Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg–Landau equations

Abstract

This article proposes and analyzes explicit and easily implementable temporal numerical approximation schemes for additive noise-driven stochastic partial differential equations (SPDEs) with polynomial nonlinearities such as, e.g., stochastic Ginzburg-Landau equations. We prove essentially sharp strong convergence rates for the considered approximation schemes. Our analysis is carried out for abstract stochastic evolution equations on separable Banach and Hilbert spaces including the above mentioned SPDEs as special cases. We also illustrate our strong convergence rate results by means of a numerical simulation in Matlab. H_dim = 2ˆ10; N_ref = H_dimˆ2; N_apx = 2.ˆ(6:18); MC_runs = 25; 2 A = -(1:H_dim).ˆ2 . * piˆ2; F = @(x) x -x.ˆ3; rng('default'); 3 X = zeros(1, H_dim); Y = zeros(numel(N_apx), H_dim); 4 Z = zeros(numel(N_apx), H_dim); W = zeros(numel(N_apx), H_dim); 5 Err_Y = zeros (1,numel(N_apx)); Err_Z = zeros(1,numel(N_apx));

Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation

“…For SPDEs with non-Lipschitz coefficients, there only exist a few results about the strong convergence rates of numerical schemes (see e.g. [2,3,8,9,12]). The strong convergence rates of numerical schemes, especially the temporal discretization, is far from being understood and it is still an open problem to derive general strong convergence rates of numerical schemes for SPDEs with non-globally Lipschitz coefficients.…”

Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Allen–Cahn Equation with Additive Noise

“…Over the past decades, there have been plenty of research articles analyzing numerical discretizations of parabolic stochastic partial differential equations (SPDEs), see, e.g., monographs [21,25] and references therein. In contrast to an overwhelming majority of literature focusing on numerical analysis of SPDEs with globally Lipschitz nonlinearity, only a limited number of papers investigated numerical SPDEs in the non-globally Lipschitz regime [1][2][3][4][5]10,11,13,15,16,19,24] and it is still far from well-understood. As a typical example of parabolic SPDEs with non-globally Lipschitz nonlinearity, stochastic Allen-Cahn equations, perturbed by additive or multiplicative noises, have received increasing attention in the last few years.…”

“…The nonlinear mapping F is assumed to be a Nemytskij operator, given by F (u)(x) = f (u(x)), x ∈ D, with f (v) = v − v 3 , v ∈ R. Such problem is often referred to as stochastic Allen-Cahn equation. Under further assumptions specified later, particularly including A γ−1 2 Q 1 2 L 2 < ∞, for some γ ∈ [1,2], (1.2) it is proved in Theorems 2.5, 2.6 that the problem (1.1) possesses a unique mild solution,…”

“…Particularly, the authors of [3] used exponential integrability properties of exact and numerical solutions to identify a strong convergence rate of order 1, but only valid in one space dimension, when the additive noise is moderately smooth, i.e., γ = 2 in (1.2). For the space-time white noise case, various discretizations were investigated in other contributions [1,2,24,30].…”

“…In the sequel we treat the above four terms one by one. Thanks to (2.4), (2.18), (2.14) and (2.13), we derive, for γ ∈ [1,2] and any fixed δ 0 ∈ ( 3 2 , 2),…”