Building generalized open boundary conditions for fluid dynamics problems

Abstract

SUMMARY 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 Uloc is computed from the coarse solution Uext, thanks to an OBC formulated as BhUlocMathClass-rel=BHUext, where Bh and BH are discretizations of the same differential operator scriptB (Bh being defined on the fine grid and BH 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 BhUlocMathClass-rel=BHUextMathClass-bin+g, where g is a correction term. When Bh and BH 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 1D Laplace equation and on a 1D shallow water system illustrate the improved efficiency of such a generalized OBC compared with usual ones. Copyright © 2012 John Wiley & Sons, Ltd.