Analysis of upwind method for piecewise deterministic Markov processes

In this paper we investigate the behaviour of the numerical solution of the Liouville---Master Equation (LME) for time dependent distributions, that arises by the statistical characterisation of a class of piecewise deterministic stochastic processes (PDP). This is a system of linear hyperbolic PDEs with non-constant coefficients. The numerical solution is found by means of upwind and forward Euler scheme. We find a Courant---Friedrichs---Lewy condition ensuring both convergence and monotonicity of the numerical solution. In particular, the global error is shown to be bounded by a linear increasing in the integration time, under an appropriate norm. Some numerical tests for known analytical solutions of practical problems verify the theoretical findings