High-Order Implicit Shock Tracking for Time-Dependent Flows

  • Naudet, Charles (Matthew)

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Shock tracking, as an alternative method to shock capturing, aims to generate a mesh such that element faces align with shock surfaces and other non-smooth features to perfectly represent them with the inter- element jumps in the solution basis, e.g., in the context of a finite volume or discontinuous Galerkin (DG) discretization. These methods lead to high-order approximations of high-speed flows and do not require nonlinear stabilization or extensive refinement in non-smooth regions because, once the non-smooth features are tracked by the mesh, the high-order solution basis approximates the remaining smooth features. In previous work, we introduced the High-Order Implicit Shock Tracking (HOIST) method that recasts the geometrically complex problem of generating a mesh that conforms to all discontinuity surfaces as a PDE-constrained optimization problem. The optimization problem seeks to determine the flow solution and nodal coordinates of the mesh that simultaneously minimize an error-based indicator function and satisfy the discrete flow equations. A DG discretization of the governing equations is used as the PDE constraint to equip the discretization with desirable properties: conservation, stability, and high-order accuracy. By using high-order elements, curved meshes are obtained that track curved shock surfaces to high-order accuracy. The optimization problem is solved using a sequential quadratic programming method that simultaneously converges the mesh and DG solution, which is critical to avoid nonlinear stability issues that would come from computing a DG solution on an unconverged (non-aligned) mesh. In this work, the HOIST method is further extended to simulate time-dependent, inviscid flows. We use a space-time formulation of the governing equations and perform shock tracking over a space-time slab for two reasons: 1) it reduces to a steady shock tracking problem over the (d + 1)-dimensional space- time domain, which allows most of the HOIST framework to be recycled and 2) complex shock-shock interactions simply manifest as triple points. The overall approach is equipped with adaptive time slabs, a procedure to extrude d-dimensional simplex elements over a time slab and split the resulting (d + 1)- dimensional prism into simplices, and resolution-based h-refinement. The method is demonstrated on a series of increasingly complex unsteady inviscid flows in one and two dimensions.