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An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation

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  • Wang, Xiaojie

Abstract

In Becker and Jentzen (2019) and Becker et al. (2017), an explicit temporal semi-discretization scheme and a space–time full-discretization scheme were, respectively, introduced and analyzed for the additive noise-driven stochastic Allen–Cahn type equations, with strong convergence rates recovered. The present work aims to propose a different explicit full-discrete scheme to numerically solve the stochastic Allen–Cahn equation with cubic nonlinearity, perturbed by additive space–time white noise. The approximation is easily implementable, performing the spatial discretization by a spectral Galerkin method and the temporal discretization by a kind of nonlinearity-tamed accelerated exponential integrator scheme. Error bounds in a strong sense are analyzed for both the spatial semi-discretization and the spatio-temporal full discretization, with convergence rates in both space and time explicitly identified. It turns out that the obtained convergence rate of the new scheme is, in the temporal direction, twice as high as existing ones in the literature. Numerical results are finally reported to confirm the previous theoretical findings.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:10:p:6271-6299
    DOI: 10.1016/j.spa.2020.05.011
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    References listed on IDEAS

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    1. Becker, Sebastian & Jentzen, Arnulf, 2019. "Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg–Landau equations," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 28-69.
    2. 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.
    3. Lord, Gabriel J. & Tambue, Antoine, 2018. "A modified semi–implicit Euler–Maruyama scheme for finite element discretization of SPDEs with additive noise," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 105-122.
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    Cited by:

    1. Cui, Jianbo & Hong, Jialin & Sun, Liying, 2021. "Weak convergence and invariant measure of a full discretization for parabolic SPDEs with non-globally Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 55-93.
    2. Chen, Chuchu & Dang, Tonghe & Hong, Jialin & Zhou, Tau, 2023. "CLT for approximating ergodic limit of SPDEs via a full discretization," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 1-41.

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