A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows

Abstract

Abstract. In this paper we present a kinetic relaxation scheme for the Euler equations of gas dynamics in one space dimension. The method is easily applicable to solve any complex system of conservation laws. The numerical scheme is based on a relaxation approximation for conservation laws viewed as a discrete velocity model of the Boltzmann equation of kinetic theory. The discrete kinetic equation is solved by a splitting method consisting of a convection phase and a collision phase. The convection phase involves only the solution of linear transport equations and the collision phase instantaneously relaxes the distribution function to an equilibrium distribution. We prove that the first order accurate method is conservative, preserves the positivity of mass density and pressure and entropy stable. An anti-diffusive Chapman-Enskog distribution is used to derive a second order accurate method. The results of numerical experiments on some benchmark problems confirm the efficiency and robustness of the proposed scheme.

A new flux‐limiting approach–based kinetic scheme for the Euler equations of gas dynamics

“…An exact periodic solution for 1D Euler equations is used as given in the work of Arun et al A computational domain [ −1,1] is taken and the computation stops at t = 0.5. The grid is refined successively and error norm for density is calculated.…”

A Boltzmann scheme with physically relevant discrete velocities for Euler equations

“…1. Consistency of the discrete kinetic approximation (15) with the system of Euler equations ( 9) in the limit → 0, leading to the moment relations:…”

“…Aregba-Driollet and Natalini [11] suggest that in the general case, we can check for the positive-definiteness of (Γ + Γ T ) which is a symmetric matrix. However, this criterion does not give an explicit condition for the stability of approximation (15). We therefore use another simpler but stronger stability condition for the approximation given by Bouchut [18].…”

“…where Ω denotes the eigenspectrum. Let us now apply the stability condition (78) for approximation (15). Before evaluating the Jacobian ∂f eq ∂U , we first note the relation between f eq in ( 78) and the equilibrium distribution function vectors f eq l given in (76):…”

“…The above schemes utilize the continuous molecular velocity for introducing upwinding, together with either Maxwellian distribution functions, Dirac delta functions or compactly supported distributions. Discrete velocity Boltzmann schemes were introduced by Natalini [10], Aregba-Driollet & Natalini [11], Raghurama Rao & Balakrishna [12], Raghurama Rao & Subba Rao [13], Arun et al [14,15,16], among others. The discrete velocity Boltzmann schemes present certain advantages compared to the continuous molecular velocity based upwind schemes by being simpler in design and analysis of numerical schemes.…”