A Runge-Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise

TL;DR

This paper introduces a new Runge-Kutta type numerical scheme for nonlinear SPDEs with multiplicative trace class noise, offering higher convergence order and easier implementation compared to existing methods.

Contribution

The paper presents a novel Runge-Kutta type scheme for SPDEs that improves convergence and reduces computational complexity over previous Milstein-type schemes.

Findings

Scheme converges faster than linear implicit Euler.

Easier to implement than Milstein scheme.

Numerical examples confirm theoretical advantages.

Abstract

In this paper a new Runge-Kutta type scheme is introduced for nonlinear stochastic partial differential equations (SPDEs) with multiplicative trace class noise. The proposed scheme converges with respect to the computational effort with a higher order than the well-known linear implicit Euler scheme. In comparison to the infinite dimensional analog of Milstein type scheme recently proposed in [Jentzen & R\"{o}ckner (2012); A Milstein scheme for SPDEs, Arxiv preprint arXiv:1001.2751v4], our scheme is easier to implement and needs less computational effort due to avoiding the derivative of the diffusion function. The new scheme can be regarded as an infinite dimensional analog of Runge-Kutta method for finite dimensional stochastic ordinary differential equations (SODEs). Numerical examples are reported to support the theoretical results.