Sub-cell shock capturing and spacetime discontinuity tracking for nonlinear conservation laws

We describe two approaches to resolving shocks and other discontinuities in spacetime discontinuous Galerkin (SDG) methods fo nonlinear conservation laws. The first is an adaptation of the sub-cell shock-capturing technique, recently introduced by Persson and Peraire, to the special circumstances of SDG solutions constructed on causal spacetime grids. We restrict the stabilization operator to spacetime element interiors, thereby preserving the O(N) computational complexity and the element-wise conservation properties of the basic SDG method, and use a special discontinuity indicator to limit the stabilization to elements traversed by discontinuous solution features. The method resolves discontinuities within individual spacetime elements having a sufficiently high-order basis. Numerical studies demonstrate the combination of sub-cell shock capturing with h-adaptive spacetime meshing that circumvents the projection errors inherent to purely spatial remeshing procedures. In a second method, we use adaptive spacetime meshing operations to track the trajectories of singular surfaces while maintaining the quality of the surrounding mesh. We present a series of feasibility studies where we track shocks and contact discontinuities in solutions to the inviscid Euler equations, including an example where the trajectories are not known a priori. The SDG-tracking method sharply resolves discontinuities without mesh refinement and requires very little stabilization.