Green Function Discretization for Numerical Solution of the Helmholtz Equation

Caruthers, J. E.; French, Jonathan; Raviprakash, G. K.

doi:10.1006/jsvi.1995.0544

Cited by 19 publications

(23 citation statements)

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“…Various choices of approximating functions are available: propagating plane waves [35,14,17,27], evanescent plane waves [29,45,46], Bessel functions [13], Green's functions [47]. Some elements of comparisons of these methods can be found in [48,36,49,19,37].…”

Section: Interpolation Basismentioning

confidence: 99%

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

Lieu

Gabard

Bériot

2016

Journal of Computational Physics

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The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.

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Section: Interpolation Basismentioning

confidence: 99%

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

Lieu

Gabard

Bériot

2016

Journal of Computational Physics

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“…This discretization methods of fer an enormous improvement in computational efficiency by reducing the number of points per wavelength required to oVjtain accurate solutions and generate sparse system matrices which have a high bandw idth especially in 3D. The discretization is perhaps optimal in the sense th a t it does not seem reasonable to expect th a t anything less than two points per wavelength, the classic Nyquist limit in DSP, is feasible [83]. It was shown [83,85] th at the GFD and WED method fail precisely at a point corresponding to this limit in two dimensions for square grids.…”

Section: 6 8 3d P E Tech N Iq U Esmentioning

confidence: 99%

“…The discretization is perhaps optimal in the sense th a t it does not seem reasonable to expect th a t anything less than two points per wavelength, the classic Nyquist limit in DSP, is feasible [83]. It was shown [83,85] th at the GFD and WED method fail precisely at a point corresponding to this limit in two dimensions for square grids.…”

Section: 6 8 3d P E Tech N Iq U Esmentioning

confidence: 99%

A forward-advancing wave expansion method for numerical solution of large-scale sound propagation problems

Rolla

Rice

2006

Journal of Sound and Vibration

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“…The Green's function based "nite di!erence scheme proposed by Caruthers et al [1] shows considerable potential as an e$cient modelling tool at high frequencies. As it is formulated using fundamental solutions of the wave equation, it can maintain its accuracy at nodal spacings as low as two per wavelength.…”

Section: Introductionmentioning

confidence: 99%

“…In this letter, a formulation using plane wave based solutions is applied to a 3-D acoustic problem (plane wave functions were suggested in reference [1] as a viable alternative to the Green's functions and used later in references [3,4]). The use of plane waves may, for many problems, simplify the implementation of the solution procedure and an example of an application in plate vibration by the authors is given in reference [5].…”

Section: Introductionmentioning

confidence: 99%

An Implementation of a Wave-Based Finite Difference Scheme for a 3-D Acoustic Problem

Ruiz¹,

Rice²

2002

Journal of Sound and Vibration

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Green Function Discretization for Numerical Solution of the Helmholtz Equation

Cited by 19 publications

References 0 publications

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

A forward-advancing wave expansion method for numerical solution of large-scale sound propagation problems

An Implementation of a Wave-Based Finite Difference Scheme for a 3-D Acoustic Problem

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