Schemes of temporal adaptability in wavelet context

Abstract

In the last decades many space adaptive methods have been developed in the context of wavelets for time dependent partial differential equations. More recently, these techniques are also used in combination with time adaptive schemes, because the space refinement requires small time steps for stability reasons while using explicity or semi-explicity schemes.The challenge of these fully space and time adaptive techniques is to compute the time evolution of the numerical solution on a locally refined mesh with as large as possible time steps which are varying with the local step size in the space.The aim is to fully resolve the dynamics of the modeled non--linear phenomena in space and time and to benefit from their intermittent behaviour. This research project proposed to develop and to analyse the adaptive space discretization of PDEs originally proposed by Ami Harten in the middle of 90 and some recent techniques for time adaptivity which have been developed in the context of adaptive wavelet methods.The main ideas are to develop new time adaptive techniques which choose automatically the time step and allow a local error estimation.Hence this advanced numerical algoritms will permit to control the error in time without loosing computational efficiency.The outcome of this research project has different large scale applications in atmospheric and space sciences like weather forecasting, magneto-hydrodynamics and combustion.