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On the discretisation in time of the stochastic Allen-Cahn equation

Journal article
Authors Mihaly Kovacs
Stig Larsson
F. Lindgren
Published in Mathematische Nachrichten
Volume 291
Issue 5-6
Pages 966-995
ISSN 0025-584X
Publication year 2018
Published at Department of Mathematical Sciences
Pages 966-995
Language en
Links doi.org/10.1002/mana.201600283
Keywords Additive noise, Allen-Cahn equation, Euler method, splitting method, stochastic partial differential equation, strong convergence, time discretisation, Wiener process, partial-differential-equations, evolution equations, convergence, approximation, schemes, spdes, Mathematics
Subject categories Mathematics

Abstract

We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a bounded spatial domain with smooth boundary in dimension d3, and study the semidiscretisation in time of the equation by an Euler type split-step method with step size k>0. We show that the method converges strongly with a rate O(k(1/2)). By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.

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