High-order Implicit Shock Tracking for High-speed Flows

Shock tracking or shock fitting, where the computational mesh is moved to align mesh element faces with discontinuities, represents them perfectly with the inter-element jump in the solution basis without requiring additional stabilization when addressing shocks. In our previous work, we introduced an implicit shock tracking framework that discretizes conservation laws on a mesh without knowledge of discontinuities and solves a PDE-constrained optimization problem over the discrete solution variables and nodal coordinates of the mesh. A Discontinuous Galerkin (DG) discretization of the governing equation is applied and implicit tracking is formulated as an optimization problem constrained by the DG residual to endow the method with desirable properties of DG: consistency, conservation, and stability. The optimization problem is solved using a sequential quadratic programming method that simultaneously converges the mesh and flow solution to their optimal values.

In this talk, we present a $p$-adaptive implicit shock tracking method aimed at solving steady and unsteady high-speed viscous flows. The polynomial degree is adapted used an indicator based on the enriched residual, which tends to increase the polynomial degree within shocks and boundary layers. A series of shock-dominated numerical experiments demonstrate the potential of the method to efficiently produce accurate solutions to these challenging problems.