Overresolving in the Laplace Domain for Convolution Quadrature Methods

Betcke, Timo; Salles, Nicolas; Śmigaj, Wojciech

doi:10.1137/16m106474x

Cited by 12 publications

(31 citation statements)

References 28 publications

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“…The characteristics of a particular implementation of the CQ algorithm is determined by the choice made for time-domain finite-difference discretization, the spectral character of the discrete frequency-domain solver used [15], and the methods utilized for numerical inversion of the Z-transform. Existing CQ approaches have primarily utilized the second-order-accurate BDF2 time discretization [10], but recent work [8] proposes the use of higher-order m-stage Runge-Kutta schemes.…”

Section: Previous Hybrid Methods: Cqmentioning

confidence: 99%

“…[45] and direct Fourier transform in time [47]. As mentioned in section 1, two hybrid time-domain methods (i.e., methods that rely on transformation of the time variable by means of Fourier or Laplace transforms) have previously been proposed, namely, the CQ method [7,8,9,10,11,15,45] and the direct Fourier transform method [47]. The CQ method employs a discrete convolution that is obtained as temporal finite-difference schemes are solved by transform methods.…”

Section: Preliminariesmentioning

confidence: 99%

“…The discrete time-domain solution is then obtained by evaluation of the inverse Z-transform of the frequency-domain solutions by means of trapezoidal rule quadrature. (References [15,45] provide further elaboration on the connections of the CQ method to Z-transforms and convolutions, respectively.) As a result, the solutions produced by this method accumulate temporal and spatial discretization errors at each time step as well as overall inversion errors arising from the approximate quadrature used in the inversion of the Z-transform.…”

Section: Preliminariesmentioning

confidence: 99%

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High-order, Dispersionless “Fast-Hybrid” Wave Equation Solver. Part I: O(1) Sampling Cost via Incident-Field Windowing and Recentering

Anderson¹,

Bruno²,

Lyon³

2020

SIAM J. Sci. Comput.

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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 timewindowing 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.

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Section: Previous Hybrid Methods: Cqmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

High-order, Dispersionless “Fast-Hybrid” Wave Equation Solver. Part I: O(1) Sampling Cost via Incident-Field Windowing and Recentering

Anderson¹,

Bruno²,

Lyon³

2020

SIAM J. Sci. Comput.

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show abstract

“…We point out here that no scattering poles associated with (6) lie inside the contour for C ⊂ {z ∈ C : Im z ≥ 0}. (An interesting discussion on analytic and numerical issues arising due to the existence of scattering poles near the contour C utilized in the practical implementation of the convolution quadrature method can be found in reference [7]. )…”

Section: Convolution Quadrature Methodsmentioning

confidence: 99%

“…is the free space Green function for the Helmholtz equation with wavenumbers k j = k j (ζ) defined in (7), which in view of Remark 3.1 are assumed to satisfy Re k j > 0 and Im k j ≥ 0.…”

Section: Windowed Green Function Methodsmentioning

confidence: 99%

Convolution quadrature methods for time-domain scattering from unbounded penetrable interfaces

2019

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This paper presents a class of boundary integral equation methods for the numerical solution of acoustic and electromagnetic time-domain scattering problems in the presence of unbounded penetrable interfaces in two-spatial dimensions. The proposed methodology relies on Convolution Quadrature (CQ) methods in conjunction with the recently introduced Windowed Green Function (WGF) method. As in standard time-domain scattering from bounded obstacles, a CQ method of the user's choice is utilized to transform the problem into a finite number of (complex) frequency-domain problems posed on the domains involving penetrable unbounded interfaces. Each one of the frequency-domain transmission problems is then formulated as a second-kind integral equation that is effectively reduced to a bounded interface by means of the WGF method-which introduces errors that decrease super-algebraically fast as the window size increases. The resulting windowed integral equations can then be solved by means of any (accelerated or unaccelerated) off-the-shelf Helmholtz boundary integral equation solver capable of handling complex wavenumbers with a large imaginary part. A high-order Nyström method based on Alpert quadrature rules is utilized here. A variety of numerical examples including wave propagation in open waveguides as well as scattering from multiply layered media, demonstrate the capabilities of the proposed approach.(CQ) methods [25,26], in particular, have effectively enabled the use of (complex) frequencydomain boundary integral equation (BIE) solvers to tackle a variety of wave propagation problems, by providing a stable procedure to discretize the associated convolution equations for the unknown time evolution of the relevant surface densities; see [33] for the mathematical foundations of the method, and [5,19] for details on the algorithmic implementation. In the case of the scalar wave equation with piecewise constant wavespeed, to which this paper is devoted to, approximate traces at discrete times are produced all at once from a finite sequence of independent Helmholtz problems that can be solved in parallel by means of BIE methods. Although this CQ-BIE approach has proven to be competitive to volume discretization methods in the context of obstacle scattering problems [3,4,34], its extension to problems involving unbounded material interfaces is severely hindered by the fact that standard BIE formulations require the knowledge of problem-specific Green functions to deal with the unboundedness of the material interfaces. These Green functions, however, are often unavailable (in terms of tractable mathematical expressions) or are given in terms of computationally expensive Sommerfeld integrals 1 [27,29,30].Recent advances on BIE methods for time-harmonic problems of scattering from unbounded material interfaces have led to the development of highly efficient solvers that completely bypass the use of problem-specific Green functions [10,12,13,24,30,37]. The windowed Green function (WGF) method, in particular, has successfully...

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A fast boundary element method using the Z‐transform and high‐frequency approximations for large‐scale three‐dimensional transient wave problems

Mavaleix-Marchessoux

Bonnet

Chaillat

et al. 2020

Numerical Meth Engineering

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Three-dimensional (3D) rapid transient acoustic problems are difficult to solve numerically when dealing with large geometries, because numerical methods based on geometry discretization (mesh), such as the boundary element method (BEM) or the finite element method (FEM), often require to solve a linear system (from the spacial discretization) 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 -transform and the convolution quadrature method, 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, 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 regard to the time discretization and O(N log N) for the spacial discretization, 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.

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Overresolving in the Laplace Domain for Convolution Quadrature Methods

Cited by 12 publications

References 28 publications

High-order, Dispersionless “Fast-Hybrid” Wave Equation Solver. Part I: O(1) Sampling Cost via Incident-Field Windowing and Recentering

High-order, Dispersionless “Fast-Hybrid” Wave Equation Solver. Part I: O(1) Sampling Cost via Incident-Field Windowing and Recentering

Convolution quadrature methods for time-domain scattering from unbounded penetrable interfaces

A fast boundary element method using the Z‐transform and high‐frequency approximations for large‐scale three‐dimensional transient wave problems

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