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

• Published in: Journal of computational physics 2022-04, Vol.454, p.110981, Article 110981
• Main Authors: Huang, Tianci, Zahr, Matthew J.
• Format: Article
• Language: English
• Subjects: Approximation; Basis functions; Compressible flow; Computational grids; Computational physics; Conservation laws; Discontinuity; Discontinuous Galerkin; Finite element method; Globalization; High speed; High-order methods; High-speed flows; Methods; Numerical methods; Numerical optimization; Optimization; Quadratic programming; Robustness (mathematics); Shock capturing; Shock fitting; Shock tracking; Shock waves; Solvers; Three dimensional flow; Tracking
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.110981

•Implicit shock tracking: high-order resolution of high-speed flows on coarse meshes•DG solution on discontinuity-aligned mesh is the solution of an optimization problem•Automated approach to preserve planar boundaries during tracking iterations•Novel SQP solver that incorporates practical robustness measures•Reliably tracks complex shock structures (e.g., curved, intersecting) in 2D and 3D High-order implicit shock tracking (fitting) is a new class of numerical methods to approximate solutions of conservation laws with non-smooth features, e.g., contact lines, shock waves, and rarefactions. 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 that employs a Levenberg-Marquardt approximation of the Hessian and line-search globalization. The robustness measures include 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.