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

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  • Mihály Kovács
  • Stig Larsson
  • Fredrik Lindgren

Abstract

We consider the stochastic Allen–Cahn equation perturbed by smooth additive Gaussian noise in a bounded spatial domain with smooth boundary in dimension d≤3, 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(k12). By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.

Suggested Citation

  • Mihály Kovács & Stig Larsson & Fredrik Lindgren, 2018. "On the discretisation in time of the stochastic Allen–Cahn equation," Mathematische Nachrichten, Wiley Blackwell, vol. 291(5-6), pages 966-995, April.
    • Handle: RePEc:bla:mathna:v:291:y:2018:i:5-6:p:966-995
    DOI: 10.1002/mana.201600283
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    Cited by:

    1. Wang, Xiaojie, 2020. "An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6271-6299.

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