A second-order, discretely well-balanced finite volume scheme for Euler equations with gravity

TL;DR

This paper introduces a second-order finite volume scheme for Euler equations with gravity that preserves hydrostatic solutions without prior knowledge, improving accuracy in complex and unknown hydrostatic states.

Contribution

The scheme is the first to be discretely well-balanced at second order without requiring a priori hydrostatic solutions, applicable to complex equations of state.

Findings

Accurately preserves hydrostatic solutions in tests

Effectively computes small perturbations around hydrostatic states

Demonstrates robustness on various test cases

Abstract

We present a well-balanced, second order, Godunov-type finite volume scheme for compressible Euler equations with gravity. By construction, the scheme admits a discrete stationary solution which is a second order accurate approximation to the exact stationary solution. Such a scheme is useful for problems involving complex equations of state and/or hydrostatic solutions which are not known in closed form expression. No \'a priori knowledge of the hydrostatic solution is required to achieve the well-balanced property. The performance of the scheme is demonstrated on several test cases in terms of preservation of hydrostatic solution and computation of small perturbations around a hydrostatic solution.