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

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Topics in multiscale modeling: numerical analysis and applications

We explore several topics in multiscale modeling, with an emphasis on numerical analysis and applications. Throughout Chapters 2 to 4, our investigation is guided by asymptotic calculations and numerical experiments based on spectral methods. In Chapter 2, we present a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, the numerical methodology that we present is based on a spectral method. We use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients in the homogenized equation. Extensions of this method are presented in Chapter 3 and 4, where they are employed for the investigation of the Desai—Zwanzig mean-field model with colored noise and the generalized Langevin dynamics in a periodic potential, respectively. In Chapter 3, we study in particular the effect of colored noise on bifurcations and phase transitions induced by variations of the temperature. In Chapter 4, we investigate the dependence of the effective diffusion coefficient associated with the generalized Langevin equation on the parameters of the equation. In Chapter 5, which is independent from the rest of this thesis, we introduce a novel numerical method for phase-field models with wetting. More specifically, we consider the Cahn—Hilliard equation with a nonlinear wetting boundary condition, and we propose a class of linear, semi-implicit time-stepping schemes for its solution.Open Acces

Impedance Transformers

Non

Uniform and Optimal Error Estimates of an Exponential Wave Integrator Sine Pseudospectral Method for the Nonlinear Schrödinger Equation with Wave Operator, preprint, arXiv:1305.6377, [math.NA

Abstract. We propose an exponential wave integrator sine pseudospectral (EWI-SP) method for the nonlinear Schrödinger equation (NLS) with wave operator (NLSW), and carry out rigorous error analysis. The NLSW is NLS perturbed by the wave operator with strength described by a dimensionless parameter ε ∈ (0, 1]. As ε → 0 +, the NLSW converges to the NLS and for the small perturbation, i.e., 0 <ε ≪ 1, the solution of the NLSW differs from that of the NLS with a function oscillating in time with O(ε2) wavelength at O(ε2)andO(ε4) amplitudes for ill-prepared and wellprepared initial data, respectively. This rapid oscillation in time brings significant difficulties in designing and analyzing numerical methods with error bounds uniformly in ε. Inthiswork,weshow that the proposed EWI-SP possesses the optimal uniform error bounds at O(τ 2)andO(τ) inτ (time step) for well-prepared initial data and ill-prepared initial data, respectively, and spectral accuracy in h (mesh size) for both cases, in the L2 and semi-H1 norms. This result significantly improves the error bounds of the finite difference methods for the NLSW. Our approach involves a careful study of the error propagation, cut-off of the nonlinearity, and the energy method. Numerical examples are provided to confirm our theoretical analysis