Efficient Time Integration Methods for Stiff Stochastic Dynamical Systems

Dynamic systems that consist of multiple time scales where the faster time scales are stable are said to be stiff. Stiff systems pose particular challenges when it comes to numerical methods of simulation that advance the system via finite time steps. In deterministic systems modeled by ordinary differential equations (ODEs), stiff systems are well understood and efficient numerical time integration methods of such systems are available. In general implicit methods are better suited for stiff systems while explicit methods are not. The main reason for this is that due to their better stability property the implicit methods are able to take large time steps of the order of the slowest time scales without compromising accuracy. Stiffness manifests in stochastic dynamical systems modeled by stochastic differential equations (SDEs) in a far more complex manner. Even implicit methods while stable are unable to capture the behavior accurately by taking time steps only on the order of the slowest time scale. This thesis introduces a new numerical method for stiff SDEs, called the Interlaced Euler Method, which consists of interlacing large implicit Euler time steps with a sequence of small explicit Euler time steps. It is shown that the asymptotic moment analysis applied to a suitable test problem provides the appropriate number of small explicit time steps to be used. It is emphasized that uniform convergence with respect to the time scale separation parameter is a desirable property of a stiff solver and it is proven that the mean and variance of the interlaced Euler method converge uniformly in the time scale separation parameter for a suitably chosen test problem. The effectiveness of this method is also illustrated via some numerical examples