Overresolving in the Laplace domain for convolution quadrature methods

Convolution quadrature (CQ) methods have enjoyed tremendous interest in recent years as an efficient tool for solving time-domain wave problems in unbounded domains via boundary integral equation techniques. In this paper we consider CQ type formulations for the parallel space-time evaluation of multistep or stiffly accurate Runge--Kutta rules for the wave equation. In particular, we decouple the number of Laplace domain solves from the number of time steps. This allows us to overresolve in the Laplace domain by computing more Laplace domain solutions than there are time steps. We use techniques from complex approximation theory to analyze the error of the CQ approximation of the underlying time-stepping rule when overresolving in the Laplace domain and show that the performance is intimately linked to the location of the poles of the solution operator. Several examples using boundary integral equation formulations in the Laplace domain are presented to illustrate the main results. Read More: http://epubs.siam.org/doi/10.1137/16M106474

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

Nystr\"om methods for high-order CQ solutions of the wave equation in two dimensions

We investigate high-order Convolution Quadratures methods for the solution of the wave equation in unbounded domains in two dimensions that rely on Nystr\"om discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. We consider two classes of CQ discretizations, one based on linear multistep methods and the other based on Runge-Kutta methods, in conjunction with Nystr\"om discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. We present a variety of accuracy tests that showcase the high-order in time convergence (up to and including fifth order) that the Nystr\"om CQ discretizations are capable of delivering for a variety of two dimensional scatterers and types of boundary conditions

High-order, Dispersionless “Fast-Hybrid” Wave Equation Solver. Part I: 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 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

Nystrom methods for high-order CQ solutions of the wave equation in two dimensions

An investigation of high order Convolution Quadratures (CQ) methods for the solution of the wave equation in unbounded domains in two dimensions is presented. These rely on Nystrom discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. Two classes of CQ discretizations are considered: one based on linear multistep methods and the other based on Runge-Kutta methods. Both are used in conjunction with Nystrom discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. CQ in conjunction with BIE is an excellent candidate to eventually explore numerical homogenization to replace a chaff cloud by a dispersive lossy dielectric that produces the same scattering. To this end, a variety of accuracy tests are presented that showcase the high-order in time convergence (up to and including fifth order) that the Nystrom CQ discretizations are capable of delivering for a variety of two dimensional single and multiple scatterers. Particular emphasis is given to Lipschitz boundaries and open arcs with both Dirichlet and Neumann boundary conditions

On hybrid convolution quadrature approaches for modelling time-domain wave problems with broadband frequency content

We propose two hybrid convolution quadrature based discretisations of the wave equation on interior domains with broadband Neumann boundary data or source terms. The convolution quadrature method transforms the time-domain wave problem into a series of Helmholtz problems with complex-valued wavenumbers, in which the boundary data and solutions are connected to those of the original problem through the Z-transform. The hybrid method terminology refers specifically to the use of different approximations of these Helmholtz problems, depending on the frequency. For lower frequencies we employ the boundary element method, while for more oscillatory problems we develop two alternative high frequency approximations based on plane wave decompositions of the acoustic field on the boundary. In the first approach we apply dynamical energy analysis to numerically approximate the plane wave amplitudes. The phases will then be reconstructed using a novel approach based on matching the boundary element solution to the plane wave ansatz in the frequency region where we switch between the low and high frequency methods. The second high frequency method is based on applying the Neumann-to-Dirichlet map for plane waves to the given boundary data. Finally, we investigate the effectiveness of both hybrid approaches across a range of numerical experiments

Hybrid convolution quadrature methods for modelling time-dependent waves with broadband frequency content

This work proposes two new hybrid convolution quadrature based discretisations of the wave equation for interior domains with broadband Neumann boundary data or source terms. The convolution quadrature method transforms the time domain wave problem into a series of Helmholtz problems with complex-valued wavenumbers, in which the boundary data and solutions are connected to those of the original problem through the Z-transform. The hybrid method terminology refers specifically to the use of different approximations of these Helmholtz problems, depending on the frequency. For lower frequencies we employ the boundary element method, while for more oscillatory problems we develop two alternative high frequency approximations based on plane wave decompositions of the acoustic field on the boundary. In the first approach we apply dynamical energy analysis to numerically approximate the plane wave amplitudes. The phases will then be reconstructed using a novel approach based on matching the boundary element solution to the plane wave ansatz in the frequency region where we switch between the low and high frequency methods. The second high frequency method is based on applying the Neumann-to Dirichlet map for plane waves to the given boundary data

A fast BEM procedure using the Z-transform and high-frequency approximations for large-scale 3D transient wave problems

