Time-integration for incompressible viscous flows: Stepsize and order selection based on the BDF

Abstract

This paper presents a procedure based on the Backward Differentiation Formulas to obtain efficient time integration of the incompressible Navier-Stokes equations. The adaptive algorithm performs both stepsize and order selection to control respectively the solution accuracy and the computational efficiency of the time integration process. The stepsize selection (h-adaptivity) is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance. The order selection (p-adaptivity) relies on the idea that low-accuracy solutions can be computed efficiently by low order time integrators while highly accurate solutions require high order time integrators to keep computational time low. Hence, the algorithm selects the most appropriate method within the formulas of order 1 to 5 based on the prescribed solution accuracy and equation stiffness. Doing so it guarantees that the variable stepsize BDF methods used always are stable during the whole time integration interval. The adaptive algorithm behaviors and performances are illustrated on the flow over a circular cylinder at low Reynolds numbers.