High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation

TL;DR

This paper introduces a high-order, explicit-like, CFL-less discontinuous Galerkin method for hyperbolic conservation laws, utilizing kinetic representations and composition methods for efficient, accurate simulations in multiple dimensions.

Contribution

It presents a novel high-order implicit DG scheme that is CFL-less, matrix-free, and applicable to complex hyperbolic systems with arbitrary order in space and time.

Findings

Validated on 1D test cases

Successfully applied to 2D and 3D problems

Achieved high accuracy with CFL-less implicit scheme

Abstract

We construct a high order discontinuous Galerkin method for solving general hyperbolic systems of conservation laws. The method is CFL-less, matrix-free, has the complexity of an explicit scheme and can be of arbitrary order in space and time. The construction is based on: (a) the representation of the system of conservation laws by a kinetic vectorial representation with a stiff relaxation term; (b) a matrix-free, CFL-less implicit discontinuous Galerkin transport solver; and (c) a stiffly accurate composition method for time integration. The method is validated on several one-dimensional test cases. It is then applied on two-dimensional and three-dimensional test cases: flow past a cylinder, magnetohydrodynamics and multifluid sedimentation.