Analysis of second order and unconditionally stable BDF2-AB2 method for the Navier-Stokes equations with nonlinear time relaxation

Abstract

In this study, we first consider a second order time stepping finite element BDF2-AB2 method for Navier-Stokes equations (NSE). We prove that the method is unconditionally stable and O(Delta t(2)) accurate. Second, we consider a nonlinear time relaxation model which consists of adding a term "kappa vertical bar u - (u) over bar vertical bar(u - (u) over bar)" to the Navier-Stokes Equations with the algorithm depends on BDF2-AB2 method. We prove that this method is unconditionally stable, too. We applied the BDF2-AB2 method to several numeral experiments including flow around the cylinder. We have also applied BDF2-AB2 method with nonlinear time relaxation to some problems. It is observed that when the equilibrium errors are high, applying BDF2-AB2 with nonlinear time relaxation method to the problem yields lower equilibrium errors.