APPLICATION OF GENERALIZED WAVELETS - AN ADAPTIVE MULTIRESOLUTION SCHEME

In this paper we add a new member to the family of adaptive multiresolution schemes. These schemes employ adaptive data-dependent reconstruction techniques which use the expensive ENO (essentially nonoscillatory) interpolation only near discontinuities and a simple central stencil everywhere else. In addition, by recursive diadic coarsening of the original fine grid, in smooth regions the fluxes can be interpolated from the coarser mesh, instead of being computed directly. We first present semi-discrete versions of Harten's multiresolution schemes (Harten (1994; to appear)), compare their performance in the nonlinear cases to their fully discrete counterparts, and then test them in the linear case as well. For the latter we find an unexpected oscillatory behavior, which, after a closer examination, is shown to be related to the linear smearing of discontinuities. We shall therefore discuss numerical issues related to the original adaptive multiresolution schemes. In doing so, we propose a new switch between ENO and central interpolation, which provides a truly ''tolerably oscillatory'' interpolation. The resulting algorithm is termed the modified adaptive multiresolution scheme. The new second order central/ENO switch is also proved to ensure that wherever the central stencil is used, it will still result is an essentially nonoscillatory, interpolation.