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 introduce a robust implicit shock tracking framework specialized for problems with parameter-dependent lead shocks (i.e., shocks separating a farfield condition from the downstream flow), which commonly arise in high-speed aerodynamics and astrophysics applications. After a shock-aligned mesh is produced at one parameter configuration, all elements upstream of the lead shock are removed and the nodes on the lead shock are positioned for new parameter configurations using the implicit shock tracking solver. The proposed framework can be used for most many-query applications involving parametrized lead shocks such as optimization, uncertainty quantification, parameter sweeps, "what-if" scenarios, or parameter-based continuation. We demonstrate the robustness and flexibility of the framework using a one-dimensional space-time Riemann problem, and two- and three-dimensional supersonic and hypersonic benchmark problems.

翻译：高精度隐式激波追踪（拟合）是一类基于优化的高精度数值方法，通过将计算网格单元与非光滑特征对齐来近似守恒律中带非光滑特征的解。这种方法确保非光滑特征完全由单元间跳跃表示，同时高阶基函数无需非线性稳定化即可近似解的光滑区域，从而在传统粗网格上实现精确近似。本文针对参数依赖的前缘激波（即区分远场条件与下游流动的激波）问题，提出了一种鲁棒的隐式激波追踪框架，这类问题常见于高速空气动力学和天体物理学应用中。在某一参数配置下生成激波对齐网格后，移除前缘激波上游的所有单元，并利用隐式激波追踪求解器将前缘激波节点调整至新参数配置的位置。所提框架可适用于大多数涉及参数化前激波的多查询场景，如优化、不确定性量化、参数扫描、“what-if”分析或参数延拓。通过一维时空黎曼问题以及二维/三维超音速和高超音速基准问题，验证了该框架的鲁棒性与灵活性。