Analysis of a cellvertex finite volume method for convectiondiffusion problems
Authors:
K. W. Morton, Martin Stynes and Endre Süli
Journal:
Math. Comp. 66 (1997), 13891406
MSC (1991):
Primary 65N99, 65L10; Secondary 76M25
MathSciNet review:
1432132
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A cellvertex finite volume approximation of elliptic convectiondominated diffusion equations is considered in two dimensions. The scheme is shown to be stable and secondorder convergent in a meshdependent norm.
 1.
P. Balland and E. Süli, Analysis of the cell vertex finite volume method for hyperbolic equations with variable coefficients, SIAM J. Numer. Anal. 34, No. 3, June 1997.
 2.
P.
I. Crumpton, J.
A. Mackenzie, and K.
W. Morton, Cell vertex algorithms for the compressible
NavierStokes equations, J. Comput. Phys. 109 (1993),
no. 1, 1–15. MR 1244209
(94e:76081), http://dx.doi.org/10.1006/jcph.1993.1194
 3.
A. Jameson, Acceleration of transonic potential flow calculations on arbitrary meshes by the multiple grid method, AIAA Paper 79, p. 1458, 1979.
 4.
Herbert
B. Keller, A new difference scheme for parabolic problems,
1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970)
Academic Press, New York, 1971, pp. 327–350. MR 0277129
(43 #2866)
 5.
K. W. Morton, Numerical Solution of ConvectionDiffusion Problems, Applied Mathematics and Mathematical Computation, 12, Chapman and Hall, London, 1996.
 6.
K.W. Morton, P.I. Crumpton and J.A. Mackenzie, Cell vertex methods for inviscid and viscous flows, Computers Fluids, 22 (1993), 91102.
 7.
J.
A. Mackenzie and K.
W. Morton, Finite volume solutions of
convectiondiffusion test problems, Math.
Comp. 60 (1993), no. 201, 189–220. MR 1153168
(93d:76065), http://dx.doi.org/10.1090/S00255718199311531680
 8.
K. W. Morton and M. F. Paisley, A finite volume scheme with shock fitting for the steady Euler equations, Journal of Computational Physics, 80 (1989), 168203.
 9.
K.
W. Morton and M.
Stynes, An analysis of the cell vertex method, RAIRO
Modél. Math. Anal. Numér. 28 (1994),
no. 6, 699–724. MR 1302420
(95h:65072)
 10.
K.
W. Morton and E.
Süli, Finite volume methods and their analysis, IMA J.
Numer. Anal. 11 (1991), no. 2, 241–260. MR 1105229
(93e:65145), http://dx.doi.org/10.1093/imanum/11.2.241
 11.
R. H. Ni, A multiple grid method for solving the Euler equations, AIAA J. 20 (1982), 15651571.
 12.
L.A. Oganesian and L.A. Ruhovec, Variationaldifference methods for the solution of elliptic equations, Publ. of the Armenian Academy of Sciences, Yerevan, 1979. (In Russian).
 13.
A. Preissmann, Propagation des intumescences dans les canaux et rivieras, Paper presented at the First Congress of the French Association for Computation, held at Grenoble, France, 1961.
 14.
H. G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer Computational Mathematics, 24, SpringerVerlag, 1996.
 15.
E. Süli, Finite volume methods on distorted partitions: stability, accuracy, adaptivity, Technical Report NA89/6, Oxford University Computing Laboratory, 1989.
 16.
Endre
Süli, The accuracy of finite volume methods on distorted
partitions, The mathematics of finite elements and applications, VII
(Uxbridge, 1990), Academic Press, London, 1991, pp. 253–260. MR 1132503
(92i:65171)
 17.
Endre
Süli, The accuracy of cell vertex finite
volume methods on quadrilateral meshes, Math.
Comp. 59 (1992), no. 200, 359–382. MR 1134740
(93a:65158), http://dx.doi.org/10.1090/S0025571819921134740X
 18.
H. A. Thomas, Hydraulics of Flood Movements in Rivers, Carnegie Institute of Technology, Pittsburgh, Pennsylvania, 1937.
 19.
Burton
Wendroff, On centered difference equations for hyperbolic
systems, J. Soc. Indust. Appl. Math. 8 (1960),
549–555. MR 0116472
(22 #7259)
 1.
