Strong convergence analysis of the stochastic exponential Rosenbrock scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

TL;DR

This paper introduces a new stochastic exponential Rosenbrock scheme for semilinear SPDEs with strong nonlinearities, demonstrating its stability and convergence through theoretical analysis and numerical experiments.

Contribution

The paper proposes a novel stochastic exponential Rosenbrock scheme tailored for strongly nonlinear semilinear SPDEs, with proven strong convergence and stability.

Findings

The scheme achieves strong convergence in the root-mean-square $L^2$ norm.

Numerical experiments confirm the theoretical convergence results.

The method maintains stability for stochastic reactive dominated transport equations.

Abstract

In this paper, we consider the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations. Most numerical techniques, including current stochastic exponential integrators lose their good stability properties on such equations. Using finite element for space discretization, we propose a new scheme appropriated on such equations, called stochastic exponential Rosenbrock scheme (SERS) based on local linearization at every time step of the semi-discrete equation obtained after space discretization. We consider noise that is in a trace class and give a strong convergence proof of the new scheme toward the exact solution in the root-mean-square $L^2$ norm. Numerical experiments to sustain theoretical results are provided.