A multilevel wavelet collocation method for solving partial differential equations in a finite domain (Q1902643)

Numerical solution of partial differential equations using wavelet bases allows combining the advantages of both spectral and difference (finite element) bases. There are two difficulties in applying this method: the treatment of nonlinearities and that of some boundary conditions. In order to tackle the last problem, the authors propose two different approaches, based on a wavelet interpolation technique developed by them. The following three issues are addressed in the paper: How to incorporate the idea of collocation with wavelet bases to construct an effective algorithm of solving partial differential equations, how to deal with boundary conditions and how to construct a stable, accurate, and efficient numerical algorithm. The methods are tested on the one-dimensional Burgers equation with small viscosity and the results indicate that they are competitive with well- established numerical algorithms.