Computational Subsurface Hydrology pp 27-176 | Cite as

# Numerical Methods Applied to Subsurface Hydrology

## Abstract

Numerical methods, as shall be described in this book, are merely tools used to enable one to replace differential equations governing the subsurface processes with approximation sets of algebraic equations or matrix equations, which are subsequently solved using the methods of linear algebra and requiring the manipulation of computers (Fig. 2.1). If the differential equations were solved exactly by analytic procedures, the solution would appear as some combination of mathematical functions. Subsequent interest in the value of the solution at various locations within a domain of interest would require that the functions be evaluated. Often, when the functions are of a complex form, the computer must be used to determine the values of the function at the points of interest. In many cases the analytical solution will be in terms of an infinite series or some transcendental functions that can be evaluated only approximately. Nevertheless, it is often possible to control the accuracy of the evaluation by careful use of the computer. The steps outlined above do require some facility with number manipulation on the computer and do yield an approximate value of the solution at points of interest. However, the actual steps involve numerical evaluation of an analytical solution to a differential equation rather than numerical solution to the differential equation. The differences between these concepts is the presence of an exact analytical expression as an intermediate step in the former case and the use of an approximation to the differential equation in the latter case.

### Keywords

Permeability Vortex Porosity Enthalpy Boron## Preview

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### References

- Axelsson, O. 1974. On preconditioning and convergence acceleration in sparse matrix problems, CERN 74–10. European Organization for Nuclear Research (CERN), Data Handling Division.Google Scholar
- Bartels, R. and J. W. Daniel. 1974. A conjugategradient approach to nonlinear ellipic boundary value problems in irregular regions. Proceeding of the Conference on Numerical Solution of Differential Equations, Dundee, Scotland, 1973. New York: Springer-Verlag.Google Scholar
- Brandt, A. 1984. Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics. Department of Applied Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel.Google Scholar
- Brebbia, C. A., J. C. F. Telles, and L. C. Wrobel. 1984. Boundary Element Techniques. New York: Springger-Verlag.CrossRefGoogle Scholar
- Briggs, W. L., 1987. A Multigrid Tutorial. Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar
- Carrano, C. S. Jr. and G. T. Yeh. 1995. A fourier analysis of dynamic optimization of the Petrov-Galerkin finite element method. Int. J. Num. Methods Engg. 38:4123–4155.CrossRefGoogle Scholar
- Chandra, R., S. C. Eisenstat, and M. H. Schultz. 1977. The modified conjugate residual method for partial differential equations. In R. Vichnevetsky, ed. Advances in Computer Methods for Partial Differential Equations. Vol. II. Minneapollis: IMACS.Google Scholar
- Cheng, H. P., G. T. Yeh, M. H. Li, J. Xu, and R. Carsel. 1998. A study of incorporating the multigrid method into the three-dimensional finite element discretization. Intl. J. Numer. Methods Engg., 41:499–526.CrossRefGoogle Scholar
- Concus, P. and G. H. Golub. 1976. A generalized conjugate gradient method for nonsymetric systems of linear equations. Rep. Stan-CS-76–646, Computer Science Department, Stanford University.Google Scholar
- Concus, P., G. H. Golub, and D. P. O’Leary. 1976. A generalized conjugate gradient method for the numerical solution of ellipic partial differential equations. In J. R. Bunch and D. R. Rose, eds. Sparse Matrix Computation (pp. 309–332). New York: Academic Press.Google Scholar
