Equilibrium Preserving Space in Discontinuous Galerkin Methods for Hyperbolic Balance Laws (Communications in Computational Physics, 2025)
In this paper, we develop a general framework for the design of the arbitrary high-order well-balanced discontinuous Galerkin (DG) method for hyperbolicbalance laws, including the compressible Euler equations with gravitation and theshallow water equations with horizontal temperature gradients (referred to as the Ripamodel). Not only does the hydrostatic equilibrium include the more complicated isobaric steady state in the Ripa system, but our scheme is also well-balanced for theexact preservation of the moving equilibrium state. The strategy adopted is to approximate the equilibrium variables in the DG piecewise polynomial space, rather thanthe conservative variables, which is pivotal in the well-balanced property. Our approach provides flexibility in combination with any consistent numerical flux, and it isfree of the reference equilibrium state recovery and the special source term treatment.This approach enables the construction of a well-balanced method for non-hydrostaticequilibria in Euler equations. Extensive numerical examples such as moving or isobaric equilibria validate the high order accuracy and exact equilibrium preservationfor various flows given by hyperbolic balance laws. With a relatively coarse mesh, it isalso possible to capture small perturbations at or close to steady flow without numerical oscillations.
Keywords: Euler equations with gravitation; Ripa model; discontinuous Galerkin method; equilibrium preserving space; well-balanced.