Discussion

We have applied the standard $D10-D30$ method to this problem. We also applied an optimized $D16$ stencil in a $D10-D16$ wave tracking method. The two set of results are very close. These results with $5$ grid points per wavelength on a grid of size $(128,512)$ are compared to the Acousmod2d results at $12.5$ grid points per wave length on a grid of size $(320,1280)$. The results are similar. The Acousmod2d algorithm took $741$ seconds for $900$ time steps. The $D10-D30$ and $D10-D16$ took $805$ and $637$ seconds, resp. The maximum time step for the wavelet tracking algorithms is a factor of $0.6$ of the maximum time step for the Acousmod2d algorithm. The grid size of the Acousmod2d problem is $5/2$ the grid size of the wave-tracking algorithms.

For $D10-D30$ the speed up factor over Ac2d in two dimensions is:

$$(741/805)(0.6)(5/2)=1.3807,$$

and in three dimensions is:

$$(741/805)^{3/2}(0.6)(5/2)=1.3247.$$

For $D10-D16$ the speed up factor over Ac2d in two dimensions is:

$$(741/627)(0.6)(5/2)=1.7727,$$

and in three dimensions is:

$$(741/627)^{3/2}(0.6)(5/2)=1.9272.$$

We found that running $D10-D16$ as a standard wavelet-Galerkin method without wave tracking took $663$ seconds for 900 time steps. For this problem this time is only slightly larger than the wave tracking time. Therefore, we examined on a $(128,512)$ grid the $D10-D10$, $D10-D12$, $D10-D14$ and $D10-D16$ wavelet-Galerkin algorithms with least square optimized stencils for the $D10$, $D12$, $D14$, $D16$ components. These results are shown and they are comparable to the Acousmod2d results at $(320,1280)$. For $900$ time steps the timings are $478$, $538$, $600$, and $663$ seconds, respectively.

For $D10-D16$ the speed up factor over Ac2d in two dimensions is:

$$(741/663)(0.6)(5/2)=1.6765,$$

and in three dimensions is:

$$(741/663)^{3/2}(0.6)(5/2)=1.7723.$$

For $D10-D14$ the speed up factor over Ac2d in two dimensions is:

$$(741/600)(0.6)(5/2)=1.8525.,$$

and in three dimensions is:

$$(741/600)^{3/2}(0.6)(5/2)=2.0587.$$

For $D10-D12$ the speed up factor over Ac2d in two dimensions is:

$$(741/538)(0.6)(5/2)=2.0660,$$

and in three dimensions is:

$$(741/538)^{3/2}(0.6)(5/2)=2.4246.$$

For $D10-D10$ the speed up factor over Ac2d in two dimensions is:

$$(741/478)(0.6)(5/2)=2.3253,$$

and in three dimensions is:

$$(741/478)^{3/2}(0.6)(5/2)=2.8952.$$

In terms of the dispersion, the $D10-D10$ algorithm is better than the $D10-D12$ and $D10-D14$ algorithms. The least square approximation for $D10-D10$ is less oscillatory than the least square approximation for $D10-D12$ and $D10-D14$. This suggest that the numerics are sensitive to the details of the optimization method. This also suggest that better algorithms can be designed with better methods of optimization. We would apply the constrained least square methods commonly used for perfect reconstruction filter bank design. These methods have are useful in digital signal processing applications since they provide perfect reconstruction filter banks with stable quantization properties, and stop band attenuation that can attain $120$ Db down.