Efficient removal of boundary-divergence errors in time-splitting methods

A normal mode analysis is presented and numerical tests are performed to assess the effectiveness of a new time-splitting algorithm proposed recently in Karniadakis et al. (1990) for solving the incompressible Navier-Stokes equations. This new algorithm employs high-order explicit pressure boundary conditions and mixed explicit/implicit stiffly stable time-integration schemes, which can lead to arbitrarily high-order accuracy in time. In the current article we investigate both the time accuracy of the new scheme as well as the corresponding reduction in boundary-divergence errors for two model flow problems involving solid boundaries. The main finding is that time discretization errors, induced by the nondivergent splitting mode, scale with the order of the accuracy of the integration rule employed if a proper rotational form of the pressure boundary condition is used; otherwise a first-order accuracy in time similar to the classical splitting methods is achieved. In the former case the corresponding errors in divergence can be completely eliminated, while in the latter case they scale as O(vΔt)_1/2.