Explicit stabilized multirate method for stiff stochastic differential equations

TL;DR

This paper introduces a multirate explicit method for stiff stochastic differential equations that efficiently handles systems with a few severely stiff terms without relying on scale separation, improving computational performance.

Contribution

A novel multirate scheme based on a stochastic modified equation that isolates slow terms, overcoming limitations of traditional stabilized explicit methods in stiff SDEs.

Findings

Scheme achieves strong order 1/2 and weak order 1.

Numerical experiments demonstrate improved efficiency and accuracy.

Method applicable to spatial discretizations of stochastic PDEs on refined grids.

Abstract

Stabilized explicit methods are particularly efficient for large systems of stiff stochastic differential equations (SDEs) due to their extended stability domain. However, they loose their efficiency when a severe stiffness is induced by very few "fast" degrees of freedom, as the stiff and nonstiff terms are evaluated concurrently. Therefore, inspired by [A. Abdulle, M. J. Grote, and G. Rosilho de Souza, Preprint (2020), arXiv:2006.00744] we introduce a stochastic modified equation whose stiffness depends solely on the "slow" terms. By integrating this modified equation with a stabilized explicit scheme we devise a multirate method which overcomes the bottleneck caused by a few severely stiff terms and recovers the efficiency of stabilized schemes for large systems of nonlinear SDEs. The scheme is not based on any scale separation assumption of the SDE and therefore it is employable for problems stemming from the spatial discretization of stochastic parabolic partial differential equations on locally refined grids. The multirate scheme has strong order 1/2, weak order 1 and its stability is proved on a model problem. Numerical experiments confirm the efficiency and accuracy of the scheme.