Abstract Discontinuous Galerkin (DG) methods due to their robustness properties, e.g. local conservation, low numerical dispersion, and well-capturing strong shocks and physical discontinuities, are well-suited for the simulation of Variable Density Flow (VDF) in porous media. This paper aims at introducing, in a unified format, the general class of Interior Penalty DG (IPDG) methods to solve the VDF equations. A combination of symmetric, non-symmetric and incomplete IPDG methods is used to discretize both head and concentration variables. Compatibility analysis is performed to prevent the loss of accuracy of the IPDG methods in simulations of coupled flow and transport equations. An accurate technique is used for time integration, based on a non-iterative procedure and adaptive time stepping with embedded error control. Several benchmarks are investigated to validate the proposed DG scheme and to examine its performance in simulating VDF problems. The new DG scheme reproduces better the experimental data than the conventional SEAWAT model. Its results are in excellent agreement with a recent semi-analytical solution of the Henry problem, dealing with seawater intrusion under convection-dominating conditions. The performance of the DG scheme is examined by simulating the challenging problem of natural convection in porous enclosure. The method is compared against a finite element solution obtained with COMSOL multi-physics. The numerical experiments indicate clearly that high-order DG method is much more appropriate than standard conforming Galerkin method in simulating VDF problems while at the same time, guaranteeing a better precision and high-fidelity solutions. The proposed numerical method can be extended to 3D problems.

中文翻译：

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一种求解变密度地下水流动问题的全内罚不连续伽辽金方法

摘要 不连续伽辽金 (DG) 方法由于其鲁棒性，例如局部守恒、低数值色散以及能很好地捕获强冲击和物理不连续性，非常适合模拟多孔介质中的变密度流 (VDF)。本文旨在以统一的格式介绍求解 VDF 方程的一般类内罚 DG (IPDG) 方法。对称、非对称和不完全 IPDG 方法的组合用于离散化水头和浓度变量。执行兼容性分析是为了防止 IPDG 方法在耦合流动和传输方程的模拟中的准确性损失。基于非迭代程序和带有嵌入式误差控制的自适应时间步进，时间积分使用了一种精确的技术。研究了几个基准来验证提议的 DG 方案并检查其在模拟 VDF 问题中的性能。新的 DG 方案比传统的 SEAWAT 模型更好地再现了实验数据。其结果与最近的亨利问题的半解析解非常吻合，该解在对流主导条件下处理海水入侵。通过模拟多孔外壳中自然对流的挑战性问题来检查 DG 方案的性能。将该方法与使用 COMSOL 多物理场获得的有限元解进行比较。数值实验清楚地表明，在模拟 VDF 问题时，高阶 DG 方法比符合标准的 Galerkin 方法更合适，同时，保证更好的精度和高保真解决方案。所提出的数值方法可以扩展到 3D 问题。