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

Abstract

We consider the stochastic Allen--Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension $d\le 3$, and study the semidiscretisation in time of the equation by an Euler type split-step method. We show that the method converges strongly with a rate $O(\Delta t^{\frac{1}{2}})$. By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.

arXiv comment32 pages. Several typos fixed, the statement and proof of Lemma 3.2 are updated and so is the proof of Lemma 5.2. The proof of Lemma 5.1 is omitted

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• Strong error estimates for a fully discrete SAV scheme for the stochastic Allen--Cahn equation with multiplicative noise
• Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs
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• A convergent stochastic scalar auxiliary variable method
• Strong convergence rates of a fully discrete scheme for nonlinear stochastic partial differential equations with non-globally Lipschitz coefficients driven by multiplicative noise
• Strong convergence of the backward Euler approximation for the finite element discretization of semilinear parabolic SPDEs with non-global Lipschitz drift driven by additive noise
• Fully Discrete Mixed Finite Element Methods for the Stochastic Cahn-Hilliard Equation with Gradient-type Multiplicative Noise