A General Well-Balanced Finite Volume Scheme for Euler Equations with Gravity

Abstract

We present a second-order well-balanced Godunov-type finite volume scheme for compressible Euler equations with a gravitational source term. The scheme is designed to work for any hydrostatic equilibrium, which must be known a priori. It can be combined with any numerical flux function, time-stepping method, and grid topology. The scheme is based on the reconstruction of a special set of variables and a special source term discretization. We show the well-balanced property numerically for isothermal and polytropic equilibria in one and two dimensions using the Roe flux function and an explicit three-stage Runge–Kutta scheme. We demonstrate the superior resolution of small pressure perturbations of hydrostatic equilibria, down to an order \(10^{-10}\) and below compared to the hydrostatic background.