Technical Report 13, c4e-Preprint Series, Cambridge

Stochastic Solution of Ordinary Differential Equations

ref: Technical Report 13, c4e-Preprint Series, Cambridge

Associated Theme: Numerics

Abstract

In this paper 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 homogeneous gas-phase reaction mechanisms describing the combustion of hydrogen, carbon monoxide, methane, n-heptane, iso-octane and n-decane. Fuels like mixtures of n-heptane and iso-octane can contain up to a thousand species and several thousand reactions and are of practical relevance to industrial combustion processes. One deterministic method yielded by our correspondence principle has been proposed and studied in a previous paper and is used here in order to shed light on various aspects of the considered stochastic algorithm. In addition, we use the widespread state-of-the-art stiff ODE-solver package DASSL for comparison. Advantages of our methods include among others their exceptional simplicity of implementation, negligible start-up costs and, as shown by numerical experiments, a linear scaling behaviour of the computational time with the number of equations. It is also shown that for large systems, the proposed algorithms exhibit computational efficiency similar to conventional implicit solvers, assuming multiple runs in the stochastic case. In view of the stiffness of the considered systems and the explicit nature of our algorithms, this is rather surprising. These properties suggest as typical application large operator-splitting problems requiring moderate accuracy, such as PDF transport models.

Material from this preprint has been published in: Monte Carlo Methods and Applications 12 (1), 19-45, (2006)

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