Low-dimensional Krylov subspace iterations for enhancing stability of time-step integration schemes

Abstract

Inaconventional integration scheme of the Predictor#Corrector #PC# type, solution on the next time layer is obtained by solving the Corrector scheme equation with few #usually one# iterative steps of Richardson's method where the initial guess is taken from the Predictor scheme. Aiming to enhance stabilityofsuchascheme by performing a few optimal Krylov subspace iterations #e.g., k steps of GMRES, k 6 5# instead of Richardson's method steps, we get a family of Minimal Residual PC #MR-PC# time step integration schemes. The optimality #residual reduction# property of iterativeschemes like GMRES leads to a scheme whichis closest in the residual sense to the implicit Corrector scheme. Two particular MR-PC schemes are investigated here: Forward Euler Predictor # Backward Euler Corrector #of the #rst order# and Adams#2# Predictor # BDF2 Corrector #of the second order#. Practical aspects of using MR-PC scheme including adaptive step size control strategy will be discussed.