Nonlinear Galerkin Method with Multilevel Incremental Unknown

Abstract

Multilevel methods are indispensable for the approximation of nonlinear evolution equations when complex physical phenomena involving the interaction of many scales are present (such as in, but without being limited to fluid turbulence). Incremental unknowns of different types have been proposed as a means to develop such numerical schemes in the context of finite difference discretizations. In this article, we present several numerical schemes using the so-called multilevel wavelet-like incremental unknowns. The fully discretized explicit and semi-explicit schemes for reaction-diffusion equations are presented and analyzed. The stability conditions are improved when compared with the corresponding standard algorithms. Furthermore the complexity of the computation on each time step is comparable to the corresponding standard algorithm.