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On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation

Part of: Stochastic analysis Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  30 January 2018

Mihály Kovács ,
Stig Larsson  and
Fredrik Lindgren
Mihály Kovács*
Affiliation:
University of Otago
Stig Larsson*
Affiliation:
Chalmers University of Technology and University of Gothenburg
Fredrik Lindgren*
Affiliation:
Chalmers University of Technology and University of Gothenburg
*
∗ Postal address: Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, New Zealand. Email address: mkovacs@maths.otago.ac.nz
∗∗ Postal address: Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden.
∗∗ Postal address: Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden.
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Abstract

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We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(Δtγ) for any γ < ½. We also prove that the scheme converges uniformly in the strong Lp-sense but with no rate given.

Keywords

Stochastic partial differential equationAllen-Cahn equationadditive noiseWiener processEuler methodpathwise convergencestrong convergencefactorization method

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 60H35: Computational methods for stochastic equations 65C30: Stochastic differential and integral equations
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 323 - 338
Copyright
© Applied Probability Trust 

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