Building Generalized Open Boundary Conditions for Fluid Dynamics Problems
Résumé
This paper deals with the design of an efficient open boundary condition (OBC) for fluid dynamics problems. Such problematics arise, for instance, when one solves a local model on a fine grid that is nested in a coarser one of greater extent. Usually the local solution $U^{\hbox{\scriptsize loc}}$ is computed from the coarse solution $U^{\hbox{\scriptsize ext}}$ thanks to an OBC formulated as $B_hU^{\hbox{\scriptsize loc}}=B_HU^{\hbox{\scriptsize ext}}$, where $B_h$ and $B_H$ are discretizations of the same differential operator ${\cal B}$ ($B_h$ being defined on the fine grid and $B_H$ on the coarse grid). In this paper we show that such an OBC cannot lead to the exact solution, and we propose a generalized formulation $B_hU^{{\hbox{\scriptsize loc}}}=B_HU^{\hbox{\scriptsize ext}}+g$, where $g$ is a correction term. When $B_h$ and $B_H$ are discretizations of a transparent operator, $g$ can be computed analytically, at least for simple equations. Otherwise we propose to approximate $g$ by a Richardson extrapolation procedure. Numerical test cases on a 1-D Laplace equation and on a 1-D shallow water system illustrate the improved efficiency of such a generalized OBC compared to usual ones.
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