VECTOR PENALTY-PROJECTION METHOD FOR INCOMPRESSIBLE FLUID FLOWS WITH OPEN BOUNDARY CONDITIONS
Résumé
A new family of methods, the so-called vector penalty-projection methods (VPP$_{r,\epsilon}$), were introduced recently by Angot et al. [1, 2] to compute the solution of unsteady incompressible fluid flows and to overcome most of the drawbacks of the usual incremental projection methods. In this work, we deal with the time-dependent incompressible Stokes equations with outflow boundary conditions using the present method. The spatial discretization is based on the finite volume scheme on a MAC staggered grid and the time discretization is based on the backward difference formula of second-order BDF2 (namely also Gear's scheme). We show that the (VPP$_{r,\epsilon}$) method provides a second-order convergence 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 $0 \leq r \leq 1$ and a penalty parameter $\epsilon$ small enough; typically $\epsilon=10^{-10}$. The resulting constraint on the discrete divergence of velocity is not exactly equal to zero but is satisfied approximately as $O(\epsilon \delta t$) with a penalty parameter $\epsilon$ as small as desired. Finally, the e±ciency and the second-order accuracy of our method are illustrated by several numerical cases.
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