A fully discrete approximation of the one-dimensional stochastic heat equation
Anton, Rikard (Umeå University. Department of Mathematics and Mathematical Statistics)
Cohen, David (Umeå University. Department of Mathematics and Mathematical Statistics)
Quer i Sardanyons, Lluís (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Data:	2020
Resum:	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 Lq(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 nonglobally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.
Ajuts:	Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-422 Ministerio de Economía y Competitividad MTM2015-67802-P
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Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Matèria:	Finite difference scheme ; Lq(Ω)-convergence ; Multiplicative noise ; Stochastic exponential integrator ; Stochastic heat equation
Publicat a:	IMA Journal of Numerical Analysis, Vol. 40, Issue 1 (January 2020) , p. 247-284, ISSN 1464-3642

DOI: 10.1093/imanum/dry060

Postprint 37 p, 386.1 KB

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