A spectral order method for inverting sectorial Laplace transforms

Laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting a potential behavior are considered. A spectral order, parallelizable method for their numerical inversion is proposed. The method takes into account the available information about the errors arising in the evaluations. Several numerical illustrations are provided.Comment: 17 pages 11 figure

Fast and oblivious convolution quadrature

We give an algorithm to compute $N$ steps of a convolution quadrature approximation to a continuous temporal convolution using only $O(N \log N)$ multiplications and $O(\log N)$ active memory. The method does not require evaluations of the convolution kernel, but instead $O(\log N)$ evaluations of its Laplace transform, which is assumed sectorial. The algorithm can be used for the stable numerical solution with quasi-optimal complexity of linear and nonlinear integral and integro-differential equations of convolution type. In a numerical example we apply it to solve a subdiffusion equation with transparent boundary conditions

Transparent Boundary Conditions for Time-Dependent Problems

A new approach to derive transparent boundary conditions (TBCs) for dispersive wave, Schrödinger, heat, and drift-diffusion equations is presented. It relies on the pole condition and distinguishes between physically reasonable and unreasonable solutions by the location of the singularities of the Laplace transform of the exterior solution. Here the Laplace transform is taken with respect to a generalized radial variable. To obtain a numerical algorithm, a Möbius transform is applied to map the Laplace transform onto the unit disc. In the transformed coordinate the solution is expanded into a power series. Finally, equations for the coefficients of the power series are derived. These are coupled to the equation in the interior and yield transparent boundary conditions. Numerical results are presented in the last section, showing that the error introduced by the new approximate TBCs decays exponentially in the number of coefficients

Domain Decomposition Method for Maxwell's Equations: Scattering off Periodic Structures

We present a domain decomposition approach for the computation of the electromagnetic field within periodic structures. We use a Schwarz method with transparent boundary conditions at the interfaces of the domains. Transparent boundary conditions are approximated by the perfectly matched layer method (PML). To cope with Wood anomalies appearing in periodic structures an adaptive strategy to determine optimal PML parameters is developed. We focus on the application to typical EUV lithography line masks. Light propagation within the multi-layer stack of the EUV mask is treated analytically. This results in a drastic reduction of the computational costs and allows for the simulation of next generation lithography masks on a standard personal computer.Comment: 24 page

A Review of Artificial Boundary Conditions for the Schrödinger Equation

In this review we discuss techniques to solve numerically the time-dependent linear Schrödinger equation on unbounded domains. We present some recent approaches and describe alternative ideas pointing out the relations between these works

A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations

In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present and compare several approaches to implement the classical transparent boundary condition into finite difference and finite element discretizations. We present in detail the approaches of the authors and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case

A fast convolution algorithm for non-reflecting boundary conditions

Nichtreflektiernede Randbedingungen für die Ausbreitung von Wellen sind nichtlokal sowohl im Raum als auch in der Zeit. Sie können stetig oder diskret gegeben sein. Hier werden neue diskrete nichtreflektierende Randbedingungen für die zeitabhängige Maxwellgleichung hergeleitet. Mit Hilfe einer Fourier- oder sphährischen Entwicklung in speziellen räumlichen Geometrien, lässt sich die räumliche Nichtlokalität entkoppeln. Um die Faltungen in der Zeit, die weiterhin bei der Berechnung eine hohe Hürde darstellen, auswerten zu können wird hier ein neuer schneller Algorithmus vorgestellt. Dieser benötigt, um die Faltung von N aufeineanderfolgener Zeitschritte zu berechnen einen Aufwand von O(N log (N)) Rechenoperationen und hat einen Speicherbedarf in der Grössenordnung von O(N log (N)). In den numerischen Beispielen, wird dieser Algorithmus verwendet um die Neumann- nach Dirichletabbildung zu diskretisieren, wie sie in rechteckigen Geometrieen bei der Schrödinger- und der Wellengleichung auftritt. Die Stabilität und Konvergenz des Faltungalgorithmus wird im Fall der Schrödingergleichung bewiesen.Non-reflecting boundary conditions for problems of wave propagation are non-local in space and time. These can be formulated continously or discrete in space. New discrete non-reflecting boundary conditions for the time-dependent Maxwell equation are developed. While the non-locality in space can be efficiently handled by Fourier or spherical expansions in special geometries, the arising temporal convolutions still form a computational bottleneck. In the present dissertation, a new algorithm for the evaluation of these convolution integrals is proposed. To compute a temporal convolution over N successive time steps, the algorithm requires O(N log (N)) operations and O(log (N)) memory. In the numerical examples, this algorithm is used to discretize the Neumann-to-Dirichlet operators arising from the formulation of non-reflecting boundary conditions in rectangular geometries for Schrödinger and wave equations. Stability and convergence of the convoluton algorithm in the Schrödinger case is proven

Convergence analysis of an explicit splitting method for laser plasma interaction simulations. ETNA - Electronic Transactions on Numerical Analysis

The convergence of a triple splitting method originally proposed by Tückmantel, Pukhov, Liljo, and Hochbruck for the solution of a simple model that describes laser plasma interactions with overdense plasmas is analyzed. For classical explicit integrators it is the large density parameter that imposes a restriction on the time step size to make the integration stable. The triple splitting method contains an exponential integrator in its central component and was specifically designed for systems that describe laser plasma interactions and overcomes this restriction. We rigorously analyze a slightly generalized version of the original method. This analysis enables us to identify modifications of the original scheme such that a second-order convergent scheme is obtained