The Hopscotch Algorithm for Three‐dimensional Simulation
Publication: Journal of Hydraulic Engineering
Volume 118, Issue 3
Abstract
This paper presents results obtained for the solution of the three‐dimensional ground‐water flow and mass‐transport equations using the Hopscotch algorithm, an iterative technique that alternates explicit and implicit finite difference approximations of the governing equations. While the method is not new, it has been mostly ignored, particularly in three dimensions. Three application examples are presented: (1) Contaminant transport in a hypothetical, steady‐state, saturated ground‐water flow system; (2) infiltration of a solute in an unsaturated soil; and (3) advection and advection‐diffusion of a Gaussian hill in rotational flow fields. The results indicate that the Hopscotch algorithm is an intriguing and powerful alternative to SIP, SOR, and conjugate gradient techniques in three dimensions. Certainly it warrants more research.
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References
1.
Ames, W. P. (1983). “A survey of finite difference schemes for parabolic partial differential equations.” Numerical Properties and Methodologies in Heat Transfer, Hemisphere, Washington, D.C.‐London, U.K., 3–15.
2.
Barry, D. A. (1990). “Supercomputers and their use in modeling subsurface transport.” Rev. Geopliys., 28(3), 277–297.
3.
Bear, J. (1972). Dynamics of fluids in porous media. American Elsevier, New York, N.Y.
4.
Black, F., and Scholes, M. (1973). “The pricing of options and corporate liabilities.” J. Polit. Econ., 81, 637–659.
5.
Bresler, E. (1973). “Simultaneous transport of solutes and water under transient unsaturated flow conditions.” Water Resour. Res., 9(4), 975–986.
6.
Cooke, C. H. (1983). “A correct derivation of acceleration parameters for Hopscotch and Checkerboard P Q relaxation schemes.” Math. Comput. Simul., 25, 206–209.
7.
Freeze, R. A. (1971). “Three‐dimensional, transient, saturated‐unsaturated flow in a groundwater basin.” Water Resour. Res., 7(2), 347–366.
8.
Freeze, R. A. and Cherry, J. A. (1979). Groundwater, Prentice Hall, Englewood Cliffs, N.J.
9.
Gourlay, A. R. (1970). “Hopscotch: A fast, second‐order partial differential equation solver.” J. Inst. Math. Appl., 6(3), 375–390.
10.
Gourlay, A. R. (1977). Hopscotch: “Recent developments of the Hopscotch method.” Adv. Comput. Methods Partial Differential Equations—II, 1–6. Proc. of the Second IMACS Int. Symp. on Computer Methods for Partial Differential Equations, R. Vichnevetsky, ed., Lehigh University, Bethlehem, Pa.
11.
Gourlay, A. R., and McGuire, G. R. (1971). “General Hopscotch algorithm for the numerical solution of partial differential equations.” J. Inst. Math. Appl, 7(2), 216–227.
12.
Gourlay, A. R., and McKee, S. (1977). “The construction of Hopscotch methods for parabolic and elliptic equations in two space dimensions with a mixed derivative.” J. Comput. Appl. Math., 3(2), 201–206.
13.
Gourlay, A. R., and McKee, S. (1979). “Hopscotch methods for elliptic partial differential equations.” J. Comput. Appl. Math., 5(2), 103–110.
14.
Hill, M. (1990). “Solving groundwater flow problems by conjugate‐gradient methods and the strongly implicit procedure.” Water Resour. Res., 26(9), 1961–1969.
15.
Lapidus, L., and Pinder, G. F. (1982). Numerical solution of partial differential equations in science and engineering. John Wiley & Sons, New York, N.Y.
16.
McDonald, M. G., and Harbaugh, A. W. (1984). “A modular three‐dimensional finite‐difference ground‐water flow model,” U.S. Geological Survey Report.
17.
McKee, S., and Mitchell, A. R. (1970). “Alternating direction methods for parabolic equations in two space dimensions with a mixed derivative.” The Comput. I., 13(1), 81–86.
18.