International audience3D rapid transient acoustic problems are difficult to solve numerically when dealing with large geometries, because numerical methods based on geometry discretisation (mesh), such as the boundary element method (BEM) or the finite element method (FEM), often require to solve a linear system (from the spacial discretisation) for each time step. We propose a numerical method to efficiently deal with 3D rapid transient acoustic problems set in large exterior domains. Using the Z-transform and the convolution quadrature method (CQM), we first present a straightforward way to reframe the problem to the solving of a large amount (the number of time steps, M) of frequency-domain BEMs. Then, taking advantage of a well-designed high-frequency approximation (HFA), we drastically reduce the number of frequency-domain BEMs to be solved, with little loss of accuracy. The complexity of the resulting numerical procedure turns out to be O(1) in regards to the time discretisation and O(N log N) for the spacial discretisation, the latter being prescribed by the complexity of the used fast BEM solver. Examples of applications are proposed to illustrate the efficiency of the procedure in the case of fluid-structure interaction: the radiation of an acoustic wave into a fluid by a deformable structure with prescribed velocity, and the scattering of an abrupt wave by simple and realistic geometries

Hybrid Frequency-Time Analysis and Numerical Methods for Time-Dependent Wave Propagation

This thesis focuses on the solution of causal, time-dependent wave propagation and scattering problems, in two- and three-dimensional spatial domains. This important and long-lasting problem has attracted a great deal of interest reflecting not only its use as a model problem but also the prevalence of wave phenomena in diverse areas of modern science, technology and engineering. Essentially all prior methods rely on "time-stepping" in one form or another, which involves local-in-time approximation of the evolution of the solution of the partial differential equation (PDE) based on the immediate time history and temporal finite-difference approximation. In addition to the need to manage the accumulation of (dispersion) error and the burdensome increase in computational cost over time, there are additionally difficult issues of stability, time-domain boundary conditions, and absorbing boundary conditions which often need to be addressed. To sidestep many of these problems, this thesis develops a novel highly-efficient approach for time-dependent wave scattering problems employing the global-in-time techniques of Fourier transformation and leading to a frequency/time hybrid method for the time-dependent wave equation. Thus, relying on Fourier Transformation in time and utilizing a fixed (time-independent) number of frequency-domain solutions, the method evaluates the desired time-domain evolution with errors that both, decay faster than any negative power of the temporal sampling rate, and that, for a given sampling rate, are additionally uniform in time for all time. The fast error decay guarantees that high accuracies can be attained on the basis of relatively coarse temporal and frequency discretizations. The uniformity of the error for all time with fixed sampling rate, a property known as dispersionlessness, plays a crucial role, together with other properties of the Fourier transform, in enabling the evaluation of solutions for long times at O(1) cost. In particular, this thesis demonstrates the significant advantages enjoyed by the proposed methods over alternative approaches based on volumetric discretizations, time-domain integral equations, and convolution-quadrature. 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 in detail the Laplace-based method, and only briefly point out its essential features and associated challenges. The proposed frequency/time Fourier-transform methods for obstacle scattering problems are easily generalizable to any linear partial differential equation in the time domain for which frequency-domain solutions can readily be obtained, including e.g. the time-domain Maxwell equations, the linear elasticity equations, inhomogeneous and/or frequency-dependent dispersive media, etc. Further, the proposed approach 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 O(1)-bounded sampling cost, for arbitrarily large values of T, and without requirement of evaluation of the solution at intermediate times. In particular, effective algorithms are introduced that, relying on use of time-asymptotics, compute two-dimensional solutions at O(1) cost despite the very slow time-decay that takes place in the two-dimensional case. A significant portion of this thesis is devoted to a theoretical study of the validity of a certain stopping criterion used by the algorithm, which guarantees that certain field contributions can safely be neglected after certain stopping times. Roughly speaking, the theoretical results guarantee that, after the incident field is turned off, the magnitude of the future scattering density (and thus the magnitudes of the fields) can be estimated by the magnitude of the integral density over a time period comparable to the time required by a wave to travel a distance equal to the diameter of the scatterer. The criterion, which is crucial in ensuring the O(1) computational cost of the algorithm, is closely related to the well-known scattering theory developed in the 1960s and '70s by Lax, Morawetz, Phillips, Strauss and others. Our approach to the decay problem is based on use of frequency-domain estimates (developed previously in the context of numerical analysis of frequency-domain problems) on integral operators in the high-frequency regime for obstacles of various trapping classes. In particular, our theory yields, for the first time, decay estimates for a class of connected trapping obstacles: all previous estimates of scattered-field decay for connected obstacles are restricted to nontrapping structures. In all, the proposed approach leverages the power of the Fourier transformation together with a range of newly developed spectrally convergent numerical methods in both the frequency and time domain and a variety of novel theoretical results in the general area of scattering theory to produce a radically-new framework for the solution of time-dependent wave propagation and scattering problems.</p