Trefftz discontinuous Galerkin basis functions for a class of Friedrichs systems coming from linear transport

Abstract

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.

A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients

“…We envisage that the quasi-Trefftz method proposed here can be extended to elastic and electromagnetic wave propagation in heterogeneous materials, and more generally to a wider class of hyperbolic or Friedrichs systems. (For constant-coefficient examples of space-time Trefftz DG schemes for elastodynamics, electromagnetics and kinetic equations/transport models see [4], [7,17] and [5], respectively.) The proposed approach might be effective also for the approximation of PDEs whose nature changes in the computational domain, exemplified by the Euler-Tricomi equation (∂ 2…”

“…with γ as defined in (5). The dimensionless coefficients ξ • K measure the impact of the choices made in a concrete implementation of the DG scheme -in terms of the numerical flux parameters and the element shapes -on the convergence bounds of Theorem 4.5.…”

An asymptotic preserving method for the linear transport equation on general meshes

“…Early works in this field are based on a Discontinuous Finite Element (DFE) discretization [2,3,5,6,25,65], which has been proven to capture the diffusive regime by construction. Another method [18] uses a Treffz Discontinuous Galerkin basis. Some recent works are based on Finite Volume schemes with an approximate Riemann solver.…”

Embedded Trefftz discontinuous Galerkin methods

“…Trefftz-DG schemes for Laplace equations have been analyzed in [20,30,31]. Different time-dependent problems have been discussed recently, such as wave problems in one space dimension [28,29,44], the acoustic wave equation [3,41,43], elastoacoustics [5], a class of Friedrichs systems coming from linear transport [6,42], for time-dependent Maxwell's equation see [12,27], and the linear Schrödinger equation [16]. Very popular applications of Trefftz methods are wave propagation problems in frequency domain.…”