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

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.

Kinetic Over-Relaxation Method for the Convection Equation with Fourier Solver

“…Then ρ f f f is a first-order approximation of the solution ρ to (1) for ω < 2 and a second-order approximation if ω = 2. We refer the reader to [1,2] as regards the corresponding equivalent equation. From this equivalent equation, we can infer the so-called sub-characteristic condition that ensures the dissipativity of the secondorder term in the expansion.…”

“…The kinetic over-relaxation method [1] is a time semi-discrete method based on the approximation of a non-linear convection equation by a set of linear transport equations with constant velocities. Very efficient, CFL-less, and accurate transport solvers like Fourier methods can be used.…”

Simulation of a Liquid-Vapour Compressible Flow by a Lattice Boltzmann Method

“…It can be proved that the resulting scheme is a second order O(∆t 2 ) approximation of (1). See [3,5], for instance, and included references.…”

“…The general principles are mainly given in [8,1]. We consider a mixture of three phases (1), (2) and (3) representing the vapour, the liquid and the non-condensable gas (air), respectively. The liquid is not miscible with the two others, while the vapour and the gas are miscible.…”

Peer Review #2 of "Improving parallel executions by increasing task granularity in task-based runtime systems using acyclic DAG clustering (v0.1)"

“…• We evaluate and compare DGCA and our approach on Chukrut (Conservative Hyperbolic Upwind Kinetic Resolution of Unstructured Tokamaks) [21] that computes the transport equation on a 2D…”