Abstract
We propose a new second-order finite volume scheme for non-homogeneous and anisotropic diffusion problems based on cell to vertex reconstructions involving minimization of functionals to provide the coefficients of the cell to vertex mapping. The method handles complex situations such as large preconditioning number diffusion matrices and very distorted meshes. Numerical examples are provided to show the effectiveness of the method.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berthon, C., Coudière, Y., Desveaux, V.: Second-order MUSCL schemes based on dual mesh gradient reconstrution (DMGR), to be published in ESAIM: Mathematical Modelling and Numerical Analysis (2014)
Bertolazzi, E., Manzini, G.: Least square-based finite volumes for solving the advection diffusion of contaminants in porous media. Applied Numerical Mathematics 51, 451–461 (2004)
Bertolazzi, E., Manzini, G.: On vertex reconstructions for cell-centered finite volume approximations of 2D anisotropic diffusion problems. Math. Models and Meth. in App. Sci. 17(1), 1–32 (2007)
Costa, R., Clain, S., Machado, G.J.: New cell-vertex reconstruction for finite volume scheme: application to the convection-diffusion-reaction equation (under review)
Chandrashekar, P., Garg, A.: Vertex-centroid finite volume scheme on tetrahedral grids for conservation laws. Computers and Mathematics with Applications 65, 58–74 (2013)
Clain, S., Machado, G.J., Nóbrega, J.M., Pereira, R.M.S.: A sixth-order finite volume method for the convection-diffusion problem with discontinuous coefficients. Computer Methods in Applied Mechanics and Engineering 267, 43–64 (2013)
Coudière, Y., Vila, J.P., Villedieu, P.: Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. M2AN 3(33), 493–516 (1999)
Domelevo, K., Omnes, P.: A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, M2AN Math. Model. Numer. Anal. 39, 1203–1249 (2005)
Hermeline, F.: A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160, 481–499 (2000)
Hidalgo, A., Dumbser, M.: ADER Schemes for Nonlinear Systems of Stiff Advection-Diffusion-Reaction Equations. J. Sci. Comput. 48(1-3), 173–189 (2011)
Holmes, D.G., Connel, S.D.: Solution of the 2D Navier-Stokes equations on unstructured adaptive grids, AIAA Paper 89-1392 (1989)
Ivan, L., Groth, C.P.T.: High-order solution-adaptative central essentially non-oscillatory (CENO) method for viscous flows, AIAA Paper 2011-367 (2011)
Ollivier-Gooch, C.: High-Order ENO schemes for unstructured Meshes based on least-squares reconstruction, AIAA Paper 97-0540 (1997)
Rausch, R.D., Batina, J.T., Yang, H.T.Y.: spatial adaptation procedure on unstructured meshes for accurate unsteady aerodynamic flow computation, AIAA Paper 91-1106 (1991)
Frink, N.T.: Three-dimensional upwind scheme for solving the Euler equations on unstructured tetrahedral grids, Ph. D Dissertation, Virginia Polytechnic Institute and state university (1991)
Frink, N.T.: Upwind scheme for solving the Euler equations on unstructured tetrhedral meshes. AIAA Journal 1, 70–77 (1992)
Frink, N.T.: Recent progress toward a three-dimensional unstructured Navier-Stokes flow solver, AIAA Paper 94-0061 (1994)
Jawahar, P., Kamath, H.: A high-resolution procedure for Euler and Navier-Stokes computations on unstructured Grids. J. Comp. Phys. 164, 165–203 (2000)
Eymard, R., Gallouët, T., Herbin, R.: Finite volume approximation of elliptic problems and convergence of an approximate gradient. Applied Numerical Mathematics 37, 31–53 (2001)
Ollivier-Gooch, C., Van Altena, M.: A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation. Journal of Computational Physics Archive 181(2), 729–752 (2002)
Herbin, R., Hubert, F.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: Finite Volumes for Complex Applications V, pp. 659–692 (2008)
Gao, Z., Wu, J.: A linearity-preserving cell-centred scheme for the heterogeneous and anisotropic diffusion equations on general meshes. Int. J. Numer. Meth. Fluids 67, 2157–2183 (2011)
Manzini, G., Putti, M.: Mesh locking effects in the finite volume solution of 2-D anisotropic diffusion equations. Journal of Computational Physics 220, 751–771 (2007)
Christophe, L.P., Hai, O.T.: A Cell-Centered Scheme For Heterogeneous Anisotropic Diffusion Problems On General Meshes. IJFV International Journal On Finite Volumes 8, 1–40 (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Costa, R., Clain, S., Machado, G.J. (2014). Finite Volume Scheme Based on Cell-Vertex Reconstructions for Anisotropic Diffusion Problems with Discontinuous Coefficients. In: Murgante, B., et al. Computational Science and Its Applications – ICCSA 2014. ICCSA 2014. Lecture Notes in Computer Science, vol 8579. Springer, Cham. https://doi.org/10.1007/978-3-319-09144-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-09144-0_7
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09143-3
Online ISBN: 978-3-319-09144-0
eBook Packages: Computer ScienceComputer Science (R0)