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In this thesis, a unified error analysis for discretizations of nonlinear first- and second-order wave-type equations is provided. For this, the wave equations as well as their space discretizations are considered as nonlinear evolution equations in Hilbert spaces. The space discretizations are supplemented with Runge-Kutta time discretizations. By employing stability properties of monotone operators, abstract error bounds for the space, time, and full discretizations are derived. Further, for semilinear second-order wave-type equations, an implicit-explicit time integration scheme is presented. This scheme only requires the solution of a linear system of equations in each time step and it is stable under a step size restriction only depending on the nonlinearity. It is proven that the scheme converges with second order in time and in combination with the abstract space discretization of the unified error analysis, corresponding full discretization error bounds are derived. The abstract results are used to derive convergence rates for an isoparametric finite element space discretization of a wave equation with kinetic boundary conditions and nonlinear forcing and damping terms. For the combination of the finite element discretization with Runge-Kutta methods or the implicit-explicit scheme, respectively, error bounds of the resulting fully discrete schemes are proven. The theoretical results are illustrated by numerical experiments.

This thesis is concerned with the time integration of certain classes of nonlinear evolution equations in Hilbert spaces by exponential integrators. We aim to prove error bounds which can be established by including only quantities given by a wellposedness result. In the first part, we consider semilinear wave equations and introduce a class of first- and second-order exponential schemes. A standard error analysis is not possible due to the lack of regularity. We have to employ appropriate filter functions as well as the integration by parts and summation by parts formulas in order to obtain optimal error bounds. In the second part, we propose two exponential integrators of first and second order applied to a class of quasilinear wave-type equations. By a detailed investigation of the differentiability of the right-hand side we derive error bounds in different norms. In the framework we can treat quasilinear Maxwell’s equations in full space and on a smooth domain as well as a class of quasilinear wave equations. In both parts, we include numerical examples to confirm our theoretical findings.

This thesis provides a unified framework for the error analysis for space and time discretizations of a quite general class of quasilinear wave-type problems. For the space discretization we prove a rigorous error estimate based on semigroup theory for nonautonomous problems. Compared to previous results, which are mostly based on Banach’s fixed-point theorem, this approach allows for a better insight into the individual error contributions. Furthermore, since wellposedness results for quasilinear wave-type problems are in general based on severe regularity assumptions with respect to the boundary of the domain, we consider nonconforming space discretizations in order to allow for domain approximation. Furthermore, we provide a rigorous error analysis for the full discretization with three different one-step time-integration schemes. On the one hand, we consider the implicit midpoint rule and a linearized version thereof. On the other hand, we also investigate the leapfrog scheme, which is an explicit scheme. Throughout this thesis, we illustrate the relevance of the abstract framework by application of our results to the undamped Westervelt equation and the Maxwell equations with Kerr nonlinearity. Finally, we conclude with numerical examples.

In this thesis the discretization in space and time of a broad class of linear first order wave-type problems is investigated. The full discretization is achieved by using a method of lines approach utilizing a central-fluxes discontinuous Galerkin (dG) method in space and the Peaceman–Rachford scheme in time. Rigorous error bounds are derived, which show full order of convergence in space and time if the exact solution is sufficiently regular. Additionally, despite the fact that the Peaceman–Rachford scheme is an implicit method exhibiting unconditional stability, it is shown that by combining it with an alternating direction implicit (ADI) approach, it can be applied to certain problems at roughly the cost of an explicit time integration scheme. Problems for which this is possible are characterized precisely, and the implementation used to achieve this efficiency is described in detail. This class of problems comprises, e.g., the 2D advection and wave equations and the 3D Maxwell's equations, for which the ADI method was previously proposed in literature.

This thesis is concerned with the full discretization of Maxwell's equations in cases where the spatial discretization has to be carried out with a locally refined grid. In such situations locally implicit time integrators are an appealing choice for the time discretization since they overcome the grid-induced stiffness of these problems. We analyze such a locally implicit time integrator in the case where the space discretization stems from a central fluxes discontinuous Galerkin method. In fact, we prove its stability under a CFL condition which solely depends on the coarse part of the spatial grid and give a rigorous error analysis showing that the integrator is second order convergent. Moreover, we extend this time integrator so that it can be applied to an upwind fluxes discontinuous Galerkin space discretization. We show that this novel integrator preserves the second order temporal convergence and that it inherits the improved properties of an upwind fluxes discretization (better stability, higher spatial convergence rate) compared to the central fluxes case.

This thesis provides a unified framework for the error analysis of non-conforming space discretizations of linear wave equations in time-domain, which can be cast as symmetric hyperbolic systems or second-order wave equations. Such problems can be written as first-order evolution equations in Hilbert spaces with linear monotone operators. We employ semigroup theory for the well-posedness analysis and to obtain stability estimates for the space discretizations. To compare the finite dimensional approximations with the original solution, we use the concept of a lift from the discrete to the continuous space. Time integration with the Crank–Nicolson method is analyzed. In this framework, we derive a priori error bounds for the abstract space semi-discretization in terms of interpolation and discretization errors. These error bounds yield previously unkown convergence rates for isoparametric finite element discretizations of wave equations with dynamic boundary conditions in smooth domains. Moreover, our results allow to consider already investigated space discretizations in a unified way. Here it successfully reproduces known error bounds. Among the examples which we dicuss in this thesis are discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the scalar wave equation.

In dieser Dissertation werden numerische Zeitintegratoren für partielle Differentialgleichungen behandelt. Primär geht es um Integratoren für partielle Differentialgleichungen auf Gebieten, deren räumliche Diskretisierung eine lokale Verfeinerung erfordert. Untersucht werden die hiermit verbundenen Schwierigkeiten. Die Konstruktion, die Analyse und die Implementierung geeigneter Integratoren werden ebenfalls vorgestellt.

With standard schemes, the plasma density in the hydrodynamic model for relativistic laser-plasma interaction can become negative. Therefore, we present a new scheme that preserves the positivity of the plasma density while still satisfying the conservation of mass. The numerical results for a vacuum-plasma transition of a laser pulse in one and two dimensions compare very well to the results from a standard PIC code, but the computation time is greatly reduced.

This thesis is concerned with the numerical analysis of some well-known time integration methods, such as implicit collocation methods and exponential integrators, for linear Maxwell's equations in time-domain. The error analysis of time integrators is done both for continuous Maxwell's equations in a semigroup theory framework and for space discrete problem obtained by discretizing Maxwell's equations in space by using discontinuous Galerkin finite element method.