The influence of the mass matrix on the dispersive nature of the semi-discrete, second-order wave equation

Christon, Mark A.

doi:10.1016/s0045-7825(98)00266-7

Cited by 55 publications

(33 citation statements)

References 13 publications

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“…Nance et al [25] have used Fourier analysis to develop low-dispersion ÿnite volume aeroacoustic solvers where the dispersion and dissipation are critical to the code performance. Christon [26] considered the in uence of the ÿnite element mass matrix on the dispersion characteristics of second-order wave equation for acoustic uid-structure interaction. Christon and Voth [27,28] have applied von Neumann analyses to assess the numerical performance of reproducing kernel semi-discretizations in one-and two-dimensions and considered both hyperbolic and parabolic partial di erential equations.…”

Section: Background and Historical Perspectivementioning

confidence: 99%

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Generalized Fourier analyses of the advection–diffusion equation—Part I: one‐dimensional domains

Voth

Martinez

Christon

2004

Numerical Methods in Fluids

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SUMMARYThis paper presents a detailed multi-methods comparison of the spatial errors associated with ÿnite difference, ÿnite element and ÿnite volume semi-discretizations of the scalar advection-di usion equation. The errors are reported in terms of non-dimensional phase and group speed, discrete di usivity, artiÿcial di usivity, and grid-induced anisotropy. It is demonstrated that Fourier analysis provides an automatic process for separating the discrete advective operator into its symmetric and skew-symmetric components and characterizing the spectral behaviour of each operator. For each of the numerical methods considered, asymptotic truncation error and resolution estimates are presented for the limiting cases of pure advection and pure di usion. It is demonstrated that streamline upwind Petrov-Galerkin and its control-volume ÿnite element analogue, the streamline upwind control-volume method, produce both an artiÿcial di usivity and a concomitant phase speed adjustment in addition to the usual semi-discrete artifacts observed in the phase speed, group speed and di usivity. The Galerkin ÿnite element method and its streamline upwind derivatives are shown to exhibit super-convergent behaviour in terms of phase and group speed when a consistent mass matrix is used in the formulation. In contrast, the CVFEM method and its streamline upwind derivatives yield strictly second-order behaviour. In Part II of this paper, we consider two-dimensional semi-discretizations of the advection-di usion equation and also assess the a ects of grid-induced anisotropy observed in the non-dimensional phase speed, and the discrete and artiÿcial di usivities. Although this work can only be considered a ÿrst step in a comprehensive multimethods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common analysis framework. Published in 2004 by John Wiley & Sons, Ltd.

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Section: Background and Historical Perspectivementioning

confidence: 99%

“…[24][25][26][103][104][105][106][107][108]. The connection between Fourier analysis and truncation error provides the means to extract the leading-order of the truncation error for a method based on the asymptotic behaviour of the discrete phase error.…”

Section: Asymptotic Analysismentioning

confidence: 99%

Generalized Fourier analyses of the advection–diffusion equation—Part I: one‐dimensional domains

Voth

Martinez

Christon

2004

Numerical Methods in Fluids

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“…The HOM matrix is obtained drawing on a combination of consistent and lumped mass matrices-leading to an improvement of the dispersion characteristic of the FEM [18,53]. The HOM matrix is defined as:…”

Section: Higher-order Mass Matrixmentioning

confidence: 99%

Reducing spurious oscillations in discontinuous wave propagation simulation using high-order finite elements

Mirbagheri

Nahvi

Parvizian

et al. 2015

Computers & Mathematics with Applications

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a b s t r a c tIn this paper, a modified explicit time integration scheme is proposed for simulating the propagation of discontinuous waves. The spatial domain is discretized by the finite element method. To obtain accurate results, the standard finite element method requires a very fine mesh, which is why the computational effort can be very time consuming. The use of high-order finite element methods -such as the spectral element method based on Lagrange polynomials through Gauss-Lobatto-Legendre points or the iso-geometric analysis using non-uniform rational B-splines -can reduce the enormous computational costs significantly, compared to the standard finite element method. However, explicit time integration schemes such as the central difference method cannot eliminate the spurious oscillation near the front wave. The procedure proposed here is basically similar to the explicit method of Noh and Bathe, but the semi-discrete equation of motion is modified by introducing a damping parameter to suit the high-order FEM analysis of the discontinuous wave propagation. The performance due to this modification is tested for one-dimensional wave propagation problems using both lumped and consistent mass matrices. Then, the idea of a combination of the consistent and row sum lumped mass matrices is evaluated. The proposed method is studied also in two-dimensions by considering the Lamb problem of wave propagation. The results are promising enough to provide a better simulation of the discontinuous wave propagation problem.

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“…Considerable research efforts have been focused on the dispersion analysis of finite element solutions to the wave or Helmholtz equation [2][3][4][5][6][7][8][9]. The finite element discretization with the consistent mass matrix in general results in a faster phase velocity than the exact propagation velocity, while a lumped mass approximation leads to slower phase velocity [2][3][4][5]10]. In addition, the finite element solutions using uniform meshes show numerical anisotropy, i.e., the solution error depends on the direction considered although the exact wave propagation is the same in all directions [4][5][6][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Transient implicit wave propagation dynamics with the method of finite spheres

Kim

Bathe

2016

Computers & Structures

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The influence of the mass matrix on the dispersive nature of the semi-discrete, second-order wave equation

Cited by 55 publications

References 13 publications

Generalized Fourier analyses of the advection–diffusion equation—Part I: one‐dimensional domains

Generalized Fourier analyses of the advection–diffusion equation—Part I: one‐dimensional domains

Reducing spurious oscillations in discontinuous wave propagation simulation using high-order finite elements

Transient implicit wave propagation dynamics with the method of finite spheres

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