Strong Convergence of a Stochastic Rosenbrock-type Scheme for the Finite Element Discretization of Semilinear SPDEs Driven by Multiplicative and Additive Noise

TL;DR

This paper introduces a stochastic Rosenbrock-type scheme for finite element discretization of semilinear SPDEs with multiplicative and additive noise, demonstrating strong convergence and stability for equations with dominant nonlinear terms.

Contribution

The paper proposes a novel stochastic Rosenbrock-type scheme tailored for semilinear SPDEs with strong nonlinearities, ensuring strong convergence and improved stability over standard methods.

Findings

The scheme achieves strong convergence rates consistent with existing literature.

Numerical experiments confirm the theoretical convergence and stability properties.

The method effectively handles SPDEs with multiplicative and additive noise, especially when nonlinear terms dominate.

Abstract

This paper aims to investigate the numerical approximation of a general second order parabolic 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, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretise the SPDE in space by the finite element method and propose a new scheme in time appropriate for such equations, called stochastic Rosenbrock-Type scheme, which is based on the local linearisation of the semi-discrete problem obtained after space discretisation. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise. Our convergence rates are in agreement with results in the literature. Numerical experiments to sustain our theoretical results are provided.