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.

翻译：本文提出了一个通用框架，用于设计任意高阶的、能良好平衡双曲平衡律的间断Galerkin（DG）方法，包括带引力的可压缩欧拉方程和水平温度梯度的浅水方程（称为Ripa模型）。我们的方案不仅适用于流体静力平衡（包括Ripa系统中更复杂的等压稳态），还能精确保持移动平衡状态。所采用的策略是在DG分段多项式空间中近似平衡变量，而非守恒变量，这对保持平衡特性至关重要。该方法可灵活结合任意相容数值通量，且无需恢复参考平衡状态或特殊源项处理。该途径能够为欧拉系统中的非流体静力平衡构造平衡方法。大量数值算例（如移动平衡或等压平衡）验证了该方法对双曲平衡律所描述的各种流动具有高阶精度和精确的平衡保持能力。即使在较粗网格上，也能在稳态流动或其附近捕获小扰动而不产生数值振荡。