Adaptive time step control for the incompressible Navier–Stokes equations

Adaptive time stepping is an important tool in Computational Fluid Dynamics for controlling the accuracy of simulations and for enhancing their efficiency. This paper presents a systematic study of three classes of implicit and linearly implicit time stepping schemes with adaptive time step control applied to a 2D laminar flow around a cylinder: θθ-schemes, diagonal-implicit Runge–Kutta (DIRK) methods and Rosenbrock–Wanner (ROW) methods. The time step is controlled using embedded methods. It is shown that several ROW methods clearly outperform the more standard θθ-schemes and the DIRK methods. The results depend on a prescribed tolerance in the time step control algorithm, whose appropriate choice varies from scheme to scheme.