Finite volume solutions of convection-diffusion test problems

Authors:
J. A. Mackenzie and K. W. Morton

Journal:
Math. Comp. **60** (1993), 189-220

MSC:
Primary 76R99; Secondary 65L10, 65N99, 76M25, 76N99

DOI:
https://doi.org/10.1090/S0025-5718-1993-1153168-0

MathSciNet review:
1153168

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Abstract | References | Similar Articles | Additional Information

Abstract: The cell-vertex formulation of the finite volume method has been developed and widely used to model inviscid flows in aerodynamics: more recently, one of us has proposed an extension for viscous flows. The purpose of the present paper is two-fold: first we have applied this scheme to a well-known convection-diffusion model problem, involving flow round a 180 bend, which highlights some of the issues concerning the application of the boundary conditions in such cell-based schemes. The results are remarkably good when the boundary conditions are applied in an appropriate manner. In our efforts to explain the high quality of the results we were led to a detailed analysis of the corresponding one-dimensional problem. Our second purpose is thus to gather together various approaches to the analysis of this problem and to draw attention to the supra-convergence phenomena enjoyed by the proposed methods.

**[1]**J. W. Barrett and K. W. Morton,*Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems*, Comput. Methods Appl. Mech. Engrg.**45**(1984), 97-122. MR**759805 (86g:65180)****[2]**E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders,*Uniform numerical methods for problems with initial and boundary layers*, Boole Press, Dublin, 1980. MR**610605 (82h:65053)****[3]**B. García-Archilla and J. A. Mackenzie,*Analysis of a supraconvergent cell vertex finite volume method for one-dimensional convection-diffusion problems*, Technical Report NA91/13, Oxford University Computing Laboratory, 11 Keble Road, Oxford, OX1 3QD, 1991. (Submitted for publication)**[4]**V. A. Gushchin and V. V. Shchennikov,*A monotonic difference scheme of second order accuracy*, U.S.S.R. Comput. Math. and Math. Phys.**14**(1974), 252-256.**[5]**J. C. Heinrich, P. S. Huyakorn, A. R. Mitchell, and O. C. Zienkiewicz,*An upwind finite element scheme for two-dimensional convective transport equations*, Internat. J. Numer. Methods Engrg.**11**(1977), 131-143.**[6]**T. J. R. Hughes and A. N. Brooks,*A multi-dimensional upwind scheme with no crosswind diffusion*, Finite Element Methods for Convection Dominated Flows (T. J. R. Hughes, ed.), ASME, New York, 1985, pp. 19-35. MR**571681 (81f:76040)****[7]**A. Jameson, W. Schmidt, and E. Turkel,*Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time stepping*, AIAA Paper No. 81-1259, 1981.**[8]**R. B. Kellogg and A. Tsan,*Analysis of some difference approximations for a singular perturbation problem without turning points*, Math. Comp.**32**(1978), 1025-1039. MR**0483484 (58:3485)****[9]**H. O. Kreiss, T. A. Manteuffel, B. Swartz, B. Wendroff, and A. B. White,*Supra-convergent schemes on irregular grids*, Math. Comp.**47**(1986), 537-554. MR**856701 (88b:65082)****[10]**J. E. Lavery,*Nonoscillatory solution of the steady inviscid Burgers' equation by mathematical programming*, J. Comput. Phys.**79**(1988), 436-448. MR**973336 (89j:76020)****[11]**R. W. MacCormack and A. J. Paullay,*Computational efficiency achieved by time splitting of finite difference operators*, AIAA Paper No. 72-154, 1972.**[12]**J. A. Mackenzie,*The cell vertex method for viscous transport problems*, Technical Report NA89/4, Oxford University Computing Laboratory, 11 Keble Road, Oxford, OX1 3QD, 1989.**[13]**T. A. Manteuffel and A. B. White, Jr.,*The numerical solution of second-order boundary value problems on nonuniform meshes*, Math. Comp.**47**(1986), 511-535. MR**856700 (87m:65116)****[14]**P. W. McDonald,*The computation of transonic flow through two-dimensional gas turbine cascades*, Paper 71-GT-89, ASME, New York, 1971.**[15]**J. Moore and J. Moore,*Calculation of horseshoe vortex flow without numerical mixing*, Technical Report JM/83-11, Virginia Polytechnic Inst. and State University, Blacksburg, Virginia 24061, 1983. Prepared for presentation at the 1984 Gas Turbine Conference, Amsterdam.**[16]**K. W. Morton,*Generalised Galerkin methods for hyperbolic problems*, Comput. Methods Appl. Mech. Engrg.**52**(1985), 847-871. Presented at FENOMECH '84, Part III, IV, Stuttgart, 1984. MR**822763 (87e:65061)****[17]**-,*Finite volume methods and their analysis*, The Mathematics of Finite Elements and Applications, VII MAFELAP 1990 (J. R. Whiteman, ed.), Academic Press, London and New York, 1991, pp. 189-214. MR**1132499 (92i:76066)****[18]**K. W. Morton and M. F. Paisley,*A finite volume scheme with shock fitting for the steady Euler equations*, J. Comput. Phys.**80**(1989), 168-203.**[19]**K. W. Morton and B. W. Scotney,*Petrov-Galerkin methods and diffusion-convection problems in*2*D*, The Mathematics of Finite Elements and Applications, V MAFELAP 1984 (J. R. Whiteman, ed.), Academic Press, London and New York, 1985, pp. 343-366. MR**811047 (87i:76049)****[20]**K. W. Morton and E. Süli,*Finite volume methods and their analysis*, Technical Report NA90/14, Oxford University Computing Laboratory, 11 Keble Road, Oxford, OX1 3QD, 1989.**[21]**R. H. Ni,*A multiple grid method for solving the Euler equations*, AIAA J.**20**(1982), 1565-1571.**[22]**E. O'Riordan and M. Stynes,*An analysis of a superconvergence result for a singularly perturbed boundary value problem*, Math. Comp.**46**(1986), 81-92. MR**815833 (87b:65107)****[23]**R. M. Smith and A. G. Hutton,*The numerical treatment of convection--a performance/comparison of current methods*, Numer. Heat Transfer**5**(1982), 439-461.**[24]**M. N. Spijker,*Stability and convergence of finite-difference methods*, PhD thesis, Leiden, Rijksuniversiteit, 1968. MR**0239761 (39:1118)****[25]**F. Stummel,*Biconvergence, bistability and consistency of one-step methods for the numerical solution of initial value problems in ordinary differential equations*, Topics in Numerical Analysis II (J. J. H. Miller, ed.), Academic Press, London, 1975, pp. 197-211. MR**0421084 (54:9089)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1993-1153168-0

Keywords:
Convection-diffusion,
finite volume,
cell-vertex

Article copyright:
© Copyright 1993
American Mathematical Society