Efficient Approximate Techniques for Integrating Stochastic Differential Equations

The delicate (and computationally expensive) nature of stochastic numerical modeling naturally leads one to look for efficient and/or convenient methods for integrating stochastic differential equations. Concomitantly, one may wish to sensibly add stochastic terms to an existing deterministic model without having to rewrite that model. In this note, two possibilities in the context of the fourth-order Runge–Kutta (RK4) integration scheme are examined. The first approach entails a hybrid of deterministic and stochastic integration schemes. In these examples, the hybrid RK4 generates time series with the correct climatological probability distributions. However, it is doubtful that the resulting time series are approximate solutions to the stochastic equations at every time step. The second approach uses the standard RK4 integration method modified by appropriately scaling stochastic terms. This is shown to be a special case of the general stochastic Runge–Kutta schemes considered by Ruemelin and has global convergence of order one. Thus, it gives excellent results for cases in which real noise with small but finite correlation time is approximated as white. This restriction on the type of problems to which the stochastic RK4 can be applied is strongly compensated by its computational efficiency.