Dispersion Relation Preserving FD Schemes and Self-Affine DG Elements

The two primary directions of research in computational hyperbolic PDE are resolving high-frequency wave modes and computation in complex geometries. In this thesis, we make progress in both of these directions with regard to computation and theory.\\ First, to resolve high frequencies we introduce upwind dispersion preserving differential operator pairs. We prove for complex curvilinear geometries that general summation-by-parts dual pairings are numerically stable and provide an analysis of errors for large-scale wave propagation and dynamic rupture simulations. We find that numerical dispersion errors can completely destroy the numerical solution for dynamic rupture problems when computed with odd order SBP dual pairings, some of which worsen with grid refinement. To overcome this error we construct dispersion relation preserving schemes whose maximal dispersion error is $