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.
The Trefftz discontinuous Galerkin (TDG) method provides natural well-balanced (WB) and asymptotic preserving (AP) discretization since exact solutions are used locally in the basis functions. However, one difficult point may be the construction of such solutions which is a necessary first step in order to apply the TDG scheme. This works deals with the construction of solutions to Friedrichs systems with relaxation with application to the spherical harmonics approximation of the transport equation (the so-called PN models). Various exponential and polynomial solutions are constructed. Two numerical tests on the P3 model illustrate the good accuracy of the TDG method. They show that the exponential solutions lead to accurate schemes to capture boundary layers on a coarse mesh and that a combination of exponential and polynomial solutions is efficient in a regime with vanishing absorption coefficient.
International audienceA chain probe graph is a graph that admits an independent set $S$ of vertices and a set $F$ of pairs of elements of $S$ such that $G+F$ is a chain graph (i.e., a $2K_2$-free bipartite graph). We show that chain probe graphs are exactly the bipartite graphs that do not contain as an induced subgraph a member of a family of six forbidden subgraphs, and deduce an $O(n^2)$ recognition algorithm
We present a multi-dimensional asymptotic preserving scheme for the approximation of a mixture of compressible flows. Fluids are modeled by two Euler systems of equations coupled with a friction term. The asymptotic preserving property is mandatory for this kind of model, to derive a scheme that behaves well in all regimes (i.e. whatever the friction parameter value is). The method we propose is defined in ALE coordinates, using a Lagrange plus remap approach. This imposes a multi-dimensional definition and analysis the scheme.
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