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：高阶隐式激波跟踪(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完全由单元间跳跃表示，高阶基函数无需非线性稳定化即可逼近解的光滑区域，从而在传统粗网格上实现精确逼近。本文采用基于时空片的方法将隐式激波跟踪扩展到时间相关问题。其核心思想是将时间相关的守恒律重构成更高维度的稳态守恒律，并应用现有隐式激波跟踪技术。为避免在整个时域上进行计算以及在高维度中生成非结构网格，我们提出一种通用流程，以保持前一时空片特征(如曲边单元、加密区域)的方式，生成时空片上的共形单纯形网格。使用时空片还可通过降低时间复杂度来简化激波跟踪问题。针对时空框架，我们开发了隐式激波跟踪求解器的多种实用改进措施，包括：1) 自适应时间边界；2) 时空片的无量纲化；3) 自适应网格加密；4) 激波边界条件。这些改进使得即使在包含复杂流动特征(如弯曲激波、激波生成、激波-激波相互作用、激波-边界相互作用及三叉点)的问题中，也能在粗时空网格上获得精确解。