Analysis of the variable step method of Dahlquist, Liniger and Nevanlinna for fluid flow

Abstract: The two-step time discretization proposed by Dahlquist, Liniger and Nevanlinna is variable step $G$-stable. (In contrast, for increasing time steps, the BDF2 method loses $A$-stability and suffers non-physical energy growth in the approximate solution.) While unexplored, it is thus ideal for time accurate approximation of the Navier-Stokes equations. This report presents an analysis, for variable time-steps, of the method's stability and convergence rates when applied to the NSE. It is proven that the method is variable step, unconditionally, long time stable and second order accurate. Variable step error estimates are also proven. The results are supported by several numerical tests.