Open AccessA FINITE DIFFERENCE METHOD FOR PIECEWISE DETERMINISTIC PROCESSES WITH MEMORY. II
Author(s) -
Mario Annunziato
Publication year - 2009
Publication title -
mathematical modelling and analysis/mathematical modeling and analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.491
H-Index - 25
eISSN - 1648-3510
pISSN - 1392-6292
DOI - 10.3846/1392-6292.2009.14.139-158
Subject(s) - mathematics , piecewise , monotonic function , conservation law , quadrature (astronomy) , courant–friedrichs–lewy condition , convergence (economics) , mathematical analysis , discretization , electrical engineering , engineering , economics , economic growth
We deal with the numerical scheme for the Liouville Master Equation (LME) of a kind \udof Piecewise Deterministic Processes (PDP) with memory, analysed in [2]. The LME is \uda linear system of hyperbolic PDEs, written in non?conservative form, with non-local\ud boundary conditions. The solutions of that equation are time dependent marginal \uddistribution functions whose sum satisfies the total probability conservation law. \udIn [2] the convergence of the numerical scheme, based on the Courant-Isaacson-Rees \udjointly with a direct quadrature, has been proved under a Courant-Friedrichs-Lewy \udlike (CFL) condition. Here we show that the numerical solution is monotonic under \uda similar CFL condition. Moreover, we evaluate the conservativity of the total \udprobability for the calculated solution. Finally, an implementation of a parallel \udalgorithm by using the MPI library is described and the results of some performance \udtests are presented