High-frequency wave-propagation: error analysis for analytical and numerical approximations

In this thesis we investigate a specific type of semilinear hyperbolic systems with highly oscillatory initial data. This type of systems is numerically very challenging to treat since the solutions are highly oscillatory in space and time. The goal is to derive suitable analytical and numerical approximations. Based on the classical slowly varying envelope approximation (SVEA), an improved error estimate is proven for this analytical approximation. The envelope equation avoids oscillations in space, making this approximation attractive for numerical computations. Furthermore, more accurate analytical approximations are obtained by extending the ansatz of the SVEA. In addition to the analytical study of the SVEA two numerical time integrators are constructed and analyzed without any step-size restrictions. Numerical examples are provided to illustrate the theoretical results.
Finally, a complementary approach is presented which address both problems, the oscillations in space and time, simultaneously.