Picard iteration convergence analysis in a Galerkin finite element approximation of the one‐dimensional shallow water equations

Abstract

AbstractA Fourier method for analyzing iteration convergence is applied to four different formulations of the “simple” or Picard iteration procedure of a shallow water model. The model uses the Galerkin linear FE scheme in space and the implicit θ finite differencing in time. Although the numerical scheme is implicit and linearly stable for forward centering in time (θ > 0.5), it is shown that if the Picard iteration procedure is used, there may be additional operating restrictions which need to be observed to achieve iteration convergence. These restrictions effectively limit the time step (Δt) and take the form of upper bounds on θ times the Courant number (θℂ). In the 1D shallow water model, iteration convergence requires θℂ\documentclass{article}\pagestyle{empty}\begin{document}$ \le 1/ \sqrt 3 $\end{document} for two of the four iteration formulations, but there are no effective restrictions for the other two formulations. These results were confirmed by numerical experiments. There is a tradeoff between computer storage and run times. The model version with the shortest run time (due to unrestricted Δt) also requires most computer storage. © 1993 John Wiley & Sons, Inc.