Runge-Kutta methods for Ito stochastic differential equations with scalar noise

A general class of stochastic Runge-Kutta methods for Ito stochastic differential equation systems w.r.t. a one-dimensional Wiener process is introduced. The colored rooted tree analysis is applied to derive conditions for the coefficients of the stochastic Runge-Kutta method assuring convergence in the weak sense with a prescribed order. Some coefficients for new stochastic Runge-Kutta schemes of order two are calculated explicitly and a simulation study reveals their good performance.