Equilibrium preserving space in discontinuous Galerkin methods for hyperbolic balance laws

TL;DR

This paper introduces a high-order discontinuous Galerkin method that preserves equilibrium states exactly for hyperbolic balance laws, including complex steady states, without requiring reference state recovery or special source term treatments.

Contribution

The paper presents a novel framework for well-balanced DG methods that approximate equilibrium variables in polynomial space, enabling exact steady state preservation for complex hyperbolic systems.

Findings

Accurately preserves hydrostatic and moving equilibria.

Achieves high-order accuracy with coarse meshes.

Effectively captures small perturbations near steady states.

Abstract

In this paper, we develop a general framework for the design of the arbitrary high-order well-balanced discontinuous Galerkin (DG) method for hyperbolic balance laws, including the compressible Euler equations with gravitation and the shallow water equations with horizontal temperature gradients (referred to as the Ripa model). Not only the hydrostatic equilibrium including the more complicated isobaric steady state in Ripa system, but our scheme is also well-balanced for the exact preservation of the moving equilibrium state. The strategy adopted is to approximate the equilibrium variables in the DG piecewise polynomial space, rather than the conservative variables, which is pivotal in the well-balanced property. Our approach provides flexibility in combination with any consistent numerical flux, and it is free of the reference equilibrium state recovery and the special source term treatment. This approach enables the construction of a well-balanced method for non-hydrostatic equilibria in Euler systems. Extensive numerical examples such as moving or isobaric equilibria validate the high order accuracy and exact equilibrium preservation for various flows given by hyperbolic balance laws. With a relatively coarse mesh, it is also possible to capture small perturbations at or close to steady flow without numerical oscillations.