Solutions to the governing partial differential equations obtained from a discrete numerical scheme can have significant errors, especially near shocks when the discrete representation of the solution cannot fully capture the discontinuity in the solution. A recent approach to shock tracking [1, 2] has been to implicitly align the faces of mesh elements with the shock, yielding accurate solutions on coarse meshes. In engineering applications, the solution field is often used to evaluate a scalar functional of interest, such as lift or drag over an airfoil. While functionals are sensitive to errors in the flow solution, certain regions in the domain are more important for accurate evaluation of the functional than the rest. Using this fact, we formulate a goal-oriented implicit shock tracking approach that captures a segment of the shock that is important for evaluating the functional. Shock tracking is achieved using Lagrange-Newton-Krylov-Schur (LNKS) full space optimizer, with the objective of minimizing the adjoint-weighted residual error indicator. We also present a method to evaluate the sensitivity and the Hessian of the functional error. Using available block preconditioners for LNKS [3, 4] makes the full space approach scalable. The method is applied to test cases of two-dimensional advection and inviscid compressible flows to demonstrate functional-dependent shock tracking. Tracking the entire shock without using artificial dissipation results in the error converging at the orders of $\mathcal{O}(h^{p+1})$.

翻译：离散数值格式求解控制偏微分方程时，解可能产生显著误差，尤其在激波附近，由于解的离散表示无法完全捕捉解的间断性。近期激波捕捉方法[1,2]通过隐式对齐网格单元面与激波方向，在粗网格上即可获得精确解。工程应用中，解场常被用于评估感兴趣标量泛函（如机翼升力或阻力）。尽管泛函对流动解误差敏感，但域内某些区域对泛函精确评估的重要性远高于其他区域。基于此，我们提出一种目标导向的隐式激波捕捉方法，通过捕捉对泛函评估关键的激波段以实现目标导向。采用拉格朗日-牛顿-克雷洛夫-舒尔（LNKS）全空间优化器实现激波捕捉，优化目标为最小化伴随加权残差误差指标。同时提出评估泛函误差敏感度及海塞矩阵的方法。利用LNKS的现有块预处理器[3,4]使全空间方法具备可扩展性。该方法应用于二维平流及无粘可压缩流动测试案例，验证了泛函依赖的激波捕捉特性。在不使用人工耗散的情况下追踪完整激波，误差收敛阶可达$\mathcal{O}(h^{p+1})$。