- Daniel, J. W. 1965. The conjugate gradient method for linear and nonlinear operator equations. Doctoral thesis, Stanford University.Google Scholar
- Daniel, J. W. 1967. The conjugate gradient method for linear and nonlinear operator equations. SIAM J. Numer. Anal. 4:10–26.CrossRefGoogle Scholar
- Duff, I. S. 1977. MA28 — A set of Fortran Subroutines for Sparse Unsymmetric Linear Equations. AERE-R 8730. Harwell, Oxfordshire: AERE.Google Scholar
- Finlayson, B. A. 1972. Existence of variational principles for the Navier-Stokes equation. Phvs. Fluids. 15(6): 963–967.CrossRefGoogle Scholar
- Finalyson, B. A. and L. E. Scriven. 1967. On the search for variational principles. J. Heat Mass Transfer. 19:799–821.CrossRefGoogle Scholar
- Forsythe, G. E. and W. R. Wasow. 1960. Finite-Difference Methods for Partial Differential Equations. New York: John Wiley & Sons Inc.Google Scholar
- Gambolati, G. 1980. Fast solution to finite element flow equations by new iteration and modified conjugate gradient method. Intl. J. Numer. Methods Engg., 15:661–675.CrossRefGoogle Scholar
- Gambolati, G. and G. Volpi. 1980a. Analysis of performance of the modified conjugate gradient method for solution of sparse linear sets of finite element equations. Third International Conference on Finite Elements in Flow Problems, Banff, Canada, 1980.Google Scholar
- Gambolati, G. and G. Volpi. 1980b. An improved iterative scheme for refining the solution of ill-conditioned systems. Information Processing. 80:729–734.Google Scholar
- Gambolati, G. and A. M. Perdon. 1984. The conjugate gradients in subsurface flow and land subsidence modeling. In J. Bear and M. Y. Corapcioglu, eds. Fundamentals of Transport Phenomena in Porous Media. NATO-ASI Series (Vol. 82, pp. 953–983). The Hague: Martinus Nijoff.Google Scholar
- Gambolati, G., G Galeati, and S. P. Neuman. 1990. A Eulerian-Lagrangian finite element model for coupled groundwater transport. In G. Gambolati, A. Rinaldo, C. A. Brebbia, W. G. Gray, and G. F. Pinder, eds. Computational Methods in Subsurface Hydrology. (pp. 341–348). Eighth International Conference on Computational Methods in Water Resources, Venice, Italy, June 11–15, 1990. Southampton: Computational Mechanics Publications.Google Scholar
- Gosman, A. D. and K. Y. M. Lai. 1982. Finite difference and other approximations for the transport and Navier-Stokes equations. Presented at the IAHR Symposium on Refined Modeling of Flows, Paris.Google Scholar
- Gottlieb, D. 1984. Spectral methods for compressible flow problems. ICASE Report No. 84–29, June 22, 1984. Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA.Google Scholar
- Hageman, L. A. and D. M. Young. 1981. Applied Iterative Methods. New York: Academic Press.Google Scholar
- Hassan, A. A. 1974. Mathematical modeling of water quality for water resources management. Vol. 1. Development of the Groundwater Quality Model. District Report, Department of Water Resources, Southern District, State of California.Google Scholar
- Hayes, L., G. F. Pinder, and M. Celia. 1981. Alternating-direction collocation for rectangular regions. Comp. Meth. Appl. Meth. Eng. 17:265–277.CrossRefGoogle Scholar
- Hestenes, M. R. and E. L. Stiefel. 1952. Methods of conjugate gradients for solving linear system. Nat. Bur. Std. J. Res. 49:409–436.CrossRefGoogle Scholar
- Hilderbrand, F. B. 1968. Finite-Difference Equations and Simulations. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
- Hill, M. C. 1990. Preconditioned Conjugate-Gradient 2 (PCG2), A Computer Program for Solving Ground-Water Flow Equations. USGS Water-Resources Investigations Report 90–4048. Denver: U.S. Geological Survery.Google Scholar
- Hinton, E. and D. R. J. Owen. 1977. Finite Element Programming. New York: Academic Press.Google Scholar
