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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 5%. This improves on existing central stencil difference operators that have a maximal relative dispersion error of 100%. We prove that in 3 dimensions for time-dependent PDE, our stencils require 4% of the computational e ort needed to resolve high-frequency waves on a given computational grid compared to traditional operators. Our newly derived operators are efficiently implemented in the large-scale wave simulation software Wave-Qlab-3D, and numerical results are provided.

Second, we give the theoretical details of how to construct fractal finite elements which can be used for Discontinuous Galerkin (DG) schemes. Such elements provide a step towards the accurate simulation of wave-propagation along self-affine boundaries, contributing to simulation in complex geometries. We build off a method for approximating a non-differentiable self-affine surface with a self-affine function in a weak sense to form our scheme. We then give efficient and stable algorithms to compute the integrals of polynomials over such domains, and also compute orthonormal polynomial basis functions, which completes the theory for the interior of the element. For the element boundary, we show that the weak derivatives of refractable self-affine curves exist and can be computed efficiently. Results from geometric measure theory guarantee that the integration-by-parts formula holds for such boundaries and our elements fit within the summation-by-parts framework. Our algorithms then extend to both random fractals and graph-directed Iterated Function Systems.