A spacetime discontinuous Galerkin method for hyperbolic conservation laws

The thesis presents a spacetime discontinuous Galerkin (SDG) finite element method for hyperbolic conservation laws. The SDG method is based on a weighted residual statement, defined in terms of spacetime functions, that generates a weak form that satisfies the integral form of the conservation law on each element. The Rankine-Hugoniot conditions arise naturally as the spacetime jump conditions that couple the solutions in adjacent elements. The method is compatible with either structured or unstructured spacetime meshes. If the spacetime grid conforms to a causality constraint, then a direct patch-by-patch solution procedure with linear complexity in the number of elements (for fixed polynomial order) is possible. The causality constraint is nonlinear and solution-dependent, so we use an advancing-front solution technique that inter-leaves mesh generation with patch-wise solutions. This approach accommodates an h-adaptive scheme that simultaneously refines the mesh in space and time to track closely shock trajectories with regions of high mesh density.; The basic SDG approximation is a simple Bubnov-Galerkin projection that is not prone to global patterns of spurious oscillations. However, it does require stabilization to eliminate local overshoot and undershoot in the immediate vicinity of shocks and other discontinuous solution features. We address this requirement with a diffusion operator whose intensity is controlled by a shock indicator that measures the relative strength of the high-frequency components of the SDG approximation. Results demonstrating the performance of the SDG method, the h-adaptive refinement scheme, and the diffusion operator for applications of the inviscid Euler equations in one and two spatial dimensions are presented.