UC Berkeley

We present a framework for resolving discontinuous solutions of conservation laws using implicit tracking and a high-order discontinuous Galerkin (DG) discretization. Central tothe framework is an optimization problem and associated sequential quadratic programming solver which simultaneously solves for a discontinuity-aligned mesh and the corresponding high-order approximation to the flow that does not require explicit meshing of the a priori unknown discontinuity surface. We utilize an error-based objective function that penalizes violation of the DG residual in an enriched test space, which endows the method with r-adaptive behavior: mesh nodes move to track discontinuities with element faces and improve the conservation law approximation in smooth regions of the flow. This method is shown to deliver highly accurate solutions on coarse, high-order discretizations without nonlinear stabilization and recover optimal convergence rates O(hp+1) for problems with discontinuous solutions. We demonstrate this framework on a series of inviscid steady and unsteady conservation laws, the latter of which using both a space-time and method of lines discretization. We also develop local mesh operations for curved meshes that are required to maintain mesh and solution quality as our high-order meshes deform to track the shock.