Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes

discretized with first order finite volume schemes. This problem is representative of many transport problems, such as transfer and neutron transport, for which the small parameter ε is the ratio of two very different sound velocities and σ is the absorption or the opacity. For simplicity both ε and σ > 0 are kept constant in space in this study. The system (1.1) can also be introduced as a specific linearization of a pressure-velocity system of partial differential equations in the acoustic regime. In this work we will need the following well known energy estimates concerning the solution V of the Cauchy problem for the partial differential equation (1.1). Proposition 1.1. If Ω = R or Ω = T, then