On the Wavelet Optimized Finite Difference Method

When one considers the e ect in the physical space, Daubechies-based wavelet methods are equivalent to nite di erence methods with grid re nement in regions of the domain where small scale structure exists. Adding a wavelet basis function at a given scale and location where one has a correspondingly large wavelet coe cient is, essentially, equivalent to adding a grid point, or two, at the same location and at a grid density which corresponds to the wavelet scale. This paper introduces a waveletoptimized nite di erence method which is equivalent to a wavelet method in its multiresolution approach but which does not su er from di culties with nonlinear terms and boundary conditions, since all calculations are done in the physical space. With this method one can obtain an arbitrarily good approximation to a conservative di erence method for solving nonlinear conservation laws. This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681. Research was also supported by AFOSR grant 93-1-0090, by DARPA grant N00014-91-4016, and by NSF grant DMS-9211820.