Vector Penalty-Projection Methods for Open Boundary Conditions with Optimal Second-Order Accuracy

Abstract

Recently, a new family of splitting methods, the so-called vector penalty-projection methods (VPP) were introduced by Angot et al. [2, 3] to compute the solution of unsteady incompressible fluid flows and to overcome most of the drawbacks of the usual incremental projection methods. Two different parameters are related to the VPP methods: the augmentation parameter r ≥ 0 and the penalty parameter 0 < e ≤ 1. In this paper, we deal with the time-dependent incompressible Navier-Stokes equations with outflow boundary conditions using the VPP methods. The spatial dis-cretization is based on the finite volume scheme on a Marker and Cells (MAC) staggered grid. Furthermore, two different second-order time discretization schemes are investigated: the second-order Backward Difference Formula (BDF2) known also as Gear's scheme and the Crank-Nicolson scheme. We show that the VPP methods provide a second-order convergence rate for both velocity and pressure in space and time even in the presence of open boundary conditions with small values of the augmentation parameter r typically 0 ≤ r ≤ 1 and a penalty parameter e small enough typically e = 10 −10. The resulting constraint on the discrete divergence of velocity is not exactly equal to zero but is satisfied approximately as O(e δt) where e is the penalty parameter (taken as small as desired) and δt is the time step. The choice r = 0 requires a special attention to avoid the accumulation of round-off errors for very small values of e. Indeed, it is important in this case to directly correct the pressure gradient by taking account of the velocity correction issued from the vector penalty-projection step. Finally, the efficiency and the second-order accuracy of the method are illustrated by several numerical test cases including homogeneous or non-homogeneous given traction on the boundary.