Previous to this Phase I SBIR we have applied the Wavelet-Galerkin method to several classes of problems that exhibit phenomena of an interesting nature. Our work appears to be the first application of the compactly-supported Wavelet-Galerkin procedure to nonlinear equations. To correctly apply the Galerkin procedure with compactly supported wavelets requires methods for the exact evaluation of functionals of compact-wavelets. Based on our earlier work on the foundations of wavelets we can now precisely evaluate the functional required for implementation of wavelet methods.
For turbulence we ran simulations of the two dimensional Euler flow and examined the phenomena related to vortex merger, filamentation, the gradients of vorticity and the long time limit. Our initial results show that the Wavelet-Galerkin method is as accurate as the fully-dealiased Spectral method, is far more stable, and, for each time step, (depending on the specific wavelet basis, algorithm and state of optimization), can be faster than the Spectral method [58]. It seems that the wavelet discretization allows the implementation of numerical schemes for Euler flow that are useful for long time studies and that cannot be implemented by Spectral or finite difference methods.
The basic numerical lesson is that scaling function expansions subdue the Gibbs's phenomenon, allow the stable calculation with oscillatory approximations that occur in inviscid calculations, and allow the efficient, iterative implementation of implicit time differencing methods. Compact-support and the ability of scaling functions to exactly approximate features smoother than themselves, and hence to be able to better approximate discontinuous features, is essential. The orthogonality of the translated scaling function basis also guarantees the conservative approximation of the quadratic integrals (energy and enstrophy) by the Galerkin approximation to the Jacobian nonlinearity. The implicit time differencing with the Wavelet-Galerkin space discretization allows the control of, i.e. conserves, or monotonically increases or decreases, the Energy and/or Enstrophy. This is a quite useful property [58,60]. It does seem clear that wavelet based numerical methods for turbulence simulations hold considerable promise and warrant further development.
We have also developed a wavelet-based method for the solution of boundary value problems in arbitrary geometries. This method (the Wavelet-Capacitance Method) is defined by a nontrivial extension of the classical Capacitance Matrix method, and, unlike the classical method, can be spectrally accurate [47]. We have used the Wavelet-Capacitance method to numerically resolve the long term limit of two-dimensional Euler and Navier-Stokes flows in rectangular and L-shaped domains. This has considerable relevance to the qualitative behavior of two-dimensional turbulence [47,60] and practical applications of the method to problems in engineering. In this study we show that the Wavelet-Capacitance Matrix method defines a fast, general method for solving the compressible Euler and Navier-Stokes equations in nonseparable domains.