Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations

Abstract

The scientific literature contains a number of numerical approximation results for stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities but, to the best of our knowledge, none of them prove strong or weak convergence rates for full-discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities. In particular, in the scientific literature there exists neither a result which proves strong convergence rates nor a result which proves weak convergence rates for full-discrete numerical approximations of stochastic Allen-Cahn equations. In this article we bridge this gap and establish strong convergence rates for full-discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. Moreover, we also establish lower bounds for strong temporal and spatial approximation errors which demonstrate that our strong convergence rates are essentially sharp and can, in general, not be improved.

104 pages

References in corpus (6)

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• Optimal error estimates of Galerkin finite element methods for stochastic Allen-Cahn equation with additive noise
• Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen-Cahn Equation with multiplicative noise
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Cited by in corpus (21)

• Strong Approximation of Stochastic Allen-Cahn Equation with White Noise
• Analysis of Some Splitting Schemes for the Stochastic Allen-Cahn Equation
• 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
• $L^p$-Convergence Rate of Backward Euler Schemes for Monotone SDEs
• 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
• 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
• Strong convergence of a fully discrete scheme for stochastic Burgers equation with fractional-type noise
• On the Itô-Alekseev-Gröbner formula for stochastic differential equations
• Numerical Ergodicity and Uniform Estimate of Monotone SPDEs Driven by Multiplicative Noise
• Strong and weak convergence rates of finite element method for stochastic partial differential equation with non-sided Lipschitz coefficient
• Strong convergence for explicit space-time discrete numerical approximation for 2D stochastic Navier-Stokes equations
• Theoretical analysis of a finite-volume scheme for a stochastic Allen-Cahn problem with constraint
• Strong convergence of parabolic rate $1$ of discretisations of stochastic Allen-Cahn-type equations
• Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus
• A convergent stochastic scalar auxiliary variable method
• Monte Carlo convergence rates for $k$th moments in Banach spaces
• Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions