Discussion

We have examined the solutions when and where the sup norm differences were largest and increased the discretization to observe the rate of convergence. For the two data presented the generic $D10$ wavelet-Galerkin solution with explicit time differencing has converged at $N=512$. The Acousmod2d solution at $N=1280$ was closest to the wavelet solution. We profiled both codes on a Pentium Pro running Linux using the gprof tool. At $N=424$ the $D10$ wavelet-Galerkin took $43.39$ seconds for one hundred basic time steps. At $N=512$ the $D10$ wavelet-Galerkin took $73.48$ seconds for one hundred basic time steps. At $N=512$ the Acousmod2d algorithm took $62.5$ seconds for one hundred basic time steps. At $N=1280$ the Acousmod2d algorithm took $392.40$ seconds for one hundred basic time steps.

Due to stability considerations the basic, explicit time step for the wavelet-Galerkin method is $6/10$ of the basic time step for the Acousmod2d method.

The explicit time step D10 wavelet algorithm at $N=512$ matches the Acousmod2d algorithm at $N=1280$. For explicit time steps the speed up factor in two dimensions is:

$$(392/73)(6/10)(1280/512)=8.054$$

and in three dimensions would be

$$(392/73)^{3/2}(6/10)(1280/512)=18.6653$$

The implicit time step wavelet D10 wavelet algorithm at $N=424$ matches the Acousmod2d algorithm at $N=1280$. For implicit time steps the speed up factor in two dimensions is:

$$(392/48)(1/2)(6/10)(1280/424)=7.3962$$

and in three dimensions would be:

$$(392/48)^{3/2}(1/2)(6/10)(1280/424)=21.1365$$