Well-Balanced Schemes for the Euler Equations with Gravitation: Conservative Formulation Using Global Fluxes

TL;DR

This paper introduces a second-order conservative scheme for Euler equations with gravity, capable of exactly preserving steady states by using global fluxes and a well-balanced reconstruction, improving accuracy near equilibrium.

Contribution

A novel conservative reformulation with global fluxes and a well-balanced scheme for Euler equations with gravity, enhancing steady-state preservation and reducing numerical viscosity.

Findings

Exact steady-state preservation demonstrated in tests

Improved accuracy near equilibrium states

Effective reduction of numerical viscosity in steady flows

Abstract

We develop a second-order well-balanced central-upwind scheme for the compressible Euler equations with gravitational source term. Here, we advocate a new paradigm based on a purely conservative reformulation of the equations using global fluxes. The proposed scheme is capable of exactly preserving steady-state solutions expressed in terms of a nonlocal equilibrium variable. A crucial step in the construction of the second-order scheme is a well-balanced piecewise linear reconstruction of equilibrium variables combined with a well-balanced central-upwind evolution in time, which is adapted to reduce the amount of numerical viscosity when the flow is at (near) steady-state regime. We show the performance of our newly developed central-upwind scheme and demonstrate importance of perfect balance between the fluxes and gravitational forces in a series of one- and two-dimensional examples.