Higher Order Algorithms for the Numerical Integration of Stochastic Differential Equations.

Standard numerical integration methods for ordinary differential equations are the second order Runge-Kutta (RKII) method and the fourth order Runge-Kutta (RKIV) method. We have extended these methods to integrate additive stochastic differential equations driven by white or colored noise, resulting in stochastic Runge-Kutta methods (SRK). We have tested these new SRK methods on the Ornstein-Uhlenbeck processes. The simulation averages, variances, and correlations for these extensions show considerable improvement over the standard lower order Euler Algorithm. Mean first passage times (MFPT) are obtained for the bistable potential well with colored noise using the new SRK algorithms. These results are compared with existing theory and show good agreement.