A Wavelet Based Space-Time Adaptive Numerical Method for Partial Di erential Equations

We describe a space and time adaptive numerical method based on wavelet orthonormal bases for solving partial di erential equations. The multiresolution structure of wavelet orthonormal bases provides a simple way to adapt computational re nements to the local regularity of the solution [11] [16]. High resolution computations are performed only in regions where singularities or sharp transitions occur. For many evolution equations it is necessary to adapt the time steps to the spatial resolution in order to maintain the stability and precision of the numerical scheme. We describe an algorithm that modi es the time discretization at each resolution, depending on the structure of the solution. The stability of this space-time adaptive scheme is studied for the heat equation and the linear advection equation. We also explain how this algorithm can be used to solve the one-dimensional Burgers equation with periodic boundary conditions. We present numerical results on the accuracy and complexity of the algorithm. This research was supported by the AFOSR grant AFOSR-90-0040.