Seriani, Géza; Oliveira, Saulo Pomponet DFT modal analysis of spectral element methods for acoustic wave propagation. (English) Zbl 1257.76078 J. Comput. Acoust. 16, No. 4, 531-562 (2008). Summary: Spectral element methods are now widely used for wave propagation simulations. They are appreciated for their high order of accuracy, but are still used on a heuristic basis. In this work we present the numerical dispersion of spectral elements, which allows us to assess their error limits and to devise efficient numerical simulations, particularly for 2D and 3D problems. We propose a novel approach based on a discrete Fourier transform of both the probing plane waves and the discrete wave operators. The underlying dispersion relation is estimated by the Rayleigh quotients of the plane waves with respect to the discrete operator. Together with the Kronecker product properties, this approach delivers numerical dispersion estimates for 1D operators as well as for 2D and 3D operators, and is well suited for spectral element methods, which use nonequidistant collocation points such as Gauss-Lobatto-Chebyshev and Gauss-Lobatto-Legendre points. We illustrate this methodology with dispersion and anisotropy graphs for spectral elements with polynomial degrees from 4 to 12. These graphs confirm the rule of thumb that spectral element methods reach a safe level of accuracy at about four grid points per wavelength. Cited in 5 Documents MSC: 76M22 Spectral methods applied to problems in fluid mechanics 76Q05 Hydro- and aero-acoustics Keywords:spectral element method; dispersion analysis; wave equation PDFBibTeX XMLCite \textit{G. Seriani} and \textit{S. P. Oliveira}, J. Comput. Acoust. 16, No. 4, 531--562 (2008; Zbl 1257.76078) Full Text: DOI References: [1] DOI: 10.1002/nme.1620350604 · Zbl 0764.76033 [2] DOI: 10.1137/S0036142903423460 · Zbl 1074.65112 [3] T. Belytschko and R. Mullen, Modern Problems in Elastic Wave Propagation, eds. J. Miklowitz and J. Achenbach (John Wiley & Sons, New York, NY, 1978) pp. 67–82. [4] Brillouin L., Wave Propagation in Periodic Structures. Electric Filters and Crystal Lattices (1953) · Zbl 0050.45002 [5] Canuto C., Spectral Methods in Fluid Dynamics (1987) · Zbl 0615.65123 [6] DOI: 10.1016/S0065-2687(06)48007-9 [7] DOI: 10.1016/S0045-7825(98)00266-7 · Zbl 0964.76038 [8] DOI: 10.1016/S0168-874X(96)00075-3 · Zbl 0896.65067 [9] DOI: 10.1002/(SICI)1097-0207(19991010)46:4<471::AID-NME684>3.0.CO;2-6 · Zbl 0957.65098 [10] Dhatt G., The Finite Element Method Displayed (1984) · Zbl 0553.65070 [11] DOI: 10.1190/1.1442319 [12] DOI: 10.1002/nme.1308 · Zbl 1085.76036 [13] DOI: 10.1016/S0045-7825(96)01034-1 · Zbl 0898.76058 [14] DOI: 10.1137/S1064827502406403 · Zbl 1042.65028 [15] Horn R., Matrix Analysis (1990) · Zbl 0704.15002 [16] DOI: 10.1046/j.1365-246x.1999.00967.x [17] Lütkepohl H., Handbook of Matrices (1996) · Zbl 0856.15001 [18] DOI: 10.1016/S0024-3795(02)00730-9 · Zbl 1030.65025 [19] DOI: 10.1190/1.1441689 [20] K. Marfurt, Numerical Modeling of Seismic Wave Propagation, Geophysics Reprint Series, eds. K. Kelly and K. Marfurt (Society of Expl. Geophys., Tulsa, OK, 1990) pp. 516–520. [21] DOI: 10.1016/S0168-9274(98)00078-6 · Zbl 0933.65112 [22] DOI: 10.1002/nme.1620180103 · Zbl 0466.73102 [23] DOI: 10.1109/8.362777 [24] DOI: 10.1142/S0218396X97000058 · Zbl 1360.76194 [25] DOI: 10.1016/S0045-7825(98)00057-7 · Zbl 0962.76072 [26] Seriani G., Geophysics 72 pp SM95– [27] DOI: 10.1016/0168-874X(94)90076-0 · Zbl 0810.73079 [28] DOI: 10.1023/A:1007629609576 · Zbl 1076.76565 [29] DOI: 10.1007/BF00350228 · Zbl 0789.73076 [30] DOI: 10.1109/8.362788 · Zbl 0953.78504 [31] DOI: 10.1109/8.486299 [32] DOI: 10.1016/j.finel.2004.12.010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.