- Hood, P. 1976. Frontal solution program for unsymmetric matrices. Intl. J. Numer. Methods Engg., 10:379–400.CrossRefGoogle Scholar
- Huebner, K. H. 1975. The Finite Element Method for Engineers. New York: John Wiley & Sons, Inc.Google Scholar
- Hussaini, M. Y., C. L. Streett, and T. A. Zang. 1983. Spectral methods for partial differential equations, ICASE Report No. 83–46, August 29. Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA.Google Scholar
- Huyakorn, P. S. and G. F. Pinder. 1977. A pressure enthalpy finite elesment model for simulating hydrothermal reservoir reservoirs. Presented at the Second International Symposium on Computer Methods for Partial Differential Equations. Leigh University, Bethlehem, Pennsylvania, June 22–24.Google Scholar
- Irons, B.M. 1970. A frontal solution program. Intl. J. Numer. Methods Engg., 2:5–32, 1970.CrossRefGoogle Scholar
- Jameson, A., W. Smith, and E. Turkei, 1981. Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time-stepping scheme. AIAA Paper 81–1259, June.Google Scholar
- Kaasschieter, E. F. 1988. Guidelines for the use of preconditioned conjugate gradients in solving discretized potential flow problems. In M. A. Celia, L. A. Ferrand, C. A. Brebbia, W. G. Gray, and G. F. Pinder,, eds. Computational Methods in Water Resources Vol. 2, Numerical Methods for Transport and Hvdrological Processes, (147–152). Amsterdam: Elsevier, Amsterdam.Google Scholar
- Konikow L. F. and J. D. Bredhoeft. 1978. Computer Model of Two-Dimensional Solute Transport and Dispersion in Groundwater, (Bk. 7, Ch. C2), Techniques of Water Resources Investigations. Reston, Va: USGS.Google Scholar
- Kuiper, L. K. 1987. Computer Program for Solving Ground-Water Flow Equations by the Preconditioned Conjugate Gradient Methods. USGS Water-Resources Investigation Report 87–4091. Austin, Texas: USGS.Google Scholar
- Livesley, R. K. 1983. Finite Elements: An Introduction for Engineers. Cambridge: Cambridge University Press.Google Scholar
- Luenberger, D. G. 1973. Introduction to Linear and Nonlinear Programming. Reading, MA: Addison-Wesley.Google Scholar
- Melosh, R. J. and R. M. Bamford. 1969. Efficient solution of load-deflection equations. J. Struct.Div. ASCE, 95:661–676.Google Scholar
- Mikhlin, S. G. 1965. The Problem of the Minimum of a Quadratic Functional. San Francisco, CA: Holden-Day. (English translation of the 1957 edition)Google Scholar
- Mikhlin, S. G. 1964. Variational Methods in Mathematical Physics, New York: Macmillan Company. (English translation of the 1957 editions)Google Scholar
- Mikhlin, S. G. and K. Smolistsky. 1967. Approximate Method for the Solution of Differential and Integral Equations, Amsterdam: Elsevier.Google Scholar
- Mondkar, R. J. and G. H. Powell. 1974. Large capacity equation solver for structural analyses. Comp. Struc, 4:531–548.CrossRefGoogle Scholar
- Narasimahn, T. N. and P. A. Weatherspoon. 1976. An integrated finite element method for analyzing fluid flow in porous media. Water Resources Research, 12(1):57–64.CrossRefGoogle Scholar
- O’Leary, D. P. 1975. Hybrid conjugate gradient algorithms. Doctoral thesis, Stanford University.Google Scholar
- Orlob, G. T. and P. C. Woods. 1967. Water quality management in irrigation system. J. Irrigation and Drainage Div., ASCE, 93:49–66.Google Scholar
- Orssag, S. A. 1980. Spectral methods in complex geometries. J. Comput. Physics, 37:70–92.CrossRefGoogle Scholar
- Peric, M. 1985. A finite volume method for the prediction of three-dimensional fluid flow in complex cucts. Ph.D. dissertation. Mechanical Engineering Department, Imperial College, London.Google Scholar
- Pinder, G. F. and W. G. Gray. 1977. Finite Element Simulation in Surface and Subsurface Hydrology. New York: Academic Press.Google Scholar
