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

Abstract

A novel explicit full discrete scheme is proposed to numerically solve the stochastic Allen-Cahn equation with cubic nonlinearity, perturbed by additive space-time white noise. The approximation is easily implementable, performing spatial discretization by a spectral Galerkin method and temporal discretization by a kind of nonlinearity-tamed accelerated exponential integrator scheme. Error bounds in a strong sense are analyzed for both the spatial semidiscretization and the spatio-temporal full discretization, with convergence rates in both space and time explicitly identified. It turns out that the obtanied 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 previous theoretical findings.

Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE

“…Thus in Sections 2-4, we mainly focus on weak convergence rates of numerical methods in the case that β ∈ (0, 1]. We would like to mention that for SPDEs with non-globally Lipschitz coefficients, there already exist a lot of results on the strong convergence and strong convergence rate of numerical approximations, see [2,3,4,6,8,14,15,16,23,24,27] and references therein.…”

Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations

“…The results of this article, that is, inequalities (2) and ( 3), prove that these rates are essentially (up to an arbitrarily small polynomial order of convergence) optimal. We also refer, e.g., to [9,25,10,8,23,20,21,7,13,14,11,2,22,15,4,3,24,19] for further research articles on explicit approximation schemes for stochastic differential equations with superlinearly growing non-linearities. Furthermore, related lower bounds for approximation errors in the linear case (i.e., in the case a = b = 0 in ( 4…”

Numerical analysis of a full discretization for stochastic Cahn--Hilliard equation driven by additive noise

“…In contrast to the Lipschitz case, convergence and convergence rate of numerical approximation for SPDEs with non-globally Lipschitz continuous nonlinearity, become more involved recently (see e.g. [1,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,20,22,23,25,28,29,30,31]) and are far from well-understood. Up to now, there have been some works focusing on the strong convergence and strong convergence rates of numerical schemes for the stochastic Cahn-Hilliard equation (see e.g.…”