A space-time high-order implicit shock tracking method for shock-dominated unsteady flows

TL;DR

This paper introduces a novel space-time high-order implicit shock tracking method for unsteady flows, enabling accurate shock representation and efficient computation by reformulating time-dependent problems as steady in higher dimensions.

Contribution

It extends implicit shock tracking to unsteady flows using a space-time approach with conforming mesh generation and practical adaptations for complex shock phenomena.

Findings

Accurate shock capturing on coarse space-time grids.

Effective handling of complex flow features like shock interactions.

Reduced computational complexity in unsteady flow simulations.

Abstract

High-order implicit shock tracking (fitting) is a class of high-order, optimization-based numerical methods to approximate solutions of conservation laws with non-smooth features by aligning elements of the computational mesh with non-smooth features. This ensures the 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 extend implicit shock tracking to time-dependent problems using a slab-based space-time approach. This is achieved by reformulating a time-dependent conservation law as a steady conservation law in one higher dimension and applying existing implicit shock tracking techniques. To avoid computations over the entire time domain and unstructured mesh generation in higher dimensions, we introduce a general procedure to generate conforming, simplex-only meshes of space-time slabs in such a way that preserves features (e.g., curved elements, refinement regions) from previous time slabs. The use of space-time slabs also simplifies the shock tracking problem by reducing temporal complexity. Several practical adaptations of the implicit shock tracking solvers are developed for the space-time setting including 1) a self-adjusting temporal boundary, 2) nondimensionalization of a space-time slab, 3) adaptive mesh refinement, and 4) shock boundary conditions, which lead to accurate solutions on coarse space-time grids, even for problem with complex flow features such as curved shocks, shock formation, shock-shock and shock-boundary interaction, and triple points.