Temporal numerical dispersion.

Figure 1: The numerical dispersion.

Following Dablain [4] we consider the temporal numerical dispersion for the one-dimensional wave equation

$$\psi_{tt}=c_0^2\psi_{xx}.$$

Substitution of wave form $\psi=e^{\imath(kx+\omega t)}$ into the second order explicit time differencing

$$\psi_{n+1}-2\psi_n+\psi_{n-1}=(dt)^2c_0^2\psi_{n,xx},$$

where $\psi_n=\psi(x,ndt)$

$$c=\frac{\omega}{k}$$

$$\theta=\omega dt$$

defines the numerical dispersion relation

$$\frac{c^2}{c_0^2}=\frac{\theta^2}{2(1-cos\theta)}$$

Substitution of wave form $\psi=e^{\imath(kx+\omega t)}$ into the second order implicit time differencing

$$\psi_{n+1}-2\psi_n+\psi_{n-1}=\frac{(dt)^2c_0^2}{4}(\psi_{n+1,xx}+ 2\psi_{n,xx} + \psi_{n-1,xx}),$$

where $\psi_n=\psi(x,ndt)$ defines the numerical dispersion relation

$$\frac{c^2}{c_0^2}=\frac{\theta^2}{4}\frac{1+cos\theta}{1-cos\theta}$$

Therefore, in the range $-\pi < \theta < \pi$, the explicit temporal dispersion is faster than the exact dispersion, and the implicit temporal dispersion is slower than the exact dispersion. Dablain [4] shows that the spatial dispersion for centered finite differences is slower than the exact dispersion. We will show that the wavelet-Galerkin procedure defines a spatial dispersion that is faster than the exact dispersion.