Filtered time-stepping method for incompressible Navier-Stokes equations with variable density

Abstract

In this paper, we present a filtered time-stepping method for incompressible Navier-Stokes equations with variable density. The method increases the time accuracy from first order to second order that requires only one backward Euler (BE) solve at each time step followed by adding a time filter. The added time filter was expressed by the linear combinations of the solutions at previous time levels without extra complexity. We proved the stability of density and velocity for fully implicit BE algorithm and backward Euler plus time filter (BETF) algorithm. Moreover, we extend the approach to a variable time stepsize BETF algorithm, and construct a new adaptive BE algorithm and a novel variable stepsize variable order algorithm with low cost error estimators. Finally, numerical experiments show the stability and efficiency of our methods.