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A linearity‐preserving cell‐centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes
Author(s) -
Gao Zhiming,
Wu Jiming
Publication year - 2010
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.2496
Subject(s) - polygon mesh , discretization , mathematics , topology (electrical circuits) , anisotropic diffusion , linearity , discontinuity (linguistics) , vertex (graph theory) , finite volume method , diffusion equation , mathematical analysis , anisotropy , geometry , discrete mathematics , physics , combinatorics , graph , quantum mechanics , mechanics , economics , economy , service (business)
In this paper a finite volume scheme for the heterogeneous and anisotropic diffusion equations is proposed on general, possibly nonconforming meshes. This scheme has both cell‐centered unknowns and vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear weighted combination of the surrounding cell‐centered unknowns, which reduces the scheme to a completely cell‐centered one. We propose two types of new explicit weights which allow arbitrary diffusion tensors, and are neither discontinuity dependent nor mesh topology dependent. Both the derivation of the scheme and that of new weights satisfy the linearity‐preserving criterion which requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is called as the linearity‐preserving cell‐centered scheme and the numerical results show that it maintain optimal convergence rates for the solution and flux on general polygonal distorted meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous. Copyright © 2010 John Wiley & Sons, Ltd.