Exponential Runge-Kutta methods for stiff stochastic differential equations (Fusion of theory and practice in applied mathematics and computational science)

Abstract

It is well known that the numerical solution of stiff stochastic differential equations (SDEs) leads to a stepsize reduction when explicit methods are used. However, there are some classes of explicit methods that are well suited to solving some types of stiff SDEs. One such class is the class of stochastic orthogonal Runge-Kutta Chebyshev (SROCK) methods. SROCK methods reduce to Runge- Kutta Chebyshev methods when applied to ordinary differential equations (ODEs). Another promising class of methods is the class of explicit methods that reduce to explicit exponential Runge-Kutta (RK) methods when applied to semilinear ODEs. In the present paper, such explicit methods are considered. As a result, the stochastic exponential Euler scheme will be derived for strong approximations to the solution of stiff Itô SDEs with a semilinear drift term. In addition, stochastic exponential RK methods will be derived for weak approximations.