Adaptive Wavelet Methods for Dierential Equations
Erik Henningsson · 2010
In this thesis we study stable Wavelet Galerkin schemes for solving initial value problems. The methods use the Daubechies family of orthogonal wavelets as bases for the solution. Their order of convergence is p− 1, where p is the number of vanishing moments of the wavelet function. Their stability regions are similar to those for BDF multistep methods of the same order. Due to the nature of the wavelets and the Galerkin procedure the discretization gives a regular grid. We propose methods to use time density control to make the schemes adaptive. A few examples are solved with our new, adaptive methods. These indicate that order of convergence is preserved if the time density sequence is smooth enough. The time density can be computed during numerical integration by a local error tracker. We demonstrate with examples that this technique works well up to order four. We also advance the Wavelet Galerkin schemes by proposing a many times more efficient way of handling the right hand side function of the initial value problem. The resulting new integration schemes are shown to preserve the order of the method but, for the higher orders, lose the excellent stability properties.