High-energy-density (HED) hydrodynamics studies such as those relevant to inertial confinement fusion and astrophysics require highly disparate densities, temperatures, viscosities, and other diffusion parameters over relatively short spatial scales. This presents a challenge for high-order accurate methods to effectively resolve the hydrodynamics at these scales, particularly in the presence of highly disparate diffusion. A significant volume of engineering and physics applications use an unstructured discontinuous Galerkin (DG) method developed based on the finite element mesh generation and algorithmic framework. This work discusses the application of an affine reconstructed nodal DG method for unstructured grids of triangles. Solving the diffusion terms in the DG method is non-trivial due to the solution representations being piecewise continuous. Hence, the diffusive flux is not defined on the interface of elements. The proposed numerical approach reconstructs a smooth solution in a parallelogram that is enclosed by the quadrilateral formed by two adjacent triangle elements. The interface between these two triangles is the diagonal of the enclosed parallelogram. Similar to triangles, the mapping of parallelograms from a physical domain to a reference domain is an affine mapping, which is necessary for an accurate and efficient implementation of the numerical algorithm. Thus, all computations can still be performed on the reference domain, which promotes efficiency in computation and storage. This reconstruction does not make assumptions on choice of polynomial basis. Reconstructed DG algorithms have previously been developed for modal implementations of the convection–diffusion equations. However, to the best of the authors’ knowledge, this is the first practical guideline that has been proposed for applying the reconstructed algorithm on a nodal discontinuous Galerkin method with a focus on accuracy and efficiency. The algorithm is demonstrated on a number of benchmark cases as well as a challenging substantive problem in HED hydrodynamics with highly disparate diffusion parameters.

高能量密度（HED）流体力学研究（例如与惯性约束聚变和天体物理学有关的研究）要求在相对较短的空间尺度上具有高度不同的密度，温度，粘度和其他扩散参数。这对有效解决这些规模的流体动力学问题的高阶精确方法提出了挑战，特别是在高度分散的情况下。大量的工程和物理应用使用基于有限元网格生成方法开发的非结构化不连续Galerkin（DG）方法和算法框架。这项工作讨论了仿射重构节点DG方法在非结构三角形网格中的应用。由于解决方案表示是分段连续的，因此在DG方法中求解扩散项并非易事。因此，在元件的界面上没有定义扩散通量。所提出的数值方法重建了平行四边形的光滑解被两个相邻的三角形元素形成的四边形所包围。这两个三角形之间的界面是封闭的平行四边形的对角线。与三角形相似，平行四边形从物理域到参考域的映射是仿射映射，这对于精确高效地实现数值算法是必需的。因此，所有计算仍可以在参考域上执行，从而提高了计算和存储效率。此重构未基于多项式选择做出假设。对流扩散方程的模态实现以前已经开发了重构的DG算法。但是，据作者所知，这是已提出的将重构算法应用于节点不连续Galerkin方法的第一个实用指南，重点是准确性和效率。在许多基准案例以及具有高度不同的扩散参数的HED流体动力学中一个具有挑战性的实质性问题上证明了该算法。