Abstract
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 γ strong linear‐implicit Taylor scheme with time‐step Δ 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 λ1 ≤ λ2 ≤ ⋯ in its drift term is then estimated by 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.
Highlights
The combined truncation and global discretization error of an 7 strong linear-implicit Taylor scheme with time-step A applied to the N dimensional It6-Galerkin stochastic differential equations (SDEs) for a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues 11 _< 12 _< in its drift term is estimated by
Obtained from (1) by a Galerkin approximation was shown by Grecksch and Kloeden [2] to have a combined truncation and global discretization error of the form
Ix] denotes the integer part of the real number x and K is constant depending on the initial value and bounds on the coefficient functions f,g of the SPDE (1) as well as on the length of the time interval 0 kA T under consideration, and 1j is the jth eigenvalue of the operator -A
Summary
A numerical method for parabolic stochastic partial differential equations (SPDE) [1, 3] of the form dU {AU + f(Ut)}dt + g(Ut)dWt, (1)where {Wt, t >_ 0} is a standard scalar Wiener process, based on the application of an order 7 strong Taylor scheme [4] with constant time-step A to the N-dimensional stochastic differential equations (SDE)dU N obtained 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 depending on the initial value and bounds on the coefficient functions f,g of the SPDE (1) as well as on the length of the time interval 0 kA T under consideration, and 1j is the jth eigenvalue of the operator -A (whose eigenfunctions provide the bases for the Galerkin approximations).We refer to [1,2,3] for the functional analytical terminology and formalism of stochastic PDE (1) with a Dirichlet boundary condition for a bounded domain inNd with sufficiently smooth boundary 0, noting here that it has a unique strong solutionU e L2([0, T], 31;’ 2) N C([0, T], 2.2)3t; for each finite T > 0 and initial condition U0 E ’2 under the assumption that f and g are uniformly Lipschitz continuous from L2( into itself. Linear-implicit versions of strong Taylor numerical schemes for finite dimensional It6 stochastic differential equations (SDEs) are shown to have the same order as the original scheme. The combined truncation and global discretization error of an 7 strong linear-implicit Taylor scheme with time-step A applied to the N dimensional It6-Galerkin SDE for a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues 11 _< 12 _< in its drift term is estimated by
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