Two-stage stochastic Runge–Kutta methods for stochastic differential equations

In this paper we discuss two-stage diagonally implicit stochastic Runge–Kutta methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations. Five stochastic Runge–Kutta methods are presented in this paper. They are an explicit method with a large MS-stability region, a semi-implicit method with minimum principal error coefficients, a semi-implicit method with a large MS-stability region, an implicit method with minimum principal error coefficients and another implicit method. We also consider composite stochastic Runge–Kutta methods which are the combination of semi-implicit Runge–Kutta methods and implicit Runge–Kutta methods. Two composite methods are presented in this paper. Numerical results are reported to compare the convergence properties and stability properties of these stochastic Runge–Kutta methods.