Equilibrium preserving space in discontinuous Galerkin methods for hyperbolic balance laws

TL;DR

The paper addresses the challenge of exactly preserving equilibrium states in high-order simulations of hyperbolic balance laws, such as the Euler equations with gravity and the Ripa model. It introduces a general well-balanced DG framework that discretizes equilibrium variables in a piecewise polynomial space and uses hydrostatic reconstruction to modify numerical fluxes, avoiding reference-state recovery. The approach is proven to preserve moving equilibria in 1D and hydrostatic states in 2D, with a specialized flux treatment for isobaric equilibria in the Ripa model, and is validated through extensive 1D and 2D tests showing third-order accuracy and robust perturbation resolution. The framework offers a flexible, high-order, oscillation-free tool for near-equilibrium flows on relatively coarse meshes, with potential extensions to broader equations of state and coordinate systems.

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.

Figures (26)

• Figure 4.1: Example \ref{['euler:wb_adi']}: From left to right: the initial density $\rho$, velocity $u$ and pressure $p$ computed by (\ref{['euler:adi_ini']}) for three values of $M$.
• Figure 4.2: Example \ref{['euler:per_adi']}: Pressure perturbation and velocity of the hydrostatic atmosphere ($M=0$) for the small amplitude wave propagation $A=10^{-6}$ in (\ref{['euler:vp']}), using the well-balanced scheme with 100 and 2000 cells, the non-well-balanced scheme with 100 cells for comparison.
• Figure 4.3: Example \ref{['euler:per_adi']}: Pressure perturbation and velocity of the hydrostatic atmosphere ($M=0$) for the big amplitude wave propagation $A=0.1$ in (\ref{['euler:vp']}), using the well-balanced scheme with 100 and 2000 cells, the non-well-balanced scheme with 100 cells for comparison.
• Figure 4.4: Example \ref{['euler:per_adi']}: Pressure perturbation and velocity perturbation of the hydrostatic atmosphere ($M=0$) for the small amplitude wave propagation $A=10^{-6}$ in (\ref{['euler:adi_peradd']}).
• Figure 4.5: Example \ref{['euler:per_adi']}: Pressure perturbation and velocity perturbation of the subsonic atmosphere ($M=0.01$) for the small amplitude wave propagation $A=10^{-6}$ in (\ref{['euler:adi_peradd']}).

Theorems & Definitions (26)

• Remark 3.1
• Remark 3.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• Theorem 3.4
• Example 4.1
• Example 4.2