A finite difference method for piecewise deterministic processes with memory. II

Abstract

We deal with the numerical scheme for the Liouville Master Equation (LME) of a kind of Piecewise Deterministic Processes (PDP) with memory, analysed in [2]. The LME is a linear system of hyperbolic PDEs, written in non‐conservative form, with non‐local boundary conditions. The solutions of that equation are time dependent marginal distribution functions whose sum satisfies the total probability conservation law. In [2] the convergence of the numerical scheme, based on the Courant‐Isaacson‐Rees jointly with a direct quadrature, has been proved under a Courant‐Friedrichs‐Lewy like (CFL) condition. Here we show that the numerical solution is monotonic under a similar CFL condition. Moreover, we evaluate the conservativity of the total probability for the calculated solution. Finally, an implementation of a parallel algorithm by using the MPI library is described and the results of some performance tests are presented.

First published online: 14 Oct 2010