Second order finite volume scheme for Euler equations with gravity which is well-balanced for general equations of state and grid systems

TL;DR

This paper introduces a second order finite volume scheme for Euler equations with gravity that maintains hydrostatic equilibrium for general equations of state and flexible grid systems, enhancing accuracy and applicability.

Contribution

The scheme is the first to be well-balanced for arbitrary hydrostatic solutions without restrictions on the equation of state, applicable on curvilinear meshes.

Findings

Successfully preserves hydrostatic solutions in numerical tests

Compatible with various numerical flux functions and time stepping routines

Demonstrates robustness on complex grid geometries

Abstract

We develop a second order well-balanced finite volume scheme for compressible Euler equations with a gravitational source term. The well-balanced property holds for arbitrary hydrostatic solutions of the corresponding Euler equations without any restriction on the equation of state. The hydrostatic solution must be known a priori either as an analytical formula or as a discrete solution at the grid points. The scheme can be applied on curvilinear meshes and in combination with any consistent numerical flux function and time stepping routines. These properties are demonstrated on a range of numerical tests.