Linear-implicit versions of strong Taylor numerical schemes for finite dimensional Itô stochastic differential equations (SDEs) are shown to have the same order as the original scheme. The combined truncation and global discretization error of an gamma strong linear-implicit Taylor scheme with time-step delta applied to the N dimensional Itô-Galerkin SDE for a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues lambda 1 <= lambda 2 <= ... in its drift term is then estimated by K(lambda N -½ + 1 + delta gamma) where the constant K depends on the initial value, bounds on the other coefficients in the SPDE and the length of the time interval under consideration. 

AMS subject classifications: 35R60, 60H15, 65M15, 65U05

Kloeden, Peter E.

Shott, Stephen

English

Hochschulschriftenserver - Universität Frankfurt am Main

Journal of Applied Mathematics and Stochastic Analysis, 14:1 (2001), 47-53.LINEAR-IMPLICIT STRONG SCHEMES FORITO-GALKERIN APPROXIMATIONS OFSTOCHASTIC PDESP.E. KLOEDENJohann Wolfgan9 Goethe Universitiit, Fachbereich MathematikD-60054 Frankfurt am Main, GermanyE-maih kloeden@math.uni-frankfurt.deS. SHOTTUniversity of Tasmania, Department of MathematicsHobart, Tasmania, AustraliaE-maih Stephen.Shott@pobox.com(Received August, 1999; Revised December, 1999)Linear-implicit versions of strong Taylor numerical schemes for finitedimensional It6 stochastic differential equations (SDEs) are shown to havethe same order as the original scheme. The combined truncation andglobal discretization error of an 7 strong linear-implicit Taylor schemewith time-step A applied to the N dimensional It6-Galerkin SDE for aclass of parabolic stochastic partial differential equation (SPDE) with astrongly monotone linear operator with eigenvalues 11 _< 12 _< in itsdrift term is then estimated by1/2 A K(+I + )where the constant K depends on the initial value, bounds on the othercoefficients in the SPDE and the length of the time interval under consider-ation.Key words: Parabolic SPDE, Galerkin Approximation, Strong TaylorScheme, Linear-Implicit Scheme.AMS subject classifications: 35R60, 60H15, 65M15, 65U05.1This work was supported by the DFG Forschungschwerpunkt "Ergodentheorie,Analysis und effiziente Simulation dynamischer Systeme".Printed in the U.S.A. ()2001 by North Atlantic Science Publishing Company 4748 P.E. KLOEDEN and S. SHOTT1. IntroductionA numerical method for parabolic stochastic partial differential equations (SPDE) [1,3] of the formdU {AU + f(Ut)}dt + g(Ut)dWt, (1)where {Wt, t >_ 0} is a standard scalar Wiener process, based on the application of anorder 7 strong Taylor scheme [4] with constant time-step A to the N-dimensionalstochastic differential equations (SDE)dUNobtained from (1) by a Galerkin approximation was shown by Grecksch and Kloeden[2] to have a combined truncation and global discretization error of the formEIUkA ylKK(l/2 +]+1 )Here Ix] denotes the integer part of the real number x and K is constant dependingon the initial value and bounds on the coefficient functions f,g of the SPDE (1) aswell as on the length of the time interval 0 kA T under consideration, and 1j isthe jth eigenvalue of the operator -A (whose eigenfunctions provide the bases forthe Galerkin approximations).We refer to [1-3] for the functional analytical terminology and formalism ofstochastic PDE (1) with a Dirichlet boundary condition for a bounded domain inNd with sufficiently smooth boundary 0, noting here that it has a unique strongsolutionU e L2([0, T], 31;’ 2) N C([0, T], 2.2)for each finite T > 0 and initial condition U0 E 3t;’2 under the assumption that fand g are uniformly Lipschitz continuous from L2( into itself. The It-GalerkinSDE (2) then also has a unique solution VtN with the initial value UoN -PNUo,where PN is the projection of L2( or H’2() onto the N-dimensional subspace %Nof H’2() spanned by {1,’",N}, the first N eigenfunctions of the operator -Acorresponding to the eigenvalues ,I,...,1N. Fo__r convenience we write UNsynonomously for (uN’I,...,UN’N) NN and q__ 1uN’jdpj fN according toand gN PNgIN"Finally, context and define AN PNAIN, fN PNfIN[2 and [1" [[2 denote the norms of L2( and H’2(), respectively.For this setup, ,NOe as Noc, so a very small time-step A will be needed togive a reasonable bound in (3) for high dimensional It6-Galerkin SDEs, which maygive rise to numerical instabilities. These can be avoided by application of an impli-cit strong Taylor scheme [4], but at the expense of having to solve numerically a non-linear algebraic equation for each time-step. However, the stiffness in the It6-Galer-kin SDEs with such a structure are ideally suited to linear-implicit versions of strongTaylor schemes, which neutralize this source of stiffness and are solvable explicitly forthe next iterate by inversion of a diagonal matrix.Linear-Implicit