Discrete transparent boundary conditions for the numerical solution of Fresnel's equation

Schmidt, F.; Deuflhard, Peter

doi:10.1016/0898-1221(95)00037-y

Cited by 64 publications

(43 citation statements)

References 12 publications

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“…It should be noted that this presentation is restricted to uniform step sizes in the propagation direction, so that we can apply the simplified shift-operator technique introduced in [15]. Similar formulas can, however, be derived for non-equidistant step sizes using the more direct procedure of [14] or the algebraic approach in [13], but the analysis and the resulting formulas would be far more complicated.…”

Section: Introductionmentioning

confidence: 91%

Transparent Boundary Conditions for Split-Step Padé Approximations of the One-Way Helmholtz Equation

Schmidt¹,

Friese

Yevick

2001

Journal of Computational Physics

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Section: Introductionmentioning

confidence: 91%

Transparent Boundary Conditions for Split-Step Padé Approximations of the One-Way Helmholtz Equation

Schmidt¹,

Friese

Yevick

2001

Journal of Computational Physics

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“…Unfortunately, the exact nonreflecting boundary conditions are generally nonlocal in both space and time, so that direct implementation is highly inefficient in terms of both computation and storage cost. Hence most research has been devoted to approximations of the exact nonreflecting boundary conditions, particularly for the one-dimensional case (see, e.g., [3,4,8,17,23,24,25,26]). A few papers in the literature consider the two-dimensional setting [5,7,22].…”

Section: 1) Iu T (X T) = U(x T) + V (X T)u(x T) X ∈ Rmentioning

confidence: 99%

Efficient representation of nonreflecting boundary conditions for the time‐dependent Schrödinger equation in two dimensions

Jiang

Greengard

2007

Comm Pure Appl Math

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We present a fast algorithm for the evaluation of exact, nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions on the unit circle. After separation of variables, the exact outgoing condition for each Fourier mode contains a nonlocal term that is a convolution integral in time. The kernel for that convolution is the inverse Laplace transform of the logarithmic derivative of a modified Bessel function, and the convolution integral can be split into two parts: a local part and a history part, which can be treated separately. The local part is easily handled by an appropriate quadrature. For the history part, we show that the convolution kernel can be well approximated by a sum of exponentials. Once such a representation is available, the convolution integrals can be evaluated recursively, reducing the cost from O(N 2 ) work to O(N ), where N is the number of time steps. The main technical development lies in the uniform rational approximation of the logarithmic derivative of the modified Bessel function K ν ( √ is).

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“…In a closed structure like the interior of the car, we can use the boundary conditions that are obtained from the displacement of the structure to obtain the fluid structure interaction, in an open structure like the acoustic waves emitted by the car to the outside, we have to incorporate appropriate far field conditions or transparent boundary conditions [5]. Let u be the vector of displacements of the structure on the surface.…”

Section: Mathematical Modeling Of Interior Car Acousticsmentioning

confidence: 99%

Nonlinear eigenvalue and frequency response problems in industrial practice

Mehrmann

Schröder

2011

Mathematics in Industry

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Background: We discuss the numerical solution of large scale nonlinear eigenvalue problems and frequency response problems that arise in the analysis, simulation and optimization of acoustic fields. We report about the cooperation with the company SFE in Berlin. We present the challenges in the current industrial problems and the state-of-the-art of current methods. Results: The difficulties that arise with current off-the-shelf methods are discussed and several industrial examples are presented. Conclusions: It is documented that industrial cooperation is by no means a one-way street of transfer from academia to industry but the challenges arising in industrial practice also lead to new mathematical questions which actually change the mathematical theory and methods.Keywords Nonlinear eigenvalue problem · frequency response problem · complex symmetric linear system · block-Arnoldi method · acoustic field computation · automotive industry AMS Subject Classification 65F18 · 15A18 Background IntroductionTraffic noise emissions by transport vehicles, such as cars, trains or airplanes are one of the key factors restricting the quality of life in urban areas. Acoustic waves in V Mehrmann ( ) · C Schröder

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Discrete transparent boundary conditions for the numerical solution of Fresnel's equation

Cited by 64 publications

References 12 publications

Transparent Boundary Conditions for Split-Step Padé Approximations of the One-Way Helmholtz Equation

Transparent Boundary Conditions for Split-Step Padé Approximations of the One-Way Helmholtz Equation

Efficient representation of nonreflecting boundary conditions for the time‐dependent Schrödinger equation in two dimensions

Nonlinear eigenvalue and frequency response problems in industrial practice

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