Uniform Convergence of Interlaced Euler Method for Stiff Stochastic Differential Equations

In contrast to stiff deterministic systems of ordinary differential equations, in general, the implicit Euler method for stiff stochastic differential equations is not effective. This paper introduces a new numerical method for stiff differential equations which consists of interlacing large implicit Euler time steps with a sequence of small explicit Euler time steps. We emphasize that uniform convergence with respect to the time scale separation parameter ε is a desirable property of a stiff solver. We prove that the means and variances of this interlaced method converge uniformly in ε for a suitably chosen test problem. We also illustrate the effectiveness of this method via some numerical examples.