Numerical Diffraction by a Uniform Grid

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 schemas numeriques des equations de l’elastodynamique lineaire, Rapport INRIA No. 41, October 1980; T. Belytschko and R. Mullen, “Dispersion analysis of finite element semi-discretizations of the two-dimen...