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

The one-leg, two-step time discretization proposed by Dahlquist, Liniger and Nevanlinna is second order and variable step G-stable. G-stability for systems of ordinary differential equations (ODEs) corrresponds to unconditional, long time energy stability when applied to the Navier-Stokes equations (NSEs). In this report, we analyze the method of Dahlquist, Liniger and Nevanlinna as a variable step, time discretization of the Navier-Stokes equations. We prove that the kinetic energy is bounded for variable time-steps, show that the method is second-order accurate, characterize its numerical dissipation and prove error estimates. The theoretical results are illustrated by several numerical tests.