Stochastic Simulation Method for Linearly Implicit Ordinary Differential Equations

Numerical schemes based on the simulation of suitable Markov jump processes such as the stochastic direct simulation method and its improved variants have shown to be a good alternative to deterministic solvers when applied to semi-discrete approximations of time-dependent partial differential equations. Moreover, in contrast to deterministic explicit solvers, this class of methods turns out to be stable also on nonuniform grids, a feature which was demonstrated by applications to moving cell methods in one space dimension. In this paper we present a modified scheme based on the same basic principle, suited for approximating linearly implicit ordinary differential equations of the form Au' = F(u). They can arise for example in the context of finite-element discretizations of the corresponding partial differential equations. The results of the numerical experiments show that methods based on the principle of stochastic simulation are able to handle also this type of problems and can motivate further research in this direction, especially for more complex, higher-dimensional problems with relevant applications.