High-order implicit shock tracking is a new class of numerical methods to approximate solutions of conservation laws with non-smooth features. These methods align elements of the computational mesh with non-smooth features to represent them perfectly, allowing high-order basis functions to approximate smooth regions of the solution without the need for nonlinear stabilization, which leads to accurate approximations on traditionally coarse meshes. The hallmark of these methods is the underlying optimization formulation whose solution is a feature-aligned mesh and the corresponding high-order approximation to the flow; the key challenge is robustly solving the central optimization problem. In this work, we develop a robust optimization solver for high-order implicit shock tracking methods so they can be reliably used to simulate complex, high-speed, compressible flows in multiple dimensions. The proposed method integrates practical robustness measures into a sequential quadratic programming method, including dimension- and order-independent simplex element collapses, mesh smoothing, and element-wise solution re-initialization, which prove to be necessary to reliably track complex discontinuity surfaces, such as curved and reflecting shocks, shock formation, and shock-shock interaction. A series of nine numerical experiments -- including two- and three-dimensional compressible flows with complex discontinuity surfaces -- are used to demonstrate: 1) the robustness of the solver, 2) the meshes produced are high-quality and track continuous, non-smooth features in addition to discontinuities, 3) the method achieves the optimal convergence rate of the underlying discretization even for flows containing discontinuities, and 4) the method produces highly accurate solutions on extremely coarse meshes relative to approaches based on shock capturing.
翻译:高度隐含的冲击跟踪是一组新的数字方法,以近似具有非悬浮特征的保护法解决方案。这些方法将计算网格的元素与非超光速特性统一起来,以便完美地反映这些元素,使高阶基础功能能够接近解决方案的平坦区域,而不需要非线性稳定,从而导致对传统上粗肿的草粒进行准确的近似。这些方法的特点是基础优化配方,其解决方案是特征匹配的网格和对流动的相应高端近似;关键挑战是强有力地解决中央优化问题。在这项工作中,我们为高端的高度直线性隐含的冲击跟踪方法开发了一个强大的优化解决方案,以便它们能够可靠地用于模拟复杂、高速、可压缩的解决方案流。 拟议的方法将实用的稳健健度措施纳入一个连续的四重编程编程方法,包括大小和顺序依赖的简单要素的崩溃、中间线顺流和元素偏顺序的解决方案的重新首发式。 事实证明,为了可靠地追踪复杂的不连续断断层表面表面,例如曲线和反映不精确的冲击、冲击、冲击形成冲击形成、休整的深度的快速的快速的快速的快速的快速的轨道,以及压的不断的不断的不断的不断的轨道的轨道。