Analysis of Some Splitting Schemes for the Stochastic Allen-Cahn Equation

Abstract

We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution. We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, L^2 ($\Omega$)-convergence of order almost 1/4, localized on an event of arbitrarily large probability, then convergence in probability of order almost 1/4. The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.

References in corpus (16)

• Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients
• Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients
• On the discretisation in time of the stochastic Allen-Cahn equation
• On the backward Euler approximation of the stochastic Allen-Cahn equation
• Computing transition rates for the 1-D stochastic Ginzburg--Landau--Allen--Cahn equation for finite-amplitude noise with a rare event algorithm
• Splitting up method for the 2D stochastic Navier-Stokes equations
• Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem
• Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations
• Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients
• On the short time asymptotic of the stochastic Allen-Cahn equation
• Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients
• Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations
• On strong $L^2$ convergence of time numerical schemes for the stochastic 2D Navier-Stokes equations
• Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations
• Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities
• Exponential moments for numerical approximations of stochastic partial differential equations

Cited by in corpus (17)

• Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise
• Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
• Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations
• Strong convergence rates of semi-discrete splitting approximations for stochastic Allen--Cahn equation
• Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE
• Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes
• Parameter Estimation in Nonlinear Multivariate Stochastic Differential Equations Based on Splitting Schemes
• Error estimates of semi-discrete and fully discrete finite element methods for the Cahn-Hilliard-Cook equation
• Optimal rate of convergence for approximations of SPDEs with non-regular drift
• On the Itô-Alekseev-Gröbner formula for stochastic differential equations
• Convergence analysis of explicit stabilized integrators for parabolic semilinear stochastic PDEs
• Strong and weak convergence rates of finite element method for stochastic partial differential equation with non-sided Lipschitz coefficient
• Qualitative properties of numerical methods for the inhomogeneous geometric Brownian motion
• Strang Splitting for Parametric Inference in Second-order Stochastic Differential Equations
• A convergent stochastic scalar auxiliary variable method
• Strong convergence of parabolic rate $1$ of discretisations of stochastic Allen-Cahn-type equations
• A new efficient operator splitting method for stochastic Maxwell equations