Explicit stochastic ODE solution methods applied to high-temperature combustion

Abstract

New stochastic algorithms for the numerical solution of systems of ordinary differential equations (ODEs) are proposed. Furthermore, a correspondence principle is established between these algorithms, which are based on the theory of Markov jump processes, and deterministic schemes. For one of the proposed stochastic algorithms, a detailed numerical study of some of its properties is carried out using examples from high-temperature homogeneous gasphase combustion. One deterministic method yielded by our correspondence principle is used to shed light on various aspects of the considered stochastic algorithm. In addition, we use the widespread stiff ODE-solver package DASSL for comparison. Advantages of our methods include among others their exceptional simplicity of implementation and negligible start-up costs. For a large system at moderate accuracy requirements, the proposed stochastic algorithms exhibit computational efficiency in the same order of magnitude as implicit solvers, assuming multiple runs. In view of the stiffness of the considered systems and the explicit nature of our algorithms, this is rather surprising. Limitations of our methods concerning the choice of system, initial conditions, and accuracy requirements are also addressed.