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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 aP  N{P_{N}}    orS  N{S_{N}}    model, and we study in more details theP  1{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 forP  1{P_{1}}    model in dimension one. Numerical results in dimensions 1 and 2 illustrate the theoretical properties.

Trefftz Discontinuous Galerkin Method for Friedrichs Systems with Linear Relaxation: Application to the P 1 Model