Analysis of a splitting scheme for a class of nonlinear stochastic Schr\uf6dinger equations

We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schr\uf6dinger equations driven by additive It\uf4 noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme

Error analysis of splitting methods for wave type equations

In dieser Doktorarbeit analysieren wir Splittingverfahren für zwei wellenartige Gleichungen. Wir untersuchen das Lie- und das Strangsplitting für die kubisch nichtlineare Schrödingergleichung auf dem Ganzraum und dem Torus in bis zu drei Raumdimensionen. Wir beweisen, dass das Strangsplitting in $L^2$ mit Ordnung $1+\theta$ für Anfangsfunktionen in $H^{2+2\theta}$ mit $\theta\in (0,1)$ konvergiert und dass beide Splittingverfahren mit Ordnung eins für Anfangsfunktionen in $H^2$ konvergieren. Wir bestätigen die theoretischen Konvergenzordnungen durch numerische Experimente. Außerdem analysieren wir ein "Alternating direction implicit"-Zeitsplittingverfahren für die Maxwellgleichungen mit Quellen, Strömen und Leitfähigkeit. Wir zeigen, dass es effizient ist, dass es mit Ordnung zwei in $L^2$ und in einem schwachen Sinne konvergiert, und dass es die Divergenzbedingungen bis auf Ordnung eins in $L^2$ und in einem schwachen Sinne erhält. Wir bestätigen die $L^2$-Resultate durch numerische Experimente

Numerical Integrators for Maxwell-Klein-Gordon and Maxwell-Dirac Systems in Highly to Slowly Oscillatory Regimes

Maxwell-Klein-Gordon (MKG) and Maxwell-Dirac (MD) systems physically describe the mutual interaction of moving relativistic particles with their self-generated electromagnetic field. Solving these systems in the nonrelativistic limit regime, i.e. when the speed of light $c$ formally tends to infinity, is numerically very delicate as the solution becomes highly oscillatory in time. In order to resolve the oscillations, standard time integrations schemes require severe restrictions on the time step $\tau\sim c^{-2}$ depending on the small parameter $c^{-2}$ which leads to high computational costs. Within this thesis we propose and analyse two types of numerical integrators to efficiently integrate the MKG and MD systems in highly oscillatory nonrelativistic limit regimes to slowly oscillatory relativistic regimes. The idea for the first type relies on asymptotically expanding the exact solution in the small parameter $c^{-1}$. This results in non-oscillatory Schrödinger-Poisson (SP) limit systems which can be solved efficiently by using classical splitting schemes. We will see that standard Strang splitting schemes, applied to the latter SP systems with step size $\tau$, allow error bounds of order $\mathcal{O}(\tau^2+c^{-N})$ for $N\in \mathbb N$ without any time step restriction. Thus, in the nonrelativistic limit regime $c\rightarrow\infty$ these methods are very efficient and allow an accurate approximation to the exact solution. The second type of numerical integrator is based on "twisted variables" which have been originally introduced for the Klein-Gordon equation in [Baumstark/Faou/Schratz, 2017]. In the case of MKG and MD systems however, due to the strong nonlinear coupling between the components of the solution, the construction and analysis is much more involved. We thereby exploit the main advantage of the "twisted variables" that they have bounded derivatives with respect to $c\rightarrow\infty$. Together with a splitting approach, this allows us to construct an exponential-type splitting method which is first order accurate in time uniformly in $c$. Due to error bounds of order $\mathcal{O}(\tau)$ independent of $c$ without any restriction on the time step $\tau$, these schemes are efficient in highly to slowly oscillatory regimes