Semi-Lagrangian Algorithm for Two-Dimensional Advection-Diffusion Equation on Curvilinear Coordinate Meshes
Publication: Journal of Hydraulic Engineering
Volume 123, Issue 5
Abstract
A semi-Lagrangian method for the solution of the unsteady advection-diffusion equation in complex geometries is presented. The domain is discretized into a set of control volumes defined by an orthogonal, boundary-fitted coordinate system. A splitting strategy is employed that decouples the advective and diffusive processes. The advective update is realized by calculating back trajectories along characteristic lines and then, at the feet of these trajectories, the value of the transported field is calculated by interpolation. The interpolated values are calculated using a tensor product cubic spline function. The method is simple to program and produces results of high accuracy. Numerical errors are considerably lower than for conventional Eulerian schemes. In one-dimensional and rectangular two-dimensional domains, the present method gives comparable results to the well-known semi-Lagrangian scheme. However, the new method allows for greater flexibility in handling complex domains.
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References
1.
Bentley, L. R.(1991). “Comment on `On use of reach back characteristics method for calculation of dispersion,' by J. C. Yang and E.-L. Hsu.”Int. J. Numer. Methods in Fluids, 13, 1205–1206.
2.
Bermejo, R.(1990). “On the equivalence of semi-Lagrangian schemes and particle-in-cell finite-element methods.”Monthly Weather Rev., 118, 989–987.
3.
Celia, M. A., and Gray, W. G. (1992). Numerical methods for differential equations: Fundamental concepts for scientific and engineering applications. Prentice-Hall, Inc., Englewood Cliffs, N.J.
4.
Crockett, S. R. (1993). “A semi-Lagrangian scheme for solving the advection-diffusion equation in two-dimensional simply connected regions,” MASc. thesis, University of Toronto, Toronto, Canada.
5.
DeBoor, C. (1978). A practical guide to splines. Springer-Verlag, New York, Inc., New York.
6.
Farrashkhalvat, M., and Miles, J. P. (1990). Tensor methods for engineers and scientists. Ellis Horwood, Ltd., Chichester, England.
7.
Glass, J., and Rodi, W.(1982). “A higher order numerical scheme for scalar transport.”Comp. Methods Appl. Mech. and Engrg., 31, 337–358.
8.
Holly, F. M. Jr., and Preissmann, A.(1977). “Accurate calculation of transport in two dimensions.”J. Hydr. Div., ASCE, 103(11), 1259–1277.
9.
Lancaster, P., and Salkauskas, K. (1986). Curve and surface fitting: An introduction. Academic Press, Inc., San Diego, Calif.
10.
Lax, P. D., and Wendroff, B.(1964). “Difference schemes with high order of accuracy for solving hyperbolic equations.”Communications in Pure and Appl. Mathematics, 17, 381–398.
11.
Leonard, B. P.(1979). “A stable and accurate convection modelling procedure based on quadratic upstream interpolation.”Comp. Methods in Appl. Mech. and Engrg., 19, 59–98.
12.
Marchuk, G. I. (1980). Methods of numerical mathematics. Springer-Verlag, New York, Inc., New York.
13.
Noye, J., ed. (1987a). “Numerical methods for solving the transport equation.”Numerical modeling: Applications to marine systems, North-Holland Publishing Co., Amsterdam, The Netherlands, 195–229.
14.
Noye, J., ed. (1987b). “Finite difference methods for solving the one-dimensional transport equation.”Numerical modelling: applications to marine systems, North-Holland Publishing Co., Amsterdam, The Netherlands, 231–256.
15.
Patankar, S. V. (1980). Numerical heat transfer and fluid flow. McGraw-Hill Book Co., Inc., New York.
16.
Prenter, P. M. (1975). Splines and variational methods. John Wiley & Sons, Inc., New York.
17.
Raithby, G. D.(1976). “Skew upstream differencing schemes for problems involving fluid flow.”Comp. Methods in Appl. Mech. and Engrg., 9, 153–164.
18.
Raithby, G. D., Galpin, P. F., and Van Doormaal, J. P.(1986). “Prediction of heat and fluid flow in complex geometries using general orthogonal coordinate.”Numer. Heat Transfer, 9, 125–142.
19.
Robert, A.(1981). “A stable numerical integration scheme for the primitive meteorological equations.”Atmospheric Ocean, 19, 35–46.
20.
Schohl, G. A., and Holly Jr., F. M.(1991). “Cubic-spline interpolation in Lagrangian advection computation.”J. Hydr. Engrg., ASCE, 117, 248–253.
21.
Staniforth, A., and Cote, J.(1991). “Semi-Lagrangian integration schemes for atmospheric models—a review.”Monthly Weather Rev., 119, 2206–2223.
22.
Thompson, J. F., Warsi, Z. U. A., and Mastin, C. W. (1985). Numerical grid generation: Foundations and applications. North-Holland Publishing Co., Amsterdam, The Netherlands.
23.
Yang, J. C., and Hsu, E.-L.(1991). “On the use of the reach-back characteristics method for calculation of dispersion.”Int. J. Numer. Methods in Fluids, 12, 225–235.
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Published In
Journal of Hydraulic Engineering
Volume 123 • Issue 5 • May 1997
Pages: 389 - 401
Copyright
Copyright © 1997 American Society of Civil Engineers.
History
Published online: May 1, 1997
Published in print: May 1997
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