Explicit Stabilized Multirate Method For Stiff Stochastic Differential Equations

Stabilized explicit methods are particularly efficient, for large systems of stiff stochastic differential equations (SDEs) due to their extended stability domain. However, they lose 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, Explicit stabilized multirate method for stiff differential equations, Math. Comp., in press, 2022], we introduce a stochastic modified equation whose stiffness depends solely on vi 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 still 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. 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.