Strong Schemes 49In the next section we introduce linear-implicit versions of a strong Taylor schemesand show that they retain the strong order of their original scheme. Then in Section3 we consider their application to ItS-Galerkin SDE (2) obtained from a stochasticPDE (1) and show that the A term in their error estimate is no longer premultipliedby a power of "N + as in (3).2. Linear-Implicit Strong Taylor SchemesFollowing Kloeden and Platen [4], an order 7 strong Taylor scheme with constanttime-step A for the SDE (2) has the formN N N N Yk+l Yk + F( zX, (4) Yk )Ic,k,with coefficient functions FN and multiple stochastic integrals Ic,k, zX, where 7 takespossible values 0.5 (Euler scheme), 1.0 (Milstein scheme), 1.5, 2.0, The admissiblemulti-indices a- (Jl,’",Jl)E JtT\{v} here have components Ji- 0 or 1 correspond-ing to integration with respect to ’dt’ or ’dWt’ respectively, while the jth componentofFN is defined bywhere the operatorsandN}2 U v’ + f v, ou 2=11/2NN)gN J(uN)ouNC2+i,j-1Nj=l "" ]ou-N,Jare applied successively to Fj(YN) yN, j, the jth component of yN, and the resultevaluated at yN. For example, the Euler scheme with constant time-step A for theSDE (2) has the formN N (A N fN gN N)AW (6) Yk +1 Yk -f- NYk -+- (YNk ))A "f- (Yk k’since I(o), k A A and I(1),k,A AWk, the Wiener process increment for the kthtime-step. ee [4] for further details.For a common initial value U0N Yo N, the global strong discretization error of thenumerical scheme (4) has the expectationElN N UkA-- Yk <- KNAT, (7)where the constant KN depends on the dimension N as well as on the time interval50 P.E. KLOEDEN and S. SHOTT0 _< kA _< T under consideration and on uniform bounds on the functions f and g andtheir derivatives.Linear-implicit schemes, which are sometimes called Rosenbrock schemes in thecontext of deterministic differential equations [7], are obtained from the explicitscheme by making implicit only the linear part of the coefficient functions. In orderto ensure the consistency of It8 stochastic calculus, only the coefficients of the purelydeterministic integral terms are made implicit, that is, only those for multi-indices athat have no nonzero components, in constructing implicit strong Taylor schemes forSDEs [4]. In defining their linear-implicit counterparts here we shall make only thelinear part of the highest order purely deterministic integral term implicit (as thisavoids having to adjust some other terms as happens in the usual implicit schemes[4]). For example, the linear-implicit version of the Euler scheme with constant time-step A for the SDE (2) has the formN N fN(yNk)A gN N Yk + I (IN AAN)- I(Yk + + (Yc )AWk),where IN is the N x N identity matrix and IN -AAN is a diagonal matrix with jthdiagonal component 1 + A,j.Let l(c) be the length of a multi-index c and n(c) the number of components of cthat are equal to zero. DefineL(’),) max{/(a): l(a) n(a), a eLet c*-(0,...,0) (L(7) times) be the multi-index with n(c*)- l(c*)- L(7)./k L1(’) and it can be shown [4] that I , L(- c ,k,AThenWe define the linear-implicit version of the order 7 strong Taylor scheme for theItS-Galerkin SDE (2) byYNk+I_(IN_L(7)!I AL(,)ALN())-I(YNk + E ,}FN (y)I,I, AA.\{,,Here the N x N matricesN Yk)" (9)1 ,rL()ALN() IN1 AL()ALN()are diagonal matrices with jth diagonal component equal, respectively, to1 )L("/)AL(’,/) L(’,/) 1-- 1 L(/)/kL("/)L("/) L@)[1 ,j L(7)!(- 1) jalthough the definition (9) of a linear-implicit scheme is also valid when the matrixAN is not a diagonal matrix and the following theorem also holds in this moregeneral case. The assumptions required on the coefficients of the SDE are the sameLinear-Implicit Strong Schemes 51as those given in Theorem 10.6.3 in [4], essentially that they are sufficiently often con-tinuous differentiable with bounded derivatives.Theorem 1: The linear-implicit version (9) of an order 7 strong Taylor schemealso has strong order 7.Proof: In order to apply Theorem 11.5.1 of [4] we first rearrange the linear-impli-cit scheme (9) into the formwhereYk + Y; + E ’Na (y)Ic,k,A" (10)Na (yN (IN L(7)!lAL(’)ALN(’r))1FaN(yN)for all multi-indices c including a*, which is possible since1 ,AL(’)Av(’) YkN + L(7) IN L(). -N k J)1 ,AL(’)A(’))-1N-L(7 ). FN,(YN I k c,k,A"Then for all multi-indices a E .A.\{v}, we have-1N 1 AL(’)Av(’r) -’N (yN)--Fa (YN) I-- (IN-L(7’), )-1 AL()L(7)!