Time-dependent electromagnetic scattering from dispersive materials

This paper studies time-dependent electromagnetic scattering from metamaterials that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave-material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation that is fully given on the surface of the scatterer via a time-dependent boundary integral equation. Discretizing this equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we derive error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method

Boundary element methods for the wave equation based on hierarchical matrices and adaptive cross approximation

Time-domain Boundary Element Methods (BEM) have been successfully used in acoustics, optics and elastodynamics to solve transient problems numerically. How ever, the storage requirements are immense, since the fully populated system matrices have to be computed for a large number of time steps or frequencies. In this article, we propose a new approximation scheme for the Convolution Quadrature Method powered BEM, which we apply to scattering problems governed by the wave equa tion. We use H 2-matrix compression in the spatial domain and employ an adaptive cross approximation algorithm in the frequency domain. In this way, the storage and computational costs are reduced significantly, while the accuracy of the method is preserved

Fast, Parallel Techniques for Time-Domain Boundary Integral Equations

This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion. As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems: the need to construct many dense matrices and to evaluate many singular and near-singular integrals. The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested. The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal. In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments

High-order, Dispersionless "Fast-Hybrid" Wave Equation Solver. Part I: O(1)\mathcal{O}(1) Sampling Cost via Incident-Field Windowing and Recentering

This paper proposes a frequency/time hybrid integral-equation method for the time dependent wave equation in two and three-dimensional spatial domains. Relying on Fourier Transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate, with superalgebraically-small errors, time domain solutions for arbitrarily long times. The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily-long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally-accurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider the Laplace-based method in the present contribution. The algorithm can handle dispersive media, it can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping---that is, solution sampling at any given time $T$ at $\mathcal{O}(1)$-bounded sampling cost, for arbitrarily large values of $T$, and without requirement of evaluation of the solution at intermediate times. The proposed frequency-time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including e.g. the time-domain Maxwell equations), and which is applicable in a wide range of scientific and engineering contexts, provides significant advantages over other available alternatives such as volumetric discretization, time-domain integral equations, and convolution-quadrature approaches.Comment: 33 pages, 8 figures, revised and extended manuscript (and now including direct comparisons to existing CQ and TDIE solver implementations) (Part I of II

Generalized convolution quadrature for the fractional integral and fractional diffusion equations

We consider the application of the generalized Convolution Quadrature (gCQ) of the first order to approximate fractional integrals and associated fractional diffusion equations. The gCQ is a generalization of Lubich's Convolution Quadrature (CQ) which allows for variable steps. In this paper we analyze the application of the gCQ to fractional integrals, with a focus in the low regularity case. It is well known that in this situation the original CQ presents an order reduction close to the singularity. Moreover, the available theory for the gCQ does not cover this situation. Here we deduce error bounds for a general time mesh. We show first order of convergence under much weaker regularity requirements than previous results in the literature. We also prove that uniform first order convergence is achievable for a graded time mesh, which is appropriately refined close to the singularity, according to the order of the fractional integral and the regularity of the data. Then we study how to obtain full order of convergence for the application to fractional diffusion equations. For the implementation of this method, we use fast and oblivious quadrature and present several numerical experiments to illustrate our theoretical results.Comment: 22 pages, 18 figure

