High-order well-balanced methods for the Euler equations of gas dynamics with gravitational forces and the Ripa model

Abstract

Different well-balanced high-order finite-volume numerical methods for the one-dimensional compressible Euler equations of gas dynamics with gravitational force and for the Ripa model have been proposed in the literature. Most of them preserve either a given family of hydrostatic stationary solutions exactly or all of them approximately. The goal of this paper is to design a general methodology to obtain high-order finite-volume numerical methods for a class of one-dimensional hyperbolic systems of balance laws that preserve approximately all the hydrostatic equilibria and exactly a given family of them. Many fluid models for which the velocity is an eigenvalue of the system belong to this class, the Euler equations and the Ripa model among them. The methods proposed here are based on the design of well-balanced reconstruction operators that require the exact or the approximate computation of local hydrostatic equilibria. To check the efficiency and the well-balancedness of the methods, a number of numerical tests have been performed: the numerical results confirm the theoretical ones.