We present a novel and simple yet intuitive approach to the stabilization problem for the numerically solved Euler equations with gravity source term relying on a low-order nodal Discontinuous Galerkin Method (DGM). Instead of assuming isothermal or polytropic solutions, we only take a hydrostatic balance as a given property of the flow and use the hydrostatic equation to calculate a hydrostatic pressure reconstruction that replaces the gravity source term. We compare two environments that both solve the Euler equations using the DGM: deal.II and StormFlash. We utilize StormFlash as it allows for the use of the novel stabilization method. Without stabilization, StormFlash does not yield results that resemble correct physical behavior while the results with stabilization for StormFlash, as well as deal.II model the fluid flow more accurately. Convergence rates for deal.II do not match the expected order while the convergence rates for StormFlash with the stabilization scheme (with the exceptions for the L errors for momentum) meet the expectation. The results from StormFlash with stabilization also fit reference solutions from the literature much better than those from deal.II. We conclude that this novel scheme is a low cost approach to stabilize the Euler equations while not limiting the flow in any way other than it being in hydrostatic balance.

中文翻译：

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使用不连续伽辽金方法的欧拉方程的线性低努力稳定方法

我们提出了一种新颖、简单而直观的方法来解决依赖于低阶节点间断伽辽金法 (DGM) 的带有重力源项的数值求解欧拉方程的稳定性问题。我们不假设等温或多变解，而是仅将静水平衡作为流动的给定属性，并使用静水方程来计算代替重力源项的静水压力重建。我们比较了两个使用 DGM 求解欧拉方程的环境：deal.II 和 StormFlash。我们利用 StormFlash，因为它允许使用新颖的稳定方法。如果没有稳定性，StormFlash 不会产生类似于正确物理行为的结果，而 StormFlash 具有稳定性的结果以及 deal.II 可以更准确地对流体流动进行建模。 deal.II 的收敛速度与预期顺序不匹配，而采用稳定方案的 StormFlash 的收敛速度（L 除外）动量误差）满足预期。具有稳定性的 StormFlash 的结果也比 deal.II 的结果更符合文献中的参考解决方案。我们得出的结论是，这种新颖的方案是一种稳定欧拉方程的低成本方法，同时除了流体静力平衡之外不会以任何方式限制流量。