Strong convergence rates of a fully discrete scheme for nonlinear stochastic partial differential equations with non-globally Lipschitz coefficients driven by multiplicative noise

TL;DR

This paper introduces a fully discrete numerical scheme for nonlinear SPDEs with multiplicative noise, achieving optimal strong convergence rates without requiring commutativity of operators, and validated by numerical experiments.

Contribution

It develops a spectral method that relaxes previous restrictions and combines it with a tamed semi-implicit time scheme for improved stability and convergence.

Findings

Achieves optimal strong convergence rates in space and time.

Does not require the elliptic operator and covariance operator to commute.

Numerical experiments confirm theoretical convergence rates.

Abstract

We consider a fully discrete scheme for nonlinear stochastic partial differential equations with non-globally Lipschitz coefficients driven by multiplicative noise in a multi-dimensional setting. Our method uses a polynomial based spectral method in space, so it does not require the elliptic operator $A$ and the covariance operator $Q$ of noise in the equation commute, and thus successfully alleviates a restriction of Fourier spectral method for SPDEs pointed out by Jentzen, Kloeden and Winkel. The discretization in time is a tamed semi-implicit scheme which treats the nonlinear term explicitly while being unconditionally stable. Under regular assumptions which are usually made for SPDEs with additive noise, we establish optimal strong convergence rates in both space and time for our fully discrete scheme. We also present numerical experiments which are consistent with our theoretical results.