Second order Runge-Kutta methods for Stratonovich stochastic differential equations

The weak approximation of the solution of a system of Stratonovich stochastic differential equations with a m-dimensional Wiener process is studied. Therefore, a new class of stochastic Runge-Kutta methods is introduced. As the main novelty, the number of stages does not depend on the dimension m of the driving Wiener process which reduces the computational effort significantly. The colored rooted tree analysis due to the author is applied to determine order conditions for the new stochastic Runge-Kutta methods assuring convergence with order two in the weak sense. Further, some coefficients for second order stochastic Runge-Kutta schemes are calculated explicitly.