Strong convergence of an explicit full-discrete scheme for stochastic Burgers-Huxley equation

TL;DR

This paper proves strong convergence of a fully discrete numerical scheme for the stochastic Burgers-Huxley equation with additive noise, using spectral Galerkin and nonlinear-tamed exponential integrator methods.

Contribution

It introduces a novel fully discrete scheme combining spectral Galerkin and tamed exponential integrator methods with rigorous convergence analysis.

Findings

Established strong convergence rates in space and time.

Provided Sobolev and Hölder regularity estimates for solutions.

Validated theoretical results with a numerical example.

Abstract

The strong convergence of an explicit full-discrete scheme is investigated for the stochastic Burgers-Huxley equation driven by additive space-time white noise, which possesses both Burgers-type and cubic nonlinearities. To discretize the continuous problem in space, we utilize a spectral Galerkin method. Subsequently, we introduce a nonlinear-tamed exponential integrator scheme, resulting in a fully discrete scheme. Within the framework of semigroup theory, this study provides precise estimations of the Sobolev regularity, $L^\infty$ regularity in space, and H\"older continuity in time for the mild solution, as well as for its semi-discrete and full-discrete approximations. Building upon these results, we establish moment boundedness for the numerical solution and obtain strong convergence rates in both spatial and temporal dimensions. A numerical example is presented to validate the theoretical findings.