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

TL;DR

This work addresses the challenge of accurately and boundary-preservingly approximating semilinear SPDEs with bounded invariant domains and non-globally Lipschitz coefficients. It introduces the Lie–Trotter–Exact (LTE) scheme, which couples a finite-difference spatial discretisation with a Lie–Trotter time split, exact simulation of the diagonal SDE, and exact integration of the linear part, all ensuring the solution remains in the invariant domain. The authors prove a weak convergence rate of $1/4$ in time and $1/2$ in space for Lipschitz test functions, with a coupling $\Delta t=\Delta x^2$ yielding $\mathcal{O}(\Delta t^{1/4})$, and establish boundary preservation. Numerical experiments on stochastic Allen–Cahn, Nagumo, and SIS SPDEs validate the theory and demonstrate the superiority of LTE over standard schemes in maintaining the invariant domain and achieving the predicted convergence behavior. This provides a robust, practically implementable method for simulating SPDEs with physically meaningful bounded states and non-globally Lipschitz nonlinearities.

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.

Figures (3)

• Figure 1: Weak errors on the time interval $[0,T] = [0,1]$ of the $\textrm{LTE}$ scheme for the Allen--Cahn SPDE for test functions $F_{1}(r) = e^{-|r|}$ and $F_{2}(r) = 0.5 \left( \sqrt{1-r^2} |r| + \arcsin(|r|) \right)$ and with $\lambda=1$, $u_{0}(x) = \sin (2 \pi x)$ and $\Delta t = \Delta x^{2}$. Averaged over $2500$ samples.
• Figure 2: Weak errors on the time interval $[0,T] = [0,1]$ of the $\textrm{LTE}$ scheme for the Nagumo SPDE using the test functions $F_{1}(r) = e^{-|r|}$ and $F_{2}(r) = 0.5 \left( \sqrt{1-r^2} |r| + \arcsin(|r|) \right)$ and with $\lambda=1$, $u_{0}(x) = 0.5( \sin (\pi x) + 1/2)$ and $\Delta t = \Delta x^{2}$. Averaged over $2500$ samples.
• Figure 3: Weak errors on the time interval $[0,T] = [0,1]$ of the $\textrm{LTE}$ scheme for the SIS SPDE for the test functions $F_{1}(r) = e^{-|r|}$ and $F_{2}(r) = 0.5 \left( \sqrt{1-r^2} |x| + \arcsin(|r|) \right)$ and with $\lambda=1$, $u_{0}(x) = 0.5( \sin (\pi x) + 1/2)$ and $\Delta t = \Delta x^{2}$. Averaged over $2500$ samples.

Theorems & Definitions (21)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Lemma 5
• proof
• Proposition 6
• proof : Proof of Proposition \ref{['prop:LTE-BP']}
• Lemma 7
• proof : Proof of Lemma \ref{['lem:LTEmild']}