Numerical Diffraction by a Uniform Grid

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Abstract
In this report we analyze the influence of a spatial discretization on the propagation of a wave generated by a harmonic point source. We consider finite element and finite difference schemes on a uniform grid for the wave equation. We study the asymptotic behavior in space of the elementary solution of the associated Helmholtz equation. This study permits us to define the notions of asymptotic wave fronts (A.W.F.) and asymptotic amplitude diagrams (A.A.D.). We point out some results about the anisotropy and the dispersion of numerical scheme which complete those obtained from the more classical plane wave analysis (see [R. M. Alford, R. Kelly, and D. M. Boore, “Accuracy of finite modeling of the acoustic wave equation,” Biophysics, 6 (1974), pp. 834–842; A. Bamberger, G. Chavent, and P. Lailly, Etude de schémas numériques des équations de l’elastodynamique linéaire, Rapport INRIA No. 41, October 1980; T. Belytschko and R. Mullen, “Dispersion analysis of finite element semi-discretizations of the two-dimensional wave equation,” Internat. J. Numer. Methods Engrg., 18 (1982), pp. 11–19; L. N. Trefethen, “Group velocity in finite difference schemes,” SIAM Rev., 24 (1982), pp. 113–136; L. Nocoletis, Simultation numérique de la propagation d’ondes sismiques, Thèse de Docteur Ingénieur, Université Pierre et Marie Curie, Paris, 1981; R. Vichnevetsky and J. B. Bowles, Fourier Analysis of Numerical Approximation of Hyperbolic Equations, SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1982]), particularly the results concerning the amplitudes. Our study also points out the existence of parasitic phenomena, of complex nature, which cannot be deduced from the plane wave analysis. Thus, the notions of A.W.F. and A.A.D. are new characteristic properties of numerical schemes.

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