A Finite Difference Method for Piecewise Deterministic Processes With Memory. Ii

Abstract

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.

On the Action of a Semi-Markov Process on a System of Differential Equations

“…From the mathematical point of view PDE's with non-local boundary conditions are a less investigated subject, but interest is growing [21]. A numerical scheme, based on Courant-Isaacson-Rees jointly to a direct quadrature, for the solution of the LME was investigated in [2,3], where convergence, positivity and conser-Downloaded by [University of Connecticut] at 16:26 12 October 2014 vativity was proved under a Courant-Friederichs-Lèvy-like conditions. Finite volume schemes for the approximation of probability measures for a system of hyperbolic PDEs arising in dynamics reliability of a system have been studied in [11,16].…”