Fourier analysis of semi-discrete and space-time stabilized methods for the advective-diffusive-reactive equation: I. SUPG

The goal of this paper is twofold. One the one side, the most common transient Galerkin and SUPG methods are analyzed for the one-dimensional advection-diffusion-reaction equation. The methods analyzed include semi-discrete, time-discontinuous space-time stabilized finite element methods and several predictor multi-corrector versions of them. On the other hand, in the framework of explicit predictor multi-corrector algorithms a novel treatment of the source terms is proposed and analyzed. The technique consists of the diagonally implicit treatment of the negative or dissipative source terms. This technique increases dramatically the phase and damping accuracy of classical explicit methods and, at the same time, removes the source terms from stability considerations for low viscosity flows, thus, leading to very economic procedures.