Families of efficient second order Runge-Kutta methods for the weak approximation of Itô stochastic differential equations

Recently, a new class of second order Runge-Kutta methods for Itô stochastic differential equations with a multidimensional Wiener process was introduced by Rößler [A. Rößler, Second order Runge-Kutta methods for Itô stochastic differential equations, Preprint No. 2479, TU Darmstadt, 2006]. In contrast to second order methods earlier proposed by other authors, this class has the advantage that the number of function evaluations depends only linearly on the number of Wiener processes and not quadratically. In this paper, we give a full classification of the coefficients of all explicit methods with minimal stage number. Based on this classification, we calculate the coefficients of an extension with minimized error constant of the well-known RK32 method [J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, West Sussex, 2003] to the stochastic case. For three examples, this method is compared numerically with known order two methods and yields very promising results.