Error analysis of tailor-made time integration schemes for certain classes of wave-type equations

This thesis is concerned with the time integration of three classes of wave-type partial differential equations. Each of these equations has numerical challenges, including nonlocal-in-time material laws, nontrivial boundary conditions, and an unbounded spatial domain. We construct tailor-made schemes and provide a rigorous numerical analysis.

We consider the semiclassical magnetic Schrödinger equation on the full space $\mathbb{R}^d$ with possibly time-dependent magnetic and electric potentials. For the approximation we use an appropriate Gaussian ansatz function, which leads to a set of ordinary differential equations. We show that in a special case, this ansatz function is the exact solution to the magnetic Schrödinger equation. Furthermore, we provide error bounds with respect to the semiclassical parameter in $L^2$-norm and we improve the error bound for relevant physical quantities of interest.

The second setting is concerned with a scattering problem for Maxwell's equations on an unbounded domain. The scatterer consists of a nonlocal-in-time material, which is modeled as a convolution in Maxwell's equations. In this situation, we use a coupling on the scatterer's bounded interface and derive a boundary integral equation. ... mehr
For the discretization we employ a convolution quadrature combined with a boundary element method. Finally, we show that this numerical approach is stable and we prove convergence rates.

For a semilinear viscoacoustic wave equation with a retarded material law and kinetic boundary conditions, we construct and analyze an implicit-explicit (IMEX) scheme. The IMEX scheme is computationally efficient, since it avoids the solution of nonlinear systems. The material law is described by a convolution term with exponential kernels. After applying an appropriate shift, we couple the convolution term as an auxiliary variable to the first-order system of partial differential equations. For the kinetic boundary conditions we make use of suitable bulk-surface Sobolev spaces to show wellposedness.