A wavelet-based spatially and temporally adaptive numerical method for partial differential equations and its application to the solution of the heat flow problem in crystal growth

A wavelet-based spatially and temporally adaptive numerical method for partial differential equations has been implemented and computationally validated in two dimensions. The method has been applied to the solution of the heat equation in the context of modeling dendritic crystal growth. The multi-resolution structure of wavelet bases provides an effective framework in which to construct numerical algorithms which automatically adapt to the evolution of the solution. The implementation has been written in C++ so that the data structures used may closely model the mathematical objects they represent. The results of computations to validate the method and to simulate dendritic growth are presented. The results are also compared to those obtained by Roosen who used finite difference schemes to solve the heat-flow problem. For low grid densities, the wavelet-based method applied to crystal growth does not outperform finite difference schemes, but for sufficiently high grid densities, it does. This is the first time that a multilevel approach has been used to compute heat flow in crystal growth.