Filter bank methods for hyperbolic PDEs

We use biorthogonal filter banks to solve hyperbolic PDEs adaptively with a sparse multilevel representation of the solution. The methods described are of finite difference type, and the filter banks are used to give a sparse representation of signals and to transform between grids on different scales. We derive bounds for the error and number of coefficients in the sparse representation. These bounds also apply for filter banks that are not associated with any wavelets. We develop algorithms for fast differentiation and multiplication in detail. The strength of the method is shown in various test problems.