Finite volume solutions of convection-diffusion test problems
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- by J. A. Mackenzie and K. W. Morton PDF
- Math. Comp. 60 (1993), 189-220 Request permission
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 $180^\circ$ 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.References
- J. W. Barrett and K. W. Morton, Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems, Comput. Methods Appl. Mech. Engrg. 45 (1984), no. 1-3, 97–122. MR 759805, DOI 10.1016/0045-7825(84)90152-X
- E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders, Uniform numerical methods for problems with initial and boundary layers, Boole Press, Dún Laoghaire, 1980. MR 610605 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) 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. 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.
- T. J. R. Hughes and A. Brooks, A multidimensional upwind scheme with no crosswind diffusion, Finite element methods for convection dominated flows (Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York, 1979) AMD, vol. 34, Amer. Soc. Mech. Engrs. (ASME), New York, 1979, pp. 19–35. MR 571681 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.
- R. Bruce Kellogg and Alice Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp. 32 (1978), no. 144, 1025–1039. MR 483484, DOI 10.1090/S0025-5718-1978-0483484-9
- H.-O. Kreiss, T. A. Manteuffel, B. Swartz, B. Wendroff, and A. B. White Jr., Supra-convergent schemes on irregular grids, Math. Comp. 47 (1986), no. 176, 537–554. MR 856701, DOI 10.1090/S0025-5718-1986-0856701-5
- John E. Lavery, Nonoscillatory solution of the steady-state inviscid Burgers’ equation by mathematical programming, J. Comput. Phys. 79 (1988), no. 2, 436–448. MR 973336, DOI 10.1016/0021-9991(88)90024-1 R. W. MacCormack and A. J. Paullay, Computational efficiency achieved by time splitting of finite difference operators, AIAA Paper No. 72-154, 1972. 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.
- Thomas A. Manteuffel and Andrew B. White Jr., The numerical solution of second-order boundary value problems on nonuniform meshes, Math. Comp. 47 (1986), no. 176, 511–535, S53–S55. MR 856700, DOI 10.1090/S0025-5718-1986-0856700-3 P. W. McDonald, The computation of transonic flow through two-dimensional gas turbine cascades, Paper 71-GT-89, ASME, New York, 1971. 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.
- K. W. Morton, Generalised Galerkin methods for hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 52 (1985), no. 1-3, 847–871. FENOMECH ’84, Part III, IV (Stuttgart, 1984). MR 822763, DOI 10.1016/0045-7825(85)90017-9
- K. W. Morton, Finite volume methods and their analysis, The mathematics of finite elements and applications, VII (Uxbridge, 1990) Academic Press, London, 1991, pp. 189–214. MR 1132499 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.
- 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 (Uxbridge, 1984) Academic Press, London, 1985, pp. 343–366. MR 811047 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. R. H. Ni, A multiple grid method for solving the Euler equations, AIAA J. 20 (1982), 1565-1571.
- Eugene O’Riordan and Martin Stynes, An analysis of a superconvergence result for a singularly perturbed boundary value problem, Math. Comp. 46 (1986), no. 173, 81–92. MR 815833, DOI 10.1090/S0025-5718-1986-0815833-8 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.
- Marc Nico Spijker, Stability and convergence of finite-difference methods, Rijksuniversiteit te Leiden, Leiden, 1968 (English, with Dutch summary). Doctoral dissertation, University of Leiden, 1968. MR 0239761
- Friedrich 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 (Proc. Roy. Irish Acad. Conf., Univ. College, Dublin, 1974) Academic Press, London, 1975, pp. 197–211. MR 0421084
Additional Information
- © Copyright 1993 American Mathematical Society
- 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