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A Nominally Second-Order Cell-Centered Finite Volume Scheme for Simulating Three-Dimensional Anisotropic Diffusion Equations on Unstructured Grids

Published online by Cambridge University Press:  03 June 2015

Pascal Jacq ,
Pierre-Henri Maire  and
Rémi Abgrall
Pascal Jacq*
Affiliation:
CEA/CESTA, 15 Avenue des Sablières CS 60001, 33116 Le Barp cedex, France
Pierre-Henri Maire*
Affiliation:
CEA/CESTA, 15 Avenue des Sablières CS 60001, 33116 Le Barp cedex, France
Rémi Abgrall*
Affiliation:
Institüt für Mathematik, Universität Zürich, CH-8057 Zürich, Switzerland
*
Email:pascal.jacq@inria.fr
Corresponding author.Email:maire@celia.u-bordeaux1.fr
Email:remi.abgrall@math.uzh.ch
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Abstract

We present a finite volume based cell-centered method for solving diffusion equations on three-dimensional unstructured grids with general tensor conduction. Our main motivation concerns the numerical simulation of the coupling between fluid flows and heat transfers. The corresponding numerical scheme is characterized by cell-centered unknowns and a local stencil. Namely, the scheme results in a global sparse diffusion matrix, which couples only the cell-centered unknowns. The space discretization relies on the partition of polyhedral cells into sub-cells and on the partition of cell faces into sub-faces. It is characterized by the introduction of sub-face normal fluxes and sub-face temperatures, which are auxiliary unknowns. A sub-cellbased variational formulation of the constitutive Fourier law allows to construct an explicit approximation of the sub-face normal heat fluxes in terms of the cell-centered temperature and the adjacent sub-face temperatures. The elimination of the sub-face temperatures with respect to the cell-centered temperatures is achieved locally at each node by solving a small and sparse linear system. This systemis obtained by enforcing the continuity condition of the normal heat flux across each sub-cell interface impinging at the node under consideration. The parallel implementation of the numerical algorithm and its efficiency are described and analyzed. The accuracy and the robustness of the proposed finite volumemethod are assessed bymeans of various numerical test cases.

Keywords

65M0865F1068W1076R50Finite volume methodsunstructured gridsanisotropic diffusionparallel computing
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 4 , October 2014 , pp. 841 - 891
Copyright
Copyright © Global Science Press Limited 2014

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