We present a hybridized discontinuous Galerkin (HDG) solver for general time‐dependent balance laws. In particular, we focus on a coupling of the solution process for unsteady problems with our anisotropic mesh refinement framework. The goal is to properly resolve all relevant unsteady features with the smallest possible number of mesh elements, and hence to reduce the computational cost of numerical simulations while maintaining its accuracy. A crucial step is then to transfer the numerical solution between two meshes, as the anisotropic mesh adaptation is producing highly skewed, non‐nested sequences of triangular grids. For this purpose, we adopt the Galerkin projection for the HDG solution transfer as it preserves the conservation of physically relevant quantities and does not compromise the accuracy of high‐order method. We present numerical experiments verifying these properties of the anisotropically adaptive HDG method.

中文翻译：

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时间精确的混合不连续伽辽金方法的各向异性网格之间的保守解传递

我们提出了一种用于一般瞬态平衡定律的混合不连续伽辽金（HDG）求解器。特别是，我们专注于不稳定问题的解决过程与各向异性网格细化框架的耦合。目标是用尽可能少的网格单元正确解决所有相关的不稳定特征，从而在保持精度的同时降低数值模拟的计算成本。然后，关键的一步是在两个网格之间传递数值解，因为各向异性网格自适应会产生高度倾斜、非嵌套的三角形网格序列。为此，我们采用伽辽金投影进行 HDG 解传递，因为它保留了物理相关量的守恒，并且不会损害高阶方法的准确性。我们通过数值实验验证了各向异性自适应 HDG 方法的这些特性。