Comparison of numerical dispersion errors

The spatial numerical dispersion for the wavelet-Galerkin discretization is faster than the exact (Fourier) dispersion. Therefore, the implicit temporal and wavelet-Galerkin spatial dispersions act in the opposing directions.

The spatial numerical dispersion for the finite difference used by Dablain [4], Shubin and Baker [22] is slower than the exact dispersion. Therefore, the explicit temporal and (Taylor series) finite difference spatial dispersions act in the opposing directions.

The following figure illustrate the two-dimensional acoustic wave equation solution for the wavelet-Galerkin and Acousmod2d program. The Acousmod2d program implements the algorithm described in the paper of Shubin and Bell [22]. The solution is evolving in a constant velocity model. Therefore, the errors are caused by the effects of numerical dispersion. The source term is a Ricker wavelet with about five grid points per minimum wavelength.

Figure 7: Comparison of D10 wavelet and Acousmod2d solution in a constant velocity model.

Figure 8: Comparison of D30 wavelet and Acousmod2d solution in a constant velocity model.

Figure 9: Comparison of the Acousmod2d solution in a constant velocity model with N=128 and N=256.

Figure 10: Comparison of the D10 wavelet solution in a constant velocity model with N=128 and N=256.

Figure 11: Comparison of the D16 wavelet solution at N=128 and D10 wavelet solution at N=256.

Figure 12: Comparison of the D30 wavelet solution at N=128 and D10 wavelet solution at N=256.