consider the following equations:(6.1)dXt ;a2Xt dt ad dW,,with a: 1 and Xe 10, anddXt p2 sinh X, cosh2 Xt dt ,Bcash’ Xt dWt ,(6.2)with p l/10 and Xo l/2. These have the exact solutionsXt cosh(al/l/‘t arccosh Xc)(6.3)Xt arctanh(PWt tanh Xc),(6.4)andrespectively. These can be easily verified using Ito’s formula and are just two of many possibleexamples listed in [6].We computed approximate solutions Y, using the Euler and Mil’shtein schemes and the SABscheme from Section 3. Then we computed the following error:e IEd(sup IX, -Yn12O n N .To obtain the mean value needed, we used 500 trajectories.In the figures, the order of each scheme is given by the slope of the corresponding line. So wecan see that the orders are l/2 for Euler, 1 for Mil’shtein, and 2 for the SAB of Section 3.Note that for the SAB scheme, the stochastic integral I (s,i,i) (which is difficult to generate) wasapproximated by a normal law. The results tend to show that this does not affect the accuracy(at least in these two cases). We will try to improve this point, which seems to raise interestingprobabilistic questions, as already mentioned in the Introduction.REFERENCES1. V. Bally and D. Talay, The law of the Euler scheme for stochasticdifferentialequationsI. Convergencerateof the distributionfunction,Probab.Theory Relat. Fields 104, 43-60, (1996).2. V. Bally and D. Talay, The law of the Euler scheme for stochasticdifferentialequationsII. Convergencerateof the density,MonteCarlo Methodsand Appl. 2 (2), 93-128,(1996).3. D. Talay, Simulationof stochasticdifferentialequations,In ProbabilisticMethodsin Applied Physics,(Editedby P. Kree and W. Wedig),pp. 54-96, Springer-Verlag,Berlin,(1995).4. G.N. Mil’shtein,Approximateintegrationof stochasticdifferentialequations,TheoryProb. Appl. 19, 557562, (1974).5. W. Riimelin,Numericaltreatmentof stochasticdifferentialequations,SIAMJ. Namer.Anal. 19 (3), 604613, (June 1982).6. P.E. Kloedenand E. Platen,NumericalSolutionof StochasticDigerentialEquations,Applicationof Mathematics,Volume 23, Springer-Verlag,Berlin,(1992).7. P.E. Kloeden,E. Platen and H. Schurz, NumericalSolutionof SDE erlag,Berlin, (1994).8. P.D. Sardeshmukhand B.J. Hoskins,The generationof global rotationalflow by steady idealizedtropicaldivergence,Journalof the AtmosphericSciences45 (7), 1228-1251,(1988).9. J.G. Gainesand T.J. Lyons,Randomgenerationof stochasticarea integrals,SIAMJournalon AppliedMathematics54 (4), 1132-1146,(August1994).
Analysis of Stochastic Numerical Schemes 1227 5. A STOCHASTIC ADAMS-BASHFORTH SCHEME The following is a stochastic version of a scheme which is very effective and commonly used in computational fluid dynamics. The deterministic Adams-Bashforth scheme for the ordinary
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