A-stable Runge–Kutta methods for stiff stochastic differential equations with multiplicative noise

Abstract

In this paper, a class of one-stage semi-implicit stochastic Runge–Kutta (SISRK) methods is proposed for stiff systems with multiplicative noise. The coefficient families of SISRK methods of strong order one-half are calculated. The stability functions of these methods, applied to a scalar linear test equation with multi-dimensional multiplicative noise, are determined and their regions of stability are then compared with the corresponding stability regions of the test equation. Furthermore, we also investigate mean square stability (MS-stability) of these methods applied to two linear multi-dimensional stochastic differential test equations with multi-dimensional multiplicative noise. In particular, some SISRK methods with mean-square A-stability are derived for systems with multiplicative noise. The stability properties and numerical examples are given to illustrate the efficiency and performance of the proposed methods.