Palindromic Discontinuous Galerkin Method

Abstract

We present a high-order scheme for approximating kinetic equations with stiff relaxation. The construction is based on a high-order, implicit, upwind Discontinuous Galerkin formulation of the transport equations. In practice, because of the triangular structure of the implicit system, the computations are explicit. High order in time is achieved thanks to a palindromic composition method. The whole method is asymptotic-preserving with respect to the stiff relaxation and remains stable even with large CFL numbers.

A robust and efficient solver based on kinetic schemes for Magnetohydrodynamics (MHD) equations

“…From (10) we formally observe that the scheme is second-order in time in the over-relaxation case, when…”

“…It is valid for any hyperbolic system of conservation laws and is no more limited to low-Mach number flows. This approach is very fruitful and can be used on arbitrary unstructured meshes at any order of approximation [2,10]. In addition, when the lattice velocities are aligned with the mesh, it is possible to adopt a very simple exact solver of the transport step.…”

“…It is possible to add a small dissipation τ > 0. See [22] or [10] for the relaxation scheme in the case τ > 0. This dissipation provides a better stability in numerical cases, when sharp fronts are generated.…”

Vectorial Kinetic Relaxation Model with Central Velocity. Application to Implicit Relaxations Schemes

“…To solve the kinetic representation (2) and capture the limit system as ε → 0, we consider an implicit scheme similar to the one introduced in [8,9].…”

“…Since the first transport step of one time stage can be merged with the last transport step of the previous time stage, the second order splitting scheme is equivalent to the first order Crank-Nicolson scheme with a first and last half step of transport. Splitting schemes with higher order accuracy in time have been proposed in [8,9]. Remark 2.2.…”