Vector penalty-projection methods for outflow boundary conditions with optimal second-order accuracyReport as inadecuate

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* Corresponding author 1 I2M - Institut de Mathmatiques de Marseille 2 Analyse Applique I2M - Institut de Mathmatiques de Marseille

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 < 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 small enough typically = 10 10. The resulting constraint on the discrete divergence of velocity is not exactly equal to zero but is satisfied approximately as O t where is the penalty parameter taken as small as desired and t 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 . 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.

Keywords : Vector penalty-projection methods Navier-Stokes equations Incompressible viscous flows Open or outflow boundary conditions Traction Neumann boundary conditions Second-order accuracy

Author: Philippe Angot - Rima Cheaytou -



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