Symmetric model with nonreflecting interface

The test problem is a periodic square of size $(256,256)$ with a square central region of size $(123,123)$. The wave speed in the central region is $4$, the density is $1/4$. In the exterior region, the wave speed and density are $1$. The figure shows this velocity model when scaled to geophysical coordinates. The initial pressure field is $(\sin(x)\sin(y))^{120}$. This is a band limited function with

$$\sin^{120}(x)=$$

$$\frac{1}{2^{119}}\left(\cos(120x)-\binom{120}{1}\cos(118x) +\bino... ...}\cos(116x)-\cdots -\binom{120}{59}\cos(2x) + \frac{1}{2}\binom{120}{60}\right)$$

The nonreflecting boundary conditions are used since the errors are at the interface made obvious. The symmetric model is used for the same reason. The dynamics ideally preserve the symmetry. Any deviation from symmetry can be easily measured and is a clear indication of numerical stability. The symmetry allows a clearer visualization of the solution. With periodic boundary conditions the solution evolves from a central source, higher speed region into a lower speed exterior region. The periodic boundary conditions bring the solution back to the higher speed region with a greater level of complexity. With further cycles the solution develops a kaleidoscopic and arbitrarily complex structure. The level of complexity is controlled by the number of time cycles calculated. This is useful for quantifying dispersion errors and errors over multiple interactions at the (nonreflecting) interface.

For this problem we find there are large differences between the acousmod2d solution and the wavelet-Galerkin solution after one cycle. The wave front moves out of the higher speed region into the lower speed region. At this stage the solutions are similar. When the wave reenters the higher speed region, large differences appear. These differences are to be related to the accuracy of the interface calculation.

To resolve these differences we scale the problem data (model and initial condition) to increase the spatial-temporal discretization while solving the same underlying problem. We observe the convergence of the acoousmod2d and wavelet-Galerkin solutions with increasing levels of discretization, and continue until convergence is observed for both methods. This provides reliable and self consistent data for quantifying the relative efficiency of the different methods.

Both methods converged to the same solution. The wavelet-Galerkin method converged much faster. We apply the wavelet-Galerkin method on problems of size $(256,256)$ , $(340,340)$, $(424,424)$, and $(512,512)$. The acousmod2d algorithm was applied to problems of size $(256,256)$, $(340,340)$, $(424,424)$, $(512,512)$, $(768,768)$, $(1024,1024)$, and $(1280,1280)$.

To see the nature of the solution the next set of figures show the cross sections of the Acousmod2d, D6 and D10 wavelet-Galerkin solutions through the center of symmetry. We remark that the D10 solution (shown in blue) is converged at this discretization of $(512,512)$.

Figure 38: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Initial Data.

Figure 39: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=200.

Figure 40: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=400.

Figure 41: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=600.

Figure 42: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=800.

Figure 43: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=1000.

Figure 44: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=1200.

Figure 45: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=1400.

Figure 46: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=1600.

Figure 47: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=1800.

Figure 48: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=2000.

Figure 49: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=2200.

Figure 50: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=2400.

Figure 51: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=2600.

Figure 52: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=2800.

Figure 53: Comparison of solutions. Ac2d, Red. D6-D16 Wavelet, Green. D10-D16, Blue. N=512. Time=3000.

The solution at time step $2600$ shows the largest differences. We examine the solution at this time step for the range of discretizations. The converged solution at this time step is shown. The next series of figures show the cross sections for the the Acousmod2d, D6 and D10 solutions.

Figure 54: Reference Solution. D10-D16 wavelet-Galerkin. Time=2600.

Figure 55: Acousmod2d solution for N=256,340,424,512,768,1024,1280. Time=2600.

Figure 56: D6-D10 solution for N=256,340,424,512. Time=2600.

Figure 57: D10-D10 solution for N=256,340,424,512. Time=2600.

The Figure 58 directly compares the Acousmod2d and D10-D10 solutions. By inspection the Acousmod2d solution at $N=1280$ is closest to the D10-D10 solution at $N=424$.

Figure 58: Comparison of D10 wavelet-Galerkin and Acousmod2d solutions. D10, Red. Ac2d, Green.

Figure 59: Comparison of different methods at $N=340$. Acousmod2d, D6-D16 wavelet-Galerkin, explicit time step D10-D10 wavelet-Galerkin, implicit time step D10-D10 wavelet-Galerkin, implicit time step D10-D16 wavelet-Galerkin.

Figures 59 and 60 compare different methods at $N=340$ and $N=512$.

Figure 60: Comparison of D10-D10 wavelet-Galerkin solution with implicit time stepping at $N=424,512$ and explicit time stepping at $N=512$.

Figure 61: Comparison of Acousmod2d and D10-D10 wavelet-Galerkin solution. At $N=256,340,424,512,768$ the Ricker second derivative source term has $9.9,13.2,16.5,19.8,29.8$ grid points per minimum wavelength.

Figure 62: Comparison of Acousmod2d and D10-D10 wavelet-Galerkin solution. At $N=424,1024$ the Ricker second derivative source term has $16.4,39.7$ grid points per minimum wavelength.

Figure 63: Comparison of Acousmod2d and D10-D10 wavelet-Galerkin solution. At $N=424,1280$ the Ricker second derivative source term has $16.4,49.7$ grid points per minimum wavelength.