Discontinuous Galerkin (DG) methods have been well established for single‐material hydrodynamics. However, consistent DG discretizations for non‐equilibrium multi‐material (more than two materials) hydrodynamics have not been extensively studied. In this work, a novel reconstructed DG (rDG) method for the single‐velocity multi‐material system is presented. The multi‐material system being considered assumes stiff velocity relaxation, but does not assume pressure and temperature equilibrium between the multiple materials. A second‐order DG(P₁) method and a third‐order least‐squares based rDG(P₁P₂) are used to discretize this system in space, and a third‐order total variation diminishing (TVD) Runge‐Kutta method is used to integrate in time. A well‐balanced DG discretization of the non‐conservative system is presented and is verified by numerical test problems. Furthermore, a consistent interface treatment is implemented, which ensures strict conservation of material masses and total energy. Numerical tests indicate that the DG and rDG methods are, indeed, the second‐ and third‐order accurate. Comparisons with the second‐order finite volume method show that the DG and rDG methods are able to capture the interfaces more sharply. The DG and rDG methods are also more accurate in the single‐material regions of the flow. This work focuses on the general multidimensional rDG formulation of the non‐equilibrium multi‐material system and a study of properties of the method via one‐dimensional numerical experiments. The results from this research will be the foundation for a multidimensional high‐order rDG method for multi‐material hydrodynamics.

中文翻译：

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具有锐利界面的多材料流体动力学的重构非连续Galerkin方法

非连续Galerkin（DG）方法已经很好地用于单一材料的流体动力学。但是，对于非平衡多材料（两种以上材料）流体动力学的一致DG离散化尚未得到广泛研究。在这项工作中，提出了一种用于单速多材料系统的新颖的重构DG（rDG）方法。所考虑的多材料系统假定刚性速度松弛，但不假定多种材料之间的压力和温度平衡。基于二阶DG（P ₁）方法和基于三阶最小二乘的rDG（P ₁ P ₂）用于离散化该系统在空间中的位置，并使用三阶总变差减小（TVD）Runge-Kutta方法进行时间积分。提出了非保守系统的均衡DG离散化，并通过数值测试问题进行了验证。此外，实施了一致的界面处理，可确保严格节省材料质量和总能量。数值测试表明，DG和rDG方法确实是二阶和三阶的。与二阶有限体积方法的比较表明，DG和rDG方法能够更清晰地捕获界面。DG和rDG方法在流的单一材料区域中也更加准确。这项工作的重点是非平衡多材料系统的通用多维rDG公式，并通过一维数值实验研究该方法的性质。这项研究的结果将为多维多维流体动力学的高阶rDG方法奠定基础。