 P. Balland and E. Süli, Analysis of the cell vertex finite volume method for hyperbolic equations with variable coefficients, SIAM J. Numer. Anal. 34, No. 3, June 1997.
 2.
 P.I. Crumpton, J.A. Mackenzie and K.W. Morton, Cell vertex algorithms for the compressible NavierStokes equations, Journal of Computational Physics, 109 (1993), 115.MR 94e:76081
 3.
 A. Jameson, Acceleration of transonic potential flow calculations on arbitrary meshes by the multiple grid method, AIAA Paper 79, p. 1458, 1979.
 4.
 H. Keller, A new finite difference scheme for parabolic problems, In: Numerical Solution of Partial Differential Equations II, SYNSPADE 1970 (Ed., B. Hubbard,) Academic Press, 1971, 327350. MR 43:2866
 5.
 K. W. Morton, Numerical Solution of ConvectionDiffusion Problems, Applied Mathematics and Mathematical Computation, 12, Chapman and Hall, London, 1996.
 6.
 K.W. Morton, P.I. Crumpton and J.A. Mackenzie, Cell vertex methods for inviscid and viscous flows, Computers Fluids, 22 (1993), 91102.
 7.
 J. A. Mackenzie and K. W. Morton, Finite volume solutions of convectiondiffusion test problems, Mathematics of Computation, 60 (1992), 189220. MR 93d:76065
 8.
 K. W. Morton and M. F. Paisley, A finite volume scheme with shock fitting for the steady Euler equations, Journal of Computational Physics, 80 (1989), 168203.
 9.
 K. W. Morton and M. Stynes, An analysis of the cell vertex method, Mathematical Modelling and Numerical Analysis, 28 (1994), 699724. MR 95h:65072
 10.
 K. W. Morton and E. Süli, Finite volume methods and their analysis, IMA Journal of Numerical Analysis, 11 (1991), 241260. MR 93e:65145
 11.
 R. H. Ni, A multiple grid method for solving the Euler equations, AIAA J. 20 (1982), 15651571.
 12.
 L.A. Oganesian and L.A. Ruhovec, Variationaldifference methods for the solution of elliptic equations, Publ. of the Armenian Academy of Sciences, Yerevan, 1979. (In Russian).
 13.
 A. Preissmann, Propagation des intumescences dans les canaux et rivieras, Paper presented at the First Congress of the French Association for Computation, held at Grenoble, France, 1961.
 14.
 H. G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer Computational Mathematics, 24, SpringerVerlag, 1996.
 15.
 E. Süli, Finite volume methods on distorted partitions: stability, accuracy, adaptivity, Technical Report NA89/6, Oxford University Computing Laboratory, 1989.
 16.
 E. Süli, The accuracy of finite volume methods on distorted partitions, Mathematics of Finite Elements and Applications VII (J.R. Whiteman, ed.) Academic Press, 1991, 253260. MR 92i:65171
 17.
 E. Süli, The accuracy of cell vertex finite volume methods on quadrilateral meshes, Mathematics of Computation, 59 (1992), 359382. MR 93a:65158
 18.
 H. A. Thomas, Hydraulics of Flood Movements in Rivers, Carnegie Institute of Technology, Pittsburgh, Pennsylvania, 1937.
 19.
 B. Wendroff, On centered difference equations for hyperbolic systems, J. Soc. Indust. Appl. Math. 8 (1960), 549555. MR 22:7259
Similar Articles
Retrieve articles in Mathematics of Computation of the American Mathematical Society
with MSC (1991):
65N99,
65L10,
76M25
Retrieve articles in all journals
with MSC (1991):
65N99,
65L10,
76M25
Additional Information
K. W. Morton
Affiliation:
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom
Email:
Bill.Morton@comlab.ox.ac.uk
Martin Stynes
Affiliation:
Department of Mathematics, University College, Cork, Ireland
Email:
STMT8007@iruccvax.ucc.ie
Endre Süli
Affiliation:
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom
Email:
Endre.Suli@comlab.ox.ac.uk
DOI:
http://dx.doi.org/10.1090/S0025571897008867
PII:
S 00255718(97)008867
Keywords:
Finite volume methods,
stability,
error analysis
Received by editor(s):
November 22, 1994
Received by editor(s) in revised form:
January 26, 1996, and June 12, 1996
Additional Notes:
The authors are grateful to the British Council and Forbairt for the generous financial support of this project.
Article copyright:
© Copyright 1997 American Mathematical Society
