Pollution and accuracy of solutions of the Helmholtz equation: A novel perspective from the eigenvalues

Dwarka, Vandana; Vuik, C.

doi:10.1016/j.cam.2021.113549

Cited by 3 publications

(1 citation statement)

References 21 publications

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“…It is derived using a Taylor series of the solution which permits to define a generalized 3-point stencil. For 2d Helmholtz problems, a dispersion correction using eigenvalues has been designed in [13]. Although the matrix associated to the stencil is modified, numerical results indicate that this method heavily reduces the pollution effect.…”

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confidence: 99%

Asymptotic Dispersion Correction in General Finite Difference Schemes for Helmholtz Problems

Cocquet,

Gander

2024

SIAM J. Sci. Comput.

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Most numerical approximations of frequency-domain wave propagation problems suffer from the so-called dispersion error, which is the fact that plane waves at the discrete level oscillate at a frequency different from the continuous one. In this paper, we introduce a new technique to reduce the dispersion error in general Finite Difference (FD) schemes for frequency-domain wave propagation using the Helmholtz equation as guiding example. Our method is based on the introduction of a shifted wavenumber in the FD stencil which we use to reduce the numerical dispersion for large enough numbers of grid points per wavelength (or for small enough meshsize), and thus we call the method asymptotic dispersion correction. The advantage of this technique is that the asymptotically optimal shift can be determined in closed form by computing the extrema of a function over a compact set. For 1d Helmholtz equations, we prove that the standard 3-point stencil with shifted wavenumber does not have any dispersion error, and that the so-called pollution effect is completely suppressed. For higher dimensional Helmholtz problems, we give easy to use closed form formulas for the asymptotically optimal shift associated to the second order 5-point scheme and a sixth-order 9-point scheme in 2d, and the 7-point scheme in 3d that yield substantially less dispersion error than their standard (unshifted) version. We illustrate this also with numerical experiments.

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confidence: 99%

Asymptotic Dispersion Correction in General Finite Difference Schemes for Helmholtz Problems

Cocquet,

Gander

2024

SIAM J. Sci. Comput.

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Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences

Chen,

Dwarka,

Vuik

2024

Mathematics in Industry

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Global Sound Field Reconstruction in the Room Environment Based on Inverse Wave-Based Simulation

Wang

Zhang

Wang

et al. 2022

Shock and Vibration

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An inverse wave modeling-based method is proposed for globally reconstructing the sound field in room environments. The method builds the wave model of the sound field as prior knowledge to support the reconstruction under strong reverberation. In this method, the whole space is divided into a set of subdomains. Based on the theory of discretization-based numerical simulation, a wave model that can describe the transfer characteristic between any subdomain and the source is built. Supported by this model, the sound source is recovered based on spatial sound pressure sampling and the global sound field reconstruction can be further accomplished in the reverberant environment. In particular, the shape function with the property of sparsity is constructed in building the wave model. Then, the intensity on the point source is represented by a sparse vector over the subdomains, and then, the sparse method can be used to achieve the recovery of this vector, which reduces the sampling burden in the space. Numerical verifications are performed to evaluate the performances of the proposed method. It demonstrates that the proposed method is capable of obtaining accurate reconstructions in a strong reverberant environment. It also shows that the method is applicable to problems with complicated excitations in the low-frequency range.

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Pollution and accuracy of solutions of the Helmholtz equation: A novel perspective from the eigenvalues

Cited by 3 publications

References 21 publications

Asymptotic Dispersion Correction in General Finite Difference Schemes for Helmholtz Problems

Asymptotic Dispersion Correction in General Finite Difference Schemes for Helmholtz Problems

Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences

Global Sound Field Reconstruction in the Room Environment Based on Inverse Wave-Based Simulation

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