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

A new finite volume scheme for the Euler equations with gravity and friction source terms is presented. Classical finite volume schemes are not able to capture correctly the dynamics generated by the balance between convective terms and external forces. Our purpose is to develop a method better suited for dealing with this problem. To that end, firstly, we modify the Lagrangian+remap scheme by plugging the source terms into the fluxes using the Jin–Levermore procedure. The scheme obtained is able to capture the asymptotic limit induced by the friction (Asymptotic Preserving scheme) and to discretize with a good accuracy the steady-state linked to gravity (Well-Balanced scheme). Secondly, we present some properties about this scheme and introduce a modification for an arbitrary high order discretization of the hydrostatic steady-state.