High-order implicit shock tracking (fitting) is a class of high-order, optimization-based numerical methods to approximate solutions of conservation laws with non-smooth features by aligning elements of the computational mesh with non-smooth features. This ensures the non-smooth features are perfectly represented by inter-element jumps and high-order basis functions approximate smooth regions of the solution without nonlinear stabilization, which leads to accurate approximations on traditionally coarse meshes. In this work, we extend implicit shock tracking to time-dependent problems using a slab-based space-time approach. This is achieved by reformulating a time-dependent conservation law as a steady conservation law in one higher dimension and applying existing implicit shock tracking techniques. To avoid computations over the entire time domain and unstructured mesh generation in higher dimensions, we introduce a general procedure to generate conforming, simplex-only meshes of space-time slabs in such a way that preserves features (e.g., curved elements, refinement regions) from previous time slabs. The use of space-time slabs also simplifies the shock tracking problem by reducing temporal complexity. Several practical adaptations of the implicit shock tracking solvers are developed for the space-time setting including 1) a self-adjusting temporal boundary, 2) nondimensionalization of a space-time slab, 3) adaptive mesh refinement, and 4) shock boundary conditions, which lead to accurate solutions on coarse space-time grids, even for problem with complex flow features such as curved shocks, shock formation, shock-shock and shock-boundary interaction, and triple points.

翻译：高阶隐式激波追踪（拟合）是一类基于优化的高阶数值方法，通过将计算网格单元与解的非光滑特征对齐，来逼近具有非光滑特征的守恒律解。该方法确保非光滑特征通过单元间跳跃完美表征，同时高阶基函数在无需非线性稳定的情况下逼近解的光滑区域，从而在传统粗糙网格上实现高精度逼近。本文采用基于时空片的方法将隐式激波追踪扩展至非定常问题。具体实现过程为：将非定常守恒律重构为高一个维度的稳态守恒律，并应用现有隐式激波追踪技术。为避免在整个时间域上计算以及高维非结构化网格生成问题，我们提出了一种通用程序，能够在时空片上生成保形、仅含单纯形单元的网格，同时保留前序时空片的几何特征（如曲线单元、加密区域）。通过使用时空片，时间复杂度的降低简化了激波追踪问题。针对时空框架，我们发展了隐式激波追踪求解器的多种实际改进策略，包括：1）自适应时间边界调整，2）时空片无量纲化，3）自适应网格加密，以及4）激波边界条件。这些方法即使在处理具有复杂流场特征（如弯曲激波、激波生成、激波-激波相互作用、激波-边界相互作用及三波点）的问题时，仍能在粗糙时空网格上获得精确解。