The BDF2-Maruyama Scheme for Stochastic Evolution Equations with Monotone Drift

TL;DR

This paper introduces a BDF2-Maruyama numerical scheme for stochastic evolution equations with monotone drift, proving its convergence and demonstrating its effectiveness through numerical experiments.

Contribution

It develops and analyzes a new BDF2-Maruyama scheme combining implicit time discretization with Galerkin spatial methods for stochastic equations.

Findings

Proves well-posedness of the scheme.

Establishes strong convergence rates.

Shows improved performance over backward Euler--Maruyama.

Abstract

We study the numerical approximation of stochastic evolution equations with a monotone drift driven by an infinite-dimensional Wiener process. To discretize the equation, we combine a drift-implicit two-step BDF method for the temporal discretization with an abstract Galerkin method for the spatial discretization. After proving well-posedness of the BDF2-Maruyama scheme, we establish a convergence rate of the strong error for equations under suitable Lipschitz conditions. We illustrate our theoretical results through various numerical experiments and compare the performance of the BDF2-Maruyama scheme to the backward Euler--Maruyama scheme.