A class of weak second order split-drift stochastic Runge–Kutta schemes for stiff SDE systems

In this paper we present a family of stochastic Runge–Kutta (SRK) schemes for stiff systems of Itô stochastic differential equations (SDEs). Derivation of these schemes, called SDSRK to denote split-drift SRK, is based on expressing the drift as a sum of two terms. Then a class of SDSRK schemes with stochastic weak second order is introduced. For a subclass of these schemes that is appropriate for stiff SDEs, an analysis of mean-square stability (MS-stability) is given. Then by using the obtained stability function and the order conditions we determine the parameters of two schemes, called SDSRK4a and SDSRK4b, having deterministic and stochastic weak orders 2, from which the first one is fully drift implicit with better stability properties and the second one is semi-implicit with reasonable stability properties but with much less computational cost. Numerical examples are presented to demonstrate the efficiency and accuracy of the new schemes.