Crank--Nicolson Finite Element Approximations for a Linear Stochastic Fourth Order Equation with Additive Space-Time White Noise

Abstract

We formulate an initial-and Dirichlet boundary-value problem for a linear stochastic heat equation, in one space dimension, forced by an additive space-time white noise. First, we approximate the mild solution to the problem by the solution of the regularized second-order linear stochastic parabolic problem with random forcing proposed by Allen, Novosel and Zhang (Stochastics Stochastics Rep., 64, 1998). Then, we construct numerical approximations of the solution to the regularized problem by combining the Crank-Nicolson method in time with a standard Galerkin finite element method in space. We derive strong a priori estimates of the modeling error made in approximating the mild solution to the problem by the solution to the regularized problem, and of the numerical approximation error of the Crank-Nicolson finite element method. D G t (x, y) v 0 (y) dy ∀ (t, x) ∈ (0, T ] × D 1991 Mathematics Subject Classification. 65M60, 65M15, 65C20.

Control of diffusion-driven pattern formation behind a wave of competency

“…For the time stepping, we use an implicit-explicit scheme (IMEX) [38][39][40], where we implement the diffusion term using the implicit Crank-Nicolson method, and use the explicit Euler method for the nonlinear reaction term, with ∆t = 0.01. The Crank-Nicolson scheme has been shown to be stable even in the presence of noise [41,42]. We have verified that all described behaviour still occurs for a much finer time-step with ∆t = 0.00125 or a finer grid size with ∆x = 0.25, and that the numerical scheme indeed converges as ∆t is decreased.…”