Remark 8.

Depending on the wavelet systems used, profiling of the code reveals that seventy to ninety percent of the cpu time per time step is used to evaluate the convolutions for the wavelet-Galerkin, Laplacian operator. The size of the convolution kernel increases with the order of the wavelet system.

To economize the use of higher order wavelet systems for the solution of the wave equation, we examined the integration of compression with the the evaluation of the wavelet-Galerkin operators.

The idea is to compress the solution, throw away the high pass components, apply the wavelet-Galerkin convolution operators to the low pass component, and reassemble the convolved field with the inverse wavelet transform.

This approach did not work the two dimensional, Daubechies wavelet transform. The Daubechies wavelet transform is not homogeneous. It introduces a systematic bias in certain directions that upon iteration will distort the solution and produce an unacceptable result. For instance, consider a field with the symmetry of a square. We apply the two dimensional wavelet transform this field, and apply the inverse wavelet transform to the low pass component. The result does not have the symmetry of the square. In fact, even for a field without symmetry, low pass filtering using the two dimensional will introduce a systematic bias. This bias initially will be small. However, through iteration it can produce a distorted result. This is especially important when a wave interacts with a material interface.

Figure 13: Low pass filtering breaks the symmetry of the two dimensional Daubechies wavelet transform.

The Daubechies wavelet transform breaks symmetry because the Daubechies wavelets are themselves asymmetrical basis functions. To get around this problem we applied the symmetrical, biorthogonal wavelet transform [7]. This almost works. We tested the integration of compression with the evaluation of wavelet-Galerkin operators by solving the wave equation on region with the symmetry of a square. The central region (square) has a wave speed of four. The exterior region has a wave speed of one. The interface condition is nonreflecting.The low pass filter evaluation of the Wavelet-Galerkin Laplacian preserves symmetry and produces an accurate evolving wave front. However, after the wave front passes through the interface, the trailing edge of the wave that is still in the high speed region develops an oscillatory instability that grows with time. The instability is not dependent on the time step size. It begins at the interface and propagates back into the high speed region. The wave front continues to evolve in the low speed region without any apparent distortion. The reason for the instability is not readily apparent and would be of considerable interest.

Figure 14: Biorthogonal Low Pass wavelet-Galerkin solution.

Figure 15: Biorthogonal Low Pass wavelet-Galerkin solution. Growth of the instability.

Figure 16: Biorthogonal Low Pass wavelet-Galerkin solution. Growth of the instability.