Wavelet-Galerkin and spectral FFT calculations

We solve a series of initial value problems for the potentials:

Let

$$U1=-\vert\sin(2\pi x)\sin(2\pi y)\vert.$$

$$U3=-c3*\vert\sin^3(2\pi x)\sin^3(2\pi y)\vert.$$

$$U4=-c4*\sin^4(2\pi x)\sin^4(2\pi y).$$

The constants normalize the potentials so that $\Vert U1\Vert _2=\Vert U3\Vert _2=\Vert U4\Vert _2$.

The initial conditions are

$$\psi = -2 - \imath 2.$$

We use a $(128,128)$ wavelet-Galerkin discretization with a tolerance for the implicit time step of $1.e-7$.

For the spectral calculation we used the dealiased convolution implemented by the shifted grids procedure. [10].

The potential U1 is shown in Figure 10. The potentials U3 and U4 are shown in Figure 7.

Figure 11 compares for U1 the imaginary component of the wavefunction for the wavelet and spectral methods.

Figure 12 compares for U3 the imaginary component of the wavefunction for the wavelet and spectral methods.

Figure 13 compares for U4 the imaginary component of the wavefunction for the wavelet and spectral methods.

Figure 10: Two harmonic potential wells.

Figure 11: Potential U1: Imaginary component of wavefunction to time step 500. Wavelet and spectral.

Figure 12: Potential U3: Imaginary component of wavefunction to time step 500. Wavelet and spectral.

Figure 13: Potential U4: Imaginary component of wavefunction to time step 500. Wavelet and spectral.