Boundary-preserving weak approximation for some semilinear stochastic partial differential equations

TL;DR

This paper introduces a boundary-preserving numerical scheme for weak approximation of certain SPDEs with bounded state-space, allowing non-globally Lipschitz coefficients and demonstrating convergence and practical advantages through numerical experiments.

Contribution

It develops a novel Lie--Trotter-Exact scheme that preserves boundaries and achieves specific weak convergence orders for SPDEs with non-globally Lipschitz coefficients.

Findings

Converges with order 1/4 in time and 1/2 in space for test functions.

Ensures boundary preservation through Lie--Trotter splitting and exact simulation.

Numerical experiments confirm theoretical convergence and advantages.

Abstract

We propose and analyse a boundary-preserving numerical scheme for the weak approximation for some stochastic partial differential equations (SPDEs) with bounded state-space. We impose regularity assumptions on the drift and diffusion coefficients only locally on the state-space. In particular, the drift and diffusion coefficients may be non-globally Lipschitz continuous and superlinearly growing. The scheme consists of a finite difference discretisation in space and a Lie--Trotter time splitting followed by exact simulation and exact integration in time. The proposed scheme converges in the weak sense of order $1/4$ in time and of order $1/2$ in space, for globally Lipschitz continuous test functions. We prove the weak convergence order in time by proving strong convergence towards a strong solution driven by a different noise process. The convergence order in space follows from known results. The boundary-preserving property is ensured by the use of Lie--Trotter time splitting followed by exact simulation and exact integration. Numerical experiments confirm the theoretical results and demonstrate the practical advantages of the proposed Lie--Trotter-Exact (LTE) scheme compared to existing schemes for SPDEs.