High-order well-balanced finite volume schemes for the Euler equations with gravitation

TL;DR

This paper introduces a high-order well-balanced finite volume scheme for the Euler equations with gravitation, capable of accurately preserving hydrostatic equilibria and applicable to various equations of state.

Contribution

It presents a novel local hydrostatic reconstruction method that achieves high-order accuracy and easy integration into existing codes for Euler equations with gravity.

Findings

The scheme preserves hydrostatic equilibrium accurately.

It achieves high-order accuracy for smooth solutions.

It is simple to implement and versatile for different equations of state.

Abstract

A high-order well-balanced scheme for the Euler equations with gravitation is presented. The scheme is able to preserve a spatially high-order accurate discrete representation of a large class of hydrostatic equilibria. It is based on a novel local hydrostatic reconstruction, which, in combination with any standard high-order accurate reconstruction procedure, achieves genuine high-order accuracy for smooth solutions close or away from equilibrium. The resulting scheme is very simple and can be implemented into any existing finite volume code with minimal effort. Moreover, the scheme is not tied to any particular form of the equation of state, which is crucial for example in astrophysical applications. Several numerical experiments demonstrate the robustness and high-order accuracy of the scheme nearby and out of hydrostatic equilibrium.