Asymptotic-preserving and well-balanced schemes for transport models using Trefftz discontinuous Galerkin method

Some solutions to the transport equation admit a diffusion limit and boundary layers which may be very costly to approximate with naive numerical methods. To address these issues, a possible approach is to consider well-balanced (WB) and asymptotic-preserving (AP) schemes. Such schemes are known, in some cases, to greatly improve the numerical solution on coarse meshes. This thesis 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. In particular, the formulation of the TDG method for the general case of Friedrichs systems is given. For the practical examples, 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. (author)