On local and structure-adapted time integration for wave-type problems

This thesis studies time integration for wave-type problems whose numerical approximation is
constrained by fast oscillations in space and time, small discretization parameters, or materials
with high wave speeds. Two settings are considered. In the first part, the constraint is spatially
localized. Two-field Friedrichs’ systems, such as Maxwell’s equations and the acoustic wave
equation, are discretized in space by discontinuous Galerkin (dG) methods on locally refined
meshes. The smallest cells then impose a severe CFL restriction on explicit time integrators,
although they represent only a small part of the mesh.

For central-flux discretizations, we construct and analyze a second-order, leapfrog-based local
time-integration framework. It combines the leapfrog method on the coarse part of the mesh
with a filtered treatment of the fine part, and it contains fully explicit local time-stepping
schemes as well as locally implicit variants. Under suitable assumptions on the filter, the
resulting stability and error estimates show convergence with a CFL restriction governed by the
coarse part of the mesh. This framework is then extended to stabilized dG discretizations, like
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upwind fluxes, where dissipative face penalties are applied on the coarse part of the mesh. The
analysis yields stability and error estimates under a CFL restriction which again is governed by
the coarse part of the mesh.

To obtain higher-order time integration, we use implicit Gauß collocation methods and solve
the arising linear systems by preconditioned Krylov subspace methods. The preconditioner uses
only the fine part of the mesh. It is shown that the number of preconditioned QMR iterations
needed to reach a fixed tolerance can be bounded independently of the fine mesh. Thus the
same local fine and coarse split is used at a different algorithmic level, namely inside the linear
solver for an implicit method.

The second part treats the semiclassical magnetic Schrödinger equation. Here, a small
semiclassical parameter appears which leads to rapid oscillations of the solution in space and
time making a mesh-based approach very expensive. A mesh-free variational Gaussian wave
packet ansatz captures the rapid oscillations and reduces the evolution to a finite-
dimensional system of parameter ODEs. For this system we extend the Boris algorithm
from classical charged-particle dynamics and propose a modified fourth-order Runge–Kutta
method. Both methods preserve the L2-norm of the Gaussian wave packet by construction.
Numerical experiments, including the Penning trap, investigate convergence, norm preservation
and energy behavior of the proposed schemes.