High-order implicit shock tracking (fitting) is a class of high-order numerical methods that use numerical optimization to simultaneously compute a high-order approximation to a conservation law solution and align elements of the computational mesh with non-smooth features. This alignment ensures that 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 devise a family of preconditioners for the saddle point linear system that defines the step toward optimality at each iteration of the optimization solver so Krylov solvers can be effectively used. Our preconditioners integrate standard preconditioners from constrained optimization with popular preconditioners for discontinuous Galerkin discretizations such as block Jacobi, block incomplete LU factorizations with minimum discarded fill reordering, and p-multigrid. Thorough studies are performed using two inviscid compressible flow problems to evaluate the effectivity of each preconditioner in this family and their sensitivity to critical shock tracking parameters such as the mesh and Hessian regularization, linearization state, and resolution of the solution space.

翻译：高阶隐式激波追踪（拟合）是一类通过数值优化同时计算守恒律解的高阶近似、并将计算网格单元与非光滑特征对齐的高阶数值方法。这种对齐确保非光滑特征能完全由单元间跳跃表示，而光滑区域则可通过无需非线性稳定的高阶基函数进行逼近，从而在传统粗网格上实现精确近似。本研究针对优化求解器每次迭代中定义最优性步长的鞍点线性系统，设计了一类预处理器族，使得Krylov求解器能够高效应用。我们的预处理器将约束优化中的标准预处理器与不连续伽辽金离散化常用预处理器相结合，包括块雅可比、采用最小弃置填充重排序的块不完全LU分解以及p多重网格。通过两个无粘可压缩流动问题开展系统研究，评估了该预处理器族中每种方法的有效性，并分析了其对关键激波追踪参数（如网格与海森正则化、线性化状态及解空间分辨率）的敏感性。