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）是一类高阶数值方法，通过数值优化同时计算守恒律解的高阶近似，并使计算网格单元与非光滑特征对齐。这种对齐确保非光滑特征通过单元间跳跃完美表示，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，从而在传统粗网格上实现高精度近似。本文针对定义优化求解器每次迭代最优步骤的鞍点线性系统，设计了一系列预处理算子，使Krylov子空间求解器能够有效应用。我们的预处理算子将约束优化中的标准预处理算子与间断伽辽金离散的常用预处理算子相结合，例如块雅可比、含最小丢弃填充重排序的块不完全LU分解以及p-多重网格。通过两个无黏可压缩流动问题进行全面研究，评估了该系列中各预处理算子的有效性，及其对关键激波追踪参数（如网格与黑塞矩阵正则化、线性化状态及解空间分辨率）的敏感性。