Time integration using wavelets
Kevin S. Amaratunga, John R. Williams · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1995
In this work, we describe how wavelets may be used for the temporal discretization of ODEs and PDEs. A major problem associated with the use of wavelets in time is that initial conditions are difficult to impose. A second problem is that a wavelet-based time integration scheme should be stable. We address both of these problems. Firstly, we describe a general method of imposing initial conditions, which follows on from some of our recent work on initial and boundary value problems. Secondly, we use wavelets of the Daubechies family as a starting point for the development of stable time integration schemes. By combining these two ideas we are able to develop schemes with a high order of accuracy. More specifically, the global error is O(hp-1), where p is the number of vanishing moments of the original wavelet. Furthermore, these time integration schemes are characterized by large regions of absolute stability, comparable to increasingly high order BDF methods. In particular, Daubechies D4 and D6 wavelets give rise to A-stable time-stepping schemes. In the present work we deal with single scale formulations. We note, however, that the standard multiresolution analysis for orthogonal wavelets on L2(R) applies here. This opens up interesting possibilities for treating BVPs and IVPs at multiple scales.