  In this paper the numerical approximation of solutions of Liouville-Master
Equations for time-dependent distribution functions of Piecewise Deterministic
Processes with memory is considered. These equations are linear hyperbolic PDEs
with non-constant coefficients, and boundary conditions that depend on
integrals over the interior of the integration domain. We construct a finite
difference method of the first order, by a combination of the upwind method,
for PDEs, and by a direct quadrature, for the boundary condition. We analyse
convergence of the numerical solution for distribution functions evolving
towards an equilibrium. Numerical results for two problems, whose analytical
solutions are known in closed form, illustrate the theoretical finding.
