Four general classes of methods for solving nonlinear partial differential equations on adaptive computational meshes have been developed by our group. Each method uses the adaptive wavelet collocation method (AWCM) based on bi-orthogonal lifted interpolating wavelets to construct a computational grid adapted to the solution. The wavelet decomposition naturally provides a set of nested multi-scale grids adapted to the solution, and we take advantage of this property in developing our methods.

In the first two methods we implement a traditional time marching scheme for parabolic and hyperbolic partial differential equations, but use AWCM to adapt the computational grid to the solution at each time step. When hyperbolic equations are solved an additional wavelet-based procedure for shock capturing is used. With this procedure the mesh is refined in the vicinity of the shock up to a-priori specified resolution and the shock is smoothed out using localized numerical viscosity. The third method simply uses the multi-scale wavelet decomposition as the basis for an adaptive multilevel method for nonlinear elliptic equations. Recently, we have begun to investigate a combination of the first three approaches to produce an adaptive simultaneous space–time method. In this case, both the space-time grid adapts locally to the solution, and the final solution is obtained simultaneously in the entire space–time domain of interest. Our current efforts are focused on further development of the parallel wavelet-based methods with mesh and anisotropy adaptation.