Trefftz Discontinuous Galerkin Method for Friedrichs Systems with Linear Relaxation: Application to the P1 Model

This work deals with the first Trefftz Discontinuous Galerkin (TDG) scheme for a model problem of transport with relaxation. The model problem is written as a {P_{N}} or {S_{N}} model, and we study in more details the {P_{1}} model in dimension 1 and 2. We show that the TDG method provides natural well-balanced and asymptotic preserving discretization since exact solutions are used locally in the basis functions. High-order convergence with respect to the mesh size in two dimensions is proved together with the asymptotic property for {P_{1}} model in dimension one. Numerical results in dimensions 1 and 2 illustrate the theoretical properties.