Non-asymptotic uniform in time error bounds for new and old numerical schemes for SPDEs

TL;DR

This paper develops a general method to establish non-asymptotic uniform in time error bounds for various numerical schemes solving SPDEs, including those with challenging nonlinearities like the stochastic Allen-Cahn equation.

Contribution

It introduces a novel proof technique for uniform error bounds and analyzes schemes that avoid blow-up in SPDEs with non-globally Lipschitz nonlinearities.

Findings

Semi-implicit Euler scheme can blow up in finite time for Allen-Cahn SPDEs.

Tamed schemes provide stable solutions with proven uniform error bounds.

Numerical experiments confirm theoretical error estimates.

Abstract

We study numerical schemes for Stochastic Partial Differential Equations (SPDEs). We introduce a general method of proof of non-asymptotic uniform in time error bounds on numerical integrators for SPDEs, ensuring the schemes capture both the transient and the long term dynamics faithfully. We then consider SPDEs with non-globally Lipshitz nonlinearities, which include for example the stochastic Allen-Cahn equation and some stochastic advection-diffusion equations. For the case of Allen-Cahn type SPDEs we show that the classic semi-implicit Euler time-discretization can exhibit finite time blow up. This motivates analysing other schemes which do not suffer from this blow-up problem. We consider three numerical schemes for SPDEs with non globally Lipshitz nonlinearity: a fully implicit scheme and two tamed schemes. For these schemes we prove non-asymptotic uniform in time error bounds by leveraging our general criterion, and provide numerical comparisons. While the main emphasis in this paper is on the properties of the time-discretization, the schemes we consider are full space-time discretization of the SPDE.