论文标题
具有重力的欧拉方程的均匀高阶结构不连续的盖尔金方法：阳性和均衡度
Uniformly High-Order Structure-Preserving Discontinuous Galerkin Methods for Euler Equations with Gravitation: Positivity and Well-Balancedness
论文作者
论文摘要
本文介绍了一类新型的高阶准确性不连续的盖尔金（DG）方案，用于在重力场下可压缩的Euler方程。这些方案的一个值得注意的特征是它们对于一般的静水平衡状态均衡，同时证明它们可以保留密度和压力的阳性。为了同时实现均衡和阳性的特性，仔细设计了一种新型的DG空间离散化，并通过合适的源项重新制定和正确修改的Harten-Lax-van Leer接触（HLLC）磁通量进行设计。基于一些技术分解以及可允许状态和HLLC通量的几个关键特性，进行了严格的阳性分析。事实证明，由此产生的均衡的DG方案，再加上强大的稳定性，可以满足时间离散的能力，满足了弱的阳性性能，这意味着人们可以应用一个简单的现有限制器来有效地执行阳性性保护特性，而不会丢失高级准确性和保护性。提出的方法和分析适用于具有一般状态方程的Euler系统。广泛的一维数值测试证明了这些方案的所需特性，包括对平衡状态的确切保留，捕获这种状态的小扰动的能力，解决涉及低密度和/或低压的问题的稳健性以及平滑和不连续解决方案的良好分辨率。
This paper presents a class of novel high-order accurate discontinuous Galerkin (DG) schemes for the compressible Euler equations under gravitational fields. A notable feature of these schemes is that they are well-balanced for a general hydrostatic equilibrium state, and at the same time, provably preserve the positivity of density and pressure. In order to achieve the well-balanced and positivity-preserving properties simultaneously, a novel DG spatial discretization is carefully designed with suitable source term reformulation and a properly modified Harten-Lax-van Leer contact (HLLC) flux. Based on some technical decompositions as well as several key properties of the admissible states and HLLC flux, rigorous positivity-preserving analyses are carried out. It is proven that the resulting well-balanced DG schemes, coupled with strong stability preserving time discretizations, satisfy a weak positivity property, which implies that one can apply a simple existing limiter to effectively enforce the positivity-preserving property, without losing high-order accuracy and conservation. The proposed methods and analyses are applicable to the Euler system with general equation of state. Extensive one- and two-dimensional numerical tests demonstrate the desired properties of these schemes, including the exact preservation of the equilibrium state, the ability to capture small perturbation of such state, the robustness for solving problems involving low density and/or low pressure, and good resolution for smooth and discontinuous solutions.
