A Practical Splitting Method for Stiff SDEs with Applications to Problems with Small Noise

Abstract

We present an easy to implement drift splitting numerical method for the approximation of stiff, nonlinear stochastic differential equations (SDEs). The method is an adaptation of the semi‐implicit backward differential formula (SBDF) multistep method for deterministic differential equations and allows for a semi‐implicit discretization of the drift term to remove high order stability constraints associated with explicit methods. For problems with small noise, of amplitude e, we prove that the method converges strongly with order $O(\Delta t^2 + \epsilon \Delta t + \epsilon^2 \Delta t^{1/2})$ and thus exhibits second order accuracy when the time step is chosen to be on the order of e or larger. We document the performance of the scheme with numerical examples and also present as an application a discretization of the stochastic Cahn–Hilliard equation which removes the high order stability constraints for explicit methods.