-11 AL()A() AL(’,/)N yN< KAL().The squared expectation of all of these terms is thus bounded by a constantmultiplied by A2", since 7 _< L(7) and, without undue restrictiveness, A _< 1 Theorem5.11.1 of [4] then gives the desired result.The analogous result for linear-implicit weak schemes can be found in [5] and analternative proof for the linear-implicit strong Euler scheme in [6].3. The Combined Error BoundThe next theorem shows that the use of a linear-implicit strong Taylor scheme (9)overcomes the stiffness in the structure of an It&Galerkin SDE (2).Theorem 2: The combined truncation and global discretization error of the order7 linear-implicit strong Taylor scheme (9) with constant time-step A applied to theN-dimensional It’a-Galerkin approximation (2) of the SPDE (1) has the formEIUA--yNI _< K(,] 1/2+1 + AT), (11)52 P.E. KLOEDEN and S. SHOTTwhere the constant K depends on E II Uo II 2, bounds on the f, g coefficients of theSPDE and the length of the time interval 0 <_ kA <_ T under consideration.Proof: The proof is similar to that for the error bound (3) in [2] for the order 7strong stochastic Taylor scheme. To determine the nature of the dependence of KNon N in (7) here it is necessary to examine the terms in the remainder of thestochastic Taylor expansion used to derive the strong Taylor scheme (4). Details aregiven in the proof of Theorem 10.6.3 in [4] and will not be repeated in full here. Inparticular, the remainder consists of multiple stochastic integrals with nonconstantintegrands Ia, k,A(FN(uN.)for multi-indices a in the remainder set %.. These areestimated in inequality (6.23) on page 364 of [4] under the assumption that the coeffi-cient functions satisfy a linear growth bound. This is true here under the assump-tions on the coefficients of the SPDE (1) and it is only the dependence of the coeffi-cient KN on N. Specifically, each application ofL contributes a single dominatingpower of AN The bound in the squared inequality (6.23) on page 364 of [4] thustakes the formKT(1 + E IUoN[2)A2Nn()A()where (c) is equal to 2(/(c)- 1) if l(c) n(c) and l(c)+ n(c)- 1 if l(c) :/= n(c). If7 is an integer, the dominant value occurs for l(a)- n(c)-7 +1 and the requiredbound is of the formKT, UoA2N7+ 2A2"Ynot  o in nt o urs +and the required bound is of the formKT, UoA2A+ 3A2-"Taking the square root and renaming the constants gives the estimate(1/2 +1/2]+1 ) E IUkA--YkNAI <KT, uo +I-t-A A (12)where Ix] is the integer part of the real number x, for the global discretization error.There are two basic differences for the order /linear-implicit strong Taylor scheme(9). The first is that all coefficients in the scheme and the remainder are multiplied1 )!AL(7)ALN(7) The discretization error by the inverse of the diagonal matrix IN -L(7estimate will thus be divided by the norm of this matrix, which is dominated by theterm A(7)-A +1/2]. The second difference is that the integrands in the remainderterms coming from the highest order purely deterministic multiple integral in thescheme are obtained by applying the L and the L1 operators to the coefficientfunction of this highest order term after the linear part has been made implicit andshifted to the other side. In particular, the highest order power A- +1/2] + 1comingfrom this linear part will no longer be present, though lower order powers may be,that is at most A-- +1/2], but we divide these terms by norm of the diagonal matrixwhich is dominated by the power A--+ 1/2]. The global discretization error bound willLinear-Implicit Strong Schemes 53thus be of the form KT, UoA’ and hence the combined truncation and global discreti-zation error bound will be as asserted in Theorem 2. DReferences[1] Da Prato, G. and Zabczyk, G., Stochastic Equations in Infinite Dimensions,Cambridge University Press, Cambridge 1992.[2] Grecksch, W. and Kloeden, P.E., Time-discretized Galerkin approximations ofparabolic stochastic PDEs, Bull. Austral. Math. Soc. 54 (1996), 79-84.[3] Grecksch, W. and Tudor, C., Stochastic Evolution Equations, Akademie-Verlag,Berlin 1995.[4] Kloeden, P.E. and Platen, E., Numerical Solution of Stochastic DifferentialEquations, Springer-Verlag, Berlin 1992.[5] Petersen, W.P. A general implicit splitting for stabilizing numerical simulationsof It6differential equations, SIAM J. Numer. Anal. 35 (1998), 1439-1451.[6] Schurz, H., Stability, Stationarity, and Boundedness of Some Implicit NumericalMethods for Stochastic Differential Equations, Dissertation, Humboldt Universi-tt, Berlin 1997.[7] Shampine, L.W. and Reichelt, M.W., The MATLAB ODE suite, SIAM J. Sci.Comput. 18 (1997), 1-22.

A general implicit splitting for stabilizing numerical simulations of It6differential equations,

Numerical Solution of Stochastic Differential Equations,

Stationarity, and Boundedness of Some Implicit Numerical Methods for Stochastic Differential Equations, Dissertation, Humboldt Universitt,

Stochastic Equations in Infinite Dimensions,

Stochastic Evolution Equations, Akademie-Verlag,

The MATLAB ODE suite,

Time-discretized Galerkin approximations of parabolic stochastic PDEs,

Linear-implicit strong schemes for Itô-Galkerin approximations of stochastic PDEs

Linear-implicit strong schemes for Itô-Galkerin approximations of stochastic PDEs

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