A robust, high-order implicit shock tracking method for simulation of complex, high-speed flows

High-order implicit shock tracking is a new class of numerical methods to approximate solutions conservation laws with non-smooth features. These align elements the computational mesh features represent them perfectly, allowing high-order basis functions smooth regions solution without need for nonlinear stabilization, which leads accurate approximations on traditionally coarse meshes. The hallmark these underlying optimization formulation whose feature-aligned and corresponding approximation flow; key challenge robustly solving central problem. In this work, we develop robust solver so they can be reliably used simulate complex, high-speed, compressible flows in multiple dimensions. proposed method integrates practical robustness measures into sequential quadratic programming method, including dimension- order-independent simplex element collapses, smoothing, element-wise re-initialization, prove necessary track complex discontinuity surfaces, such as curved reflecting shocks, formation, shock-shock interaction. A series nine experiments -- two- three-dimensional surfaces are demonstrate: 1) solver, 2) meshes produced high-quality continuous, addition discontinuities, 3) achieves optimal convergence rate discretization even containing 4) produces highly extremely relative approaches based capturing.