High-Order Adaptive Time Stepping for the Incompressible Navier--Stokes Equations

In this paper we develop a high-order time stepping technique for the incompressible Navier--Stokes equations. The method is based on an artificial compressibility perturbation made high order by using a Taylor series technique. The method is suitable for time step control. It is unconditionally stable in the case of the unsteady Stokes equations and conditionally stable for the full Navier--Stokes equations. The numerical results presented in the paper suggest that the stability condition in the second case is of CFL type; i.e., the time step should be of the order of the ratio of the meshsize and the magnitude of the velocity. In principle, the technique can be developed to any order in time. We illustrate the idea by giving the third-order version of the methodology. We numerically illustrate the third-order convergence rate of the method on a manufactured solution. The scheme converges with time steps randomly chosen at each time level as the size of the average time step decreases. We also demonstrate the efficiency of a simple time step control on a realistic incompressible flow in 2D.