Strong convergence of numerical discretizations for semilinear stochastic evolution equations driven by multiplicative white noise

TL;DR

This paper establishes strong convergence rates for numerical schemes solving semilinear stochastic evolution equations with non-Lipschitz coefficients driven by white noise, demonstrating superconvergence and optimality in specific cases.

Contribution

It introduces a new approach to analyze strong convergence for a broad class of second order parabolic SPDEs with multiplicative white noise, extending beyond classical Lipschitz conditions.

Findings

Spectral Galerkin approximation is superconvergent.

Explicit exponential integrator achieves optimal convergence rate.

Numerical experiments confirm theoretical results.

Abstract

For semilinear stochastic evolution equations whose coefficients are more general than the classical global Lipschitz, we present results on the strong convergence rates of numerical discretizations. The proof of them provides a new approach to strong convergence analysis of numerical discretizations for a large family of second order parabolic stochastic partial differential equations driven by space-time white noises. We apply these results to the stochastic advection-diffusion-reaction equation with a gradient term and multiplicative white noise, and show that the strong convergence rate of a fully discrete scheme constructed by spectral Galerkin approximation and explicit exponential integrator is exactly $\frac12$ in space and $\frac14$ in time. Compared with the optimal regularity of the mild solution, it indicates that the spetral Galerkin approximation is superconvergent and the convergence rate of the exponential integrator is optimal. Numerical experiments support our theoretical analysis.