An Exponential Stochastic Runge-Kutta Type Method of Order up to 1.5 for SPDEs of Nemytskii-type

TL;DR

This paper introduces a new exponential stochastic Runge-Kutta method for SPDEs of Nemytskii-type, achieving up to 1.5 order convergence with efficient computation, suitable for various spatial discretizations.

Contribution

The paper develops a derivative-free exponential stochastic Runge-Kutta scheme with strong convergence proof for Nemytskii-type SPDEs, extending numerical options for high-order solutions.

Findings

Achieves convergence order up to 1.5 in time.

Proves strong convergence in the root mean-square sense.

Numerical examples confirm theoretical convergence rates.

Abstract

For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge-Kutta type that allows for convergence with a temporal order of up to 3/2 and that can be combined with several spatial discretizations. The developed family of derivative-free schemes is tailored to stochastic partial differential equations of Nemytskii-type, i.e., with pointwise multiplicative noise operators. We prove the strong convergence of these schemes in the root mean-square sense and present some numerical examples that reveal the theoretical results.