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A robust and efficient finite volume scheme for the discretization of diffusive flux on extremely skewed meshes in complex geometries. (English) Zbl 1169.65344

Summary: An improved finite volume scheme to discretize diffusive flux on a non-orthogonal mesh is proposed. This approach, based on an iterative technique initially suggested by Khosla [P. K. Khosla and S. G. Rubin, Comput. Fluids 2, 207–209 (1974; Zbl 0335.76009)] and known as deferred correction, has been intensively utilized by S. Muzaferija [Adaptative finite volume method for flow prediction using unstructured meshes and multigrid approach, Ph.D. Thesis, Imperial College (1994)] and later J. H. Fergizer and M. Peric [Computational methods for fluid dynamics. Berlin: Springer (2002; Zbl 0998.76001)] to deal with the non-orthogonality of the control volumes. Using a more suitable decomposition of the normal gradient, our scheme gives accurate solutions in geometries where the basic idea of Muzaferija fails. First, the performances of both schemes are compared for a Poisson problem solved in quadrangular domains where control volumes are increasingly skewed in order to test their robustness and efficiency. It is shown that convergence properties and the accuracy order of the solution are not degraded even on extremely skewed mesh. Next, the very stable behavior of the method is successfully demonstrated on a randomly distorted grid as well as on an anisotropically distorted one. Finally, we compare the solution obtained for quadrilateral control volumes to the ones obtained with a finite element code and with an unstructured version of our finite volume code for triangular control volumes. No differences can be observed between the different solutions, which demonstrates the effectiveness of our approach.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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