A fast implementation of explicit time-stepping algorithms with the finite element method for a class of nonlinear evolution problems

In this paper a class of nonlinear evolution problems is considered. It is shown that, under special conditions, the application of the product approximation method for nonlinear problems in the finite element method results in constant (i.e., time-independent) matrices. In those cases the amount of computing required to solve these equations with an explicit time-stepping algorithm is decreased considerably compared to the standard Galerkin formulation in which the matrices are time-dependent. The method is applied to two practical two-dimensional problems: the shallow water equations and a nonlinear heat conduction problem.