Wave scattering from nontrivial boundary conditions

Die vorliegende Arbeit untersucht numerische Verfahren zur Simulation von akustischen und elektromagnetischen Wellen im Kontext von zeitabhängingen Streuproblemen, die an eine nichttriviale Randbedingung gekoppelt werden. Eine Vielzahl solcher Randbedingungen sind in der Praxis von Interesse, insbesondere wenn mehrere physikalische Skalen involviert sind. Effektive Randbedingungen beinhalten Modelle für dünne Schichten auf reflektierenden Materialien, oder beschreiben das Verhalten eines stark absorbierenden Mediums. Motiviert durch diese Anwendungen, behandelt die vorliegende Arbeit drei Klassen von Randbedingungen: 1. akustische Streuprobleme mit einer abstrakten, linearen Randbedingung, die neben den beschriebenen Anwendungen auch akustische Randbedingungen beinhaltet; 2. elektromagnetische Streuprobleme mit einer abstrakten linearen Randbedingung; 3. elektromagnetische Streuprobleme mit einer nichtlinearen Randbedingung. Zur Bearbeitung dieser Problemstellungen werden, basierend auf Repräsentationsformeln, zeit-abhängige Randintegralgleichungen hergeleitet. Diese Gleichungen sind vollständig auf dem Rand des Streuobjekts formuliert und äquivalent zum ursprünglichen Streuproblem. Essenzielle Eigenschaften der zugrunde liegenden zeitabhängigen Randintegraloperatoren und Repräsentationsformeln werden mithilfe von Transmissionsproblemen gezeigt. Mithilfe dieser fundamentalen Resultate wird die Wohlgestelltheit der Randintegralgleichungen hergeleitet, womit die Wohlgestelltheit der jeweiligen Randwertprobleme insgesamt gezeigt wird. Die Randintegralgleichungen werden in der Zeit durch Faltungsquadraturen basierend auf den Radau IIA Runge--Kutta Methoden diskretisiert. Die Stabilität der Semi-Diskretisierungen folgen aus den fundamentalen Eigenschaften der Randintegraloperatoren und allgemeinen Eigenschaften der Faltungsquadraturen. Komplementiert wird die Zeitdiskretisierung mit der Randelementmethode im Raum, um Volldiskretisierungen zu konstruieren, deren Lösungen effektiv berechnet werden können. Die resultierenden Verfahren berechnen in einem ersten Schritt die numerischen Lösungen auf dem Rand. Anschließend können die Approximationen durch diskrete Repräsentationsformeln an beliebigen Punkten im Gebiet ausgewertet werden. Fehleranalysen leiten Konvergenzraten für die numerischen Approximationen her. Die Notation der Behandlung der linearen Randbedingungen für akustische und elektromagnetische Wellen wurde entsprechend angepasst, sodass Gemeinsamkeiten und Unterschiede herausgestellt werden. Für die nichtlinearen Randbedingungen wird eine Fehleranalyse mithilfe neuer Techniken basierend auf diskreten Transmissionsproblemen durchgeführt. Alle numerischen Verfahren wurden implementiert und mit verschiedenen Parametern und Gittern getestet. Empirische Konvergenzraten illustrieren und komplementieren die theoretischen Ergebnisse. Visualisierungen der numerischen Approximationen zeigen den Nutzen der untersuchten Verfahren.This dissertation studies the numerical approximation of time-dependent acoustic and electromagnetic wave scattering problems in the presence of non-standard boundary conditions. Of particular interest is the numerical treatment of generalized impedance boundary conditions, effective models that approximate the wave-material interaction of partially penetrable obstacles. Classical applications of such boundary conditions are the scattering of highly absorbing materials and perfectly reflecting obstacles with a thin coating. Moreover, acoustic boundary conditions are discussed in the context of the acoustic wave equation. Finally, a class of nonlinear boundary conditions is covered in the context of electromagnetic scattering. Formulated on the time domain, these boundary conditions contain surface differential operators and temporal convolution operators. The resulting boundary value problems on exterior domains are reformulated to retarded boundary integral equations, which are themselves nonlocal in time and space, but fully formulated on the boundary. Several new fundamental properties of the time-harmonic classical potential operators and boundary operators for the acoustic wave equation and the Maxwell's equations are shown, in particular in view of their temporal counterparts. These theoretical results are the necessary preparations for the subsequent numerical analysis of these problems. To derive numerical methods, the boundary integral equations are then discretized in time and space. The temporal discretization is carried out using the Runge--Kutta convolution quadrature method. Fully discrete schemes are derived by combining the time discretization with appropriate boundary element methods in space. Error bounds with specific convergence rates are shown for all boundary conditions. The presentation of the linear boundary conditions is focused on emphasizing the similarities and differences of the acoustic and the electromagnetic settings. The error analysis for the nonlinear scattering problem substantially differs from the analysis of the linear boundary conditions and several new concepts are necessary to overcome the difficulties arising through the nonlinearity of the corresponding boundary integral equation. Numerical experiments illustrate the theoretical results and investigate practical aspects of the proposed methods