Mejerink, J. A., and van derVorst, H. A. (1977). “An iterative solution method for linear systems of which the coefficient matrix is a symmetric M‐matrix.” Math. of Comput., 137, 148–162.
19.
Merton, R. C. (1973). “Theory of rational option pricing.” Bell J. Econ. Mgmt. Sci., (4), 141–183.
20.
Meyer, P. D., Valocchi, A. J., Ashby, S. F., and Saylor, P. A. (1989). “Anumerical investigation of the conjugate gradient method as applied to three‐dimensional groundwater flow problems in randomly heterogeneous media.” Water Resour. Res., 25(6), 1,140‐1,146.
21.
Mitchell, A. R., and Griffiths, D. F. (1987). The finite difference method in partial differential equations. John Wiley & Sons, New York, N.Y.
22.
Neuman, S. P. (1973). “Saturated‐unsaturated seepage by finite elements.” Proc. J. Hydr. Div., ASCE, 99(12), Dec. 2,233‐2,250.
23.
Park, N. S., and Liggett, J. A. (1990). “Taylor‐least squares finite element for two‐dimensional advection‐dominated unsteady advection‐diffusion problems.” Int. I. Numer. Methods Fluids, 11(1), 21–38.
24.
Park, N. S., and Liggett, J. A. (1991). “Application of Taylor‐least squares finite element to three‐dimensional advection‐diffusion equation.” Int. J. Numer. Methods Fluids, 12(1), 1–15.
25.
Pickens, J. F., and Lennox, W. C. (1976). “Numerical simulation of waste movement in steady‐groundwater flow systems,” Water Resour. Res., 12(2), 171–180.
26.
Pinder, G. F., and Gray, W. G. (1977). Finite element simulation in surface and subsurface hydrology. Academic Press, New York, N.Y.
27.
Redell, D. L. and Sunada, D. K. (1970). “Numerical simulation of dispersion in groundwater aquifers.” Hydr. Paper No. 41, Colorado State University, Fort Collins, Colo.
28.
Ségol, G., (1976). “A three‐dimensional Galerkin‐finite element model for the analysis of contaminant transport in saturated‐unsaturated porous media.” Finite Elem. Water Resour. Proc. Int. Conf., 2.123–2.144, Princeton University, Princeton, N.J.
29.
Sleijpen, G. L. J. (1989). “Strong stability results for the Hopscotch method with applications to bending beam equations.” Computing, 41(3), 179–203.
30.
Ten Thije Boonkkamp, J. H. M. (1988). “The odd‐even pressure correction scheme for the incompressible Navier‐Stokes equations.” SI AM J. Sci. Stat. Comput., 9(2), 252–270.
31.
Tompson, A. F. B., and Gelhar, L. W. (1990). “Numerical simulation of solute transport in three‐dimensional, randomly heterogeneous porous media.” Water Resour. Res., 26(10), 2,541–2,562.
32.
van Genuchten, M. Th., and Gray, W. G. (1978). “Analysis of some dispersion corrected schemes for solution of the transport equation.” Int. J. Numer. Methods Engrg., 12, 387–404.
33.
Varga, R. S. (1962). Matrix iterative analysis. Prentice Hall, Englewood Cliffs, N.J.
34.
Wachpress, E. L. (1965). Iterative solution of elliptic systems. Prentice Hall, Englewood Cliffs, N.J.
35.
Warrick, A. W., Biggar, J. W., and Nielsen, D. R. (1971). “Simultaneous solute and water transfer for an unsaturated soil.” Water Resour. Res., 7(5), 1,216‐1,225.
36.
Weinstein, H. G., Stone, H. L., and Kwan, T. V. (1969). “Iterative procedure for solution of systems of parabolic and elliptic equations in three dimensions.” Ind. Engrg. Chem. Fundam., 8(2), 281–287.
Information & Authors
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Published In
Journal of Hydraulic Engineering
Volume 118 • Issue 3 • March 1992
Pages: 385 - 406
Copyright
Copyright © 1992 ASCE.
History
Published online: Mar 1, 1992
Published in print: Mar 1992
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