Finite volume methods for convection-diffusion problems. (English) Zbl 0862.65062
The paper is a short overview of the nature of one- and two-dimensional convection-diffusion problems and of the use of finite volume methods (FVM) (cell-center (CCM) and cell-vertex (CVM)) in their solution. The advantages and disadvantages of FVM are discussed. The author presents his arguments to answer of the question: CCM or CVM, comparing the two finite volume variants in several ways: is the number of unknows equal to the number of equations; how compact are the schemes; sensitivity to the mesh deformation; parasitic modes and stability.
Reviewer: K.Georgiev (Sofia)
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |
| 35J25 | Boundary value problems for second-order elliptic equations |
Keywords:
research survey; convection-diffusion problems; finite volume methods; sensitivity; mesh deformation; parasitic modes; stabilitySoftware:
WesselingReferences:
| [1] | Hirsch, C., (Numerical Computation of Internal and External Flows, Vol. 1 (1988), Wiley: Wiley Chichester) · Zbl 0662.76001 |
| [2] | Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method (1987), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0628.65098 |
| [3] | Mackenzie, J. A.; Morton, K. W., Finite volume solutions of convection-diffusion test problems, Math. Comp., 60, 189-220 (1992) · Zbl 0797.76072 |
| [4] | Morton, K. W., Finite volume methods and their analysis, (Whiteman, J. R., The Mathematics of Finite Elements and Applications VII, Proc. MAFELAP 1900, Brunel, 1990 (1991), Academic Press: Academic Press London), 189-214 · Zbl 0797.76072 |
| [5] | Morton, K. W.; Stynes, M., An analysis of the cell vertex scheme, Math. Modél. Anal. Numér., 28, 699-724 (1994) · Zbl 0822.65078 |
| [6] | K.W. Morton, M. Stynes and E. Süli, Analysis of cell-vertex methods for convection-diffusion problems, Oxford Univ. Computing Laboratory Report, in preparation.; K.W. Morton, M. Stynes and E. Süli, Analysis of cell-vertex methods for convection-diffusion problems, Oxford Univ. Computing Laboratory Report, in preparation. · Zbl 0885.65121 |
| [7] | H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations; H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations · Zbl 1155.65087 |
| [8] | Shishkin, G. I., Methods of constructing grid approximations for singularly perturbed boundary-value problems. Condensing grid methods, Russian J. Numer. Anal. Math. Modelling, 7, 537-562 (1992) · Zbl 0816.65072 |
| [9] | Wesseling, P., An Introduction to Multigrid Methods (1992), Wiley: Wiley Chichester · Zbl 0760.65092 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
