Discretization of linear transport models using Trefftz discontinuous Galerkin method

This work deals with the study and analysis of a Trefftz Discontinuous Galerkin (TDG) scheme for a model problem of transport with linear relaxation. We show that natural well-balanced and asymptotic-preserving discretization are provided by the TDG method since exact solutions, possibly non-polynomials, are used locally in the basis functions. A special attention is devoted to the PN approximation of the transport equation. For this two dimensional model, polynomial and exponential basis functions are constructed and the convergence of the scheme is studied. Numerical examples on the P1 and P3 models show that the TDG method outperforms the standard discontinuous Galerkin method when considering stiff coefficients. In particular, the TDG method leads to efficient schemes to capture boundary layers and the diffusion limit of the transport equation.‌