Weak Convergence Rates for an Explicit Full-Discretization of Stochastic Allen–Cahn Equation with Additive Noise

We discretize the stochastic Allen-Cahn equation with additive noise by means of a spectral Galerkin method in space and a tamed version of the exponential Euler method in time. The resulting error bounds are analyzed for the spatio-temporal full discretization in both strong and weak senses. Different from existing works, we develop a new and direct approach for the weak error analysis, which does not rely on the associated Kolmogorov equation. It turns out that the obtained weak convergence rates are, in both spatial and temporal direction, essentially twice as high as the strong convergence rates. Also, it is revealed how the weak convergence rates depend on the regularity of the noise. Numerical experiments are finally reported to confirm the theoretical conclusion.

[1]  Xiaojie Wang,et al.  Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise , 2013 .

[2]  Stig Larsson,et al.  Optimal Regularity for Semilinear Stochastic Partial Differential Equations with Multiplicative Noise , 2011, 1109.6487.

[3]  Jacques Printems,et al.  Weak order for the discretization of the stochastic heat equation , 2007, Math. Comput..

[4]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[5]  Xiaojie Wang.  An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation , 2018, 1802.09413.

[6]  Jialin Hong,et al.  Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE , 2018, 1811.04075.

[7]  T. Kurtz,et al.  Stochastic equations in infinite dimensions , 2006 .

[8]  Arnulf Jentzen,et al.  Galerkin Approximations for the Stochastic Burgers Equation , 2013, SIAM J. Numer. Anal..

[9]  S. Larsson,et al.  On the discretisation in time of the stochastic Allen–Cahn equation , 2015, 1510.03684.

[10]  Charles-Edouard Br'ehier,et al.  Weak convergence rates of splitting schemes for the stochastic Allen–Cahn equation , 2018, BIT Numerical Mathematics.

[11]  Zhonghua Qiao,et al.  Strong approximation of monotone stochastic partial differential equations driven by white noise , 2020, IMA Journal of Numerical Analysis.

[12]  Stig Larsson,et al.  Weak convergence for a spatial approximation of the nonlinear stochastic heat equation , 2012, Math. Comput..

[13]  Jialin Hong,et al.  Strong and weak convergence rates of finite element method for stochastic partial differential equation with non-globally Lipschitz coefficients , 2018 .

[14]  Arnaud Debussche,et al.  Weak approximation of stochastic partial differential equations: the nonlinear case , 2008, Math. Comput..

[15]  Zhonghua Qiao,et al.  Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise , 2018, Stochastics and Partial Differential Equations: Analysis and Computations.

[16]  Zhonghua Qiao,et al.  Strong Approximation of Stochastic Allen-Cahn Equation with White Noise , 2018 .

[17]  M. Hutzenthaler,et al.  Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities , 2019, 1903.06066.

[18]  R. Kruse,et al.  Duality in refined Watanabe-Sobolev spaces and weak approximations of SPDE , 2013 .

[19]  Charles-Edouard Br'ehier,et al.  Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient , 2017, Journal de Mathématiques Pures et Appliquées.

[20]  R. Kruse.  Strong and Weak Approximation of Semilinear Stochastic Evolution Equations , 2013 .

[21]  Jialin Hong,et al.  Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation , 2018, IMA Journal of Numerical Analysis.

[22]  Catherine E. Powell,et al.  An Introduction to Computational Stochastic PDEs , 2014 .

[23]  Arnulf Jentzen,et al.  Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients , 2014, The Annals of Applied Probability.

[24]  S. Larsson,et al.  Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes , 2012, BIT Numerical Mathematics.

[25]  István Gyöngy,et al.  Convergence of tamed Euler schemes for a class of stochastic evolution equations , 2014, Stochastics and Partial Differential Equations: Analysis and Computations.

[26]  Arnulf Jentzen,et al.  Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients , 2015, Found. Comput. Math..

[27]  G. Lord,et al.  Adaptive time-stepping for Stochastic Partial Differential Equations with non-Lipschitz drift , 2018, 1812.09036.

[28]  Arnulf Jentzen,et al.  Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg–Landau equations , 2016, Stochastic Processes and their Applications.

[29]  Charles-Edouard Br'ehier,et al.  Analysis of some splitting schemes for the stochastic Allen-Cahn equation , 2018, Discrete & Continuous Dynamical Systems - B.

[30]  P. Kloeden,et al.  Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations , 2017, Stochastics and Partial Differential Equations: Analysis and Computations.

[31]  Xiaojie Wang,et al.  Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Allen–Cahn Equation with Additive Noise , 2018, Journal of Scientific Computing.

[32]  Yi Zhang,et al.  Finite Element Methods for the Stochastic Allen-Cahn Equation with Gradient-type Multiplicative Noise , 2017, SIAM J. Numer. Anal..

[33]  Xiaojie Wang,et al.  Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus , 2014, 1408.0713.

[34]  Weak convergence rates for stochastic evolution equations and applications to nonlinear stochastic wave, HJMM, stochastic Schrödinger and linearized stochastic Korteweg–de Vries equations , 2017, Zeitschrift für angewandte Mathematik und Physik.

[35]  Arnulf Jentzen,et al.  Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities , 2015, IMA Journal of Numerical Analysis.

[36]  S. Larsson,et al.  Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise , 2012 .

[37]  S. Cerrai.  Second Order Pde's in Finite and Infinite Dimension: A Probabilistic Approach , 2001 .