- Reid, J. K. 1971. On the method of conjugate graidents for the solution of large sparse systems of linear equations. Proceeding of the Conference on Large Sparse Sets of Linear Equations, (pp. 231–254). New York: Academic Press.Google Scholar
- Reid, J. K. 1972. The use of conjugate gradients for systems of linear equations possessing Property A. SIAM J. Numer. Anal., 9:325–332.CrossRefGoogle Scholar
- Roache, P. T. 1976. Computational Fluid Dynamics. Albuquerque, NM: Hermosa.Google Scholar
- Sagan, H. 1961. Boundary and Eigenvalue Problems in Mathematical Physics. New York: John Wiley & Sons, Inc.Google Scholar
- Salvadori, M. G. and M. L. Baron. 1961. Numerical Methods in Engineering. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
- Shapiro, A. M. and G. F. Pinder. 1981. Solution of immiscible displacement in porous media using the collocation finite element method. In K. P. Holz. et al., eds. Finite Element in Water Resources (pp. 9.61–9.70). Berlin: Springer-VerlagGoogle Scholar
- Shoup, T. E. 1978. A Practical Guide to Computer Methods for Engineers. Englewood Cliffs, N J: Prentice-Hall.Google Scholar
- Spalding, D. B. 1972. A novel finite difference formulation for differential equations involving both first and second derivatives. Intl. J. Numer. Methods Engg., 4:551–559.CrossRefGoogle Scholar
- Tanji, K. K. 1970. A computer analysis on the leaching of boron from stratified soil column. Soil Sciences, 110:44–51.CrossRefGoogle Scholar
- Tewarson, R. P. 1973. Sparse Matrices. New York: Academic Press.Google Scholar
- Thacker, W. C. 1977. Irregular grid finite-difference technique: simulation of oscillations in shallow circular basin. J. Physical Oceanography, 7:284–292.CrossRefGoogle Scholar
- Thacker, W. C. 1977. Irregular grid finite-difference technique: simulation of oscillations in shallow circular basin. J. Physical Oceanography, 7:284–292.CrossRefGoogle Scholar
- Thacker, W. C, A. Gonzaleg, and G. E. Putland. 1980. A method of automating the construction of irregular computational grids for storm surge forecast models. J. Comput. Phys. 37:371–387.CrossRefGoogle Scholar
- Tonti, E. 1969. Variational formulation of nonlinear differential equations, I, II, Bull. Acad. Rov Belg. (Classe Sci.) (5), 55:262–278.Google Scholar
- Vainberg, M. M. 1964. Variational Methods for the Study of Nonlinear Operators. San Francisco, CA: Holden-Day.Google Scholar
- Varga, R. 1962. Matrix Iterative Analysis. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
- Wang, H. and M. P. Anderson. 1982. Introduction to Groundwater Modeling: Finite Difference and Finite Element Methods. San Francisco: Freeman.Google Scholar
- Westerink, J. M., M. E. Canterkin, and D. Shea. 1988. Non-diffusive N+2 degree upwinding for the finite element solution of the time dependent transport equation. In M. A. Celia, et al. eds. Development in Water Sci. 36, Vol. 2, Numerical Methods in Water Resources (pp. 57–62). Comput. Mechanics Publ., Elsevier.Google Scholar
- Wilkinson, J. H. 1965. The Algebraic Eigenvalue Problem. London: Oxford University Press.Google Scholar
- Yeh, G. T. 1981a. Numerical solution of Navier-Stokes equations with an integrated compartment method (ICM), Intl. J. Numeri. Methods Fluids, 1:207–223.CrossRefGoogle Scholar
- Yeh, G. T. 1981b. ICM: An Integrated Compartment Method for Numerically Solving Partial Differential Equations. ORNL-5684. Oak Ridge, TN: Oak Ridge National Laboratory.CrossRefGoogle Scholar
- Yeh, G. T. 1983. Solution of groundwater flow equations using an orthogonal finite element scheme. In G. Mesnard, ed., Proceeding of International 83 Summer Conference, Modeling and Simulation, Vol. 4. (pp. 329–351). Tassin, France: AMSE Press.Google Scholar
- Yeh, G. T. 1985. An orthogonal-upstream finite element approach to modeling aquifer contaminant transport. Water Resources Research, 22(6):952–964.CrossRefGoogle