Finite Volume Scheme with Local High Order Discretization of the Hydrostatic Equilibrium for the Euler Equations with External Forces

Abstract

In this note, we introduce a new finite volume scheme for Euler equations with source terms: the gravity and the friction. The classical finite volume schemes are not able to capture correctly the dynamic induced by the balance between convective terms and external forces. Firstly, by plugging the source terms in the fluxes with the Jin-Levermore procedure, we modify the Lagrangian+remap scheme to obtain a method able to capture the asymptotic limit induced by the friction (asymptotic preserving scheme) and discretize with good accuracy the steady state linked to the gravity (well-balanced scheme). Secondly we will give some properties about this scheme and introduce a modification which allows us to obtain an arbitrary high order discretization of the hydrostatic steady state.

Energy conserving and well-balanced discontinuous Galerkin methods for the Euler-Poisson equations in spherical symmetry

“…Some of these works assume that the desired equilibrium is explicitly known Klingenberg et al (2019); Wu & Xing (2021), while others only need a pre-description of the desired equilibrium Li & Xing (2018), and work for a class of equilibria. Recently, several works are established without any information of the desired equilibrium state Käppeli & Mishra (2016); Franck & Mendoza (2016); Berberich et al (2021). For the Euler-Poisson equations considered in this paper, the equilibrium states are more complicated due to the coupling with the Poisson equation.…”

Well-balanced methods for computational astrophysics

“… 2015 ; Ghosh and Constantinescu 2015 ; Li and Xing 2016b ; Käppeli and Mishra 2016 ; Li and Xing 2016a ; Touma et al. 2016 ; Ghosh and Constantinescu 2016 ; Franck and Mendoza 2016 ; Bispen et al. 2017 ; Käppeli 2017 ; Chandrashekar and Zenk 2017 ; Berberich et al.…”

An asymptotic preserving multidimensional ALE method for a system of two compressible flows coupled with friction

“…Even for simpler model, only few unstructured asymptotic preserving schemes have been developped (refer for instance to Berthon and Turpault [16] and Franck et al [17,18]). The scheme we propose in section 4 has connections with [19,20], where an Euler with friction system is studied. However, it is not a direct extension of [19] to the bifluid case.…”