A fully discrete approximation of the one-dimensional stochastic heat equation

Abstract

A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space-time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz, this explicit time integrator allows for error bounds in L q (Ω), for all q ≥ 2, improving some existing results in the literature. On top of this, we also prove almost sure convergence of the numerical scheme. In the case of non-globally Lipschitz coefficients, we provide sufficient conditions under which the numerical solution converges in probability to the exact solution. Numerical experiments are presented to illustrate the theoretical results.Mathematics Subject Classification (2010): 60H15; 60H35. Error analysis for globally Lipschitz continuous coefficientsThis section is divided into three subsections. We begin by stating the assumptions we will make and by recalling the mild solution of (1). The first subsection is dedicated to

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

“…Defining efficient numerical schemes for stochastic equations with non globally Lipschitz continuous coefficients is delicate: see for instance the recent work [33], and the monograph [32], and references therein. The general methodology has been recently applied to various examples of SPDEs: see for instance [2,4,34,40,41,42]. We also mention [30] for the analysis of a tamed Euler scheme for a class of SPDEs.…”

Spatial Discretization for Stochastic Semi-Linear Subdiffusion Equations Driven by Fractionally Integrated Multiplicative Space-Time White Noise

“…The main aim of this paper is to extend the spatial discretization schemes discussed in Gyöngy [1] and Anton et al [5] for the stochastic quasi-linear parabolic partial differential equations driven by multiplicative space-time white noise to the stochastic subdiffusion equations driven by integrated multiplicative space-time white noise. We obtain the error estimates uniformly in space for the proposed finite difference method.…”

“…which make the numerical analysis of the stochastic subdiffusion Equation (1) much more challenging than the stochastic parabolic equation discussed in [1,5]. To the best of our knowledge, there are no error estimates uniformly in space for the stochastic subdiffusion equations driven by space-time white noise in literature.…”

“…We assume that the nonlinear terms f and σ satisfy the following globally Lipschitz and the linear growth conditions [1], p. 6 and [5], that is, with some positive constant C > 0,…”

“…∂x 2 and the space-time white noise ∂ 2 W(t,x) ∂t∂x with the finite difference methods as in Gyöngy [1] and Anton et al [5] and obtain a spatial discretization scheme for approximating (1). The convergence rate in the mean-square sense is obtained, uniformly for x ∈ [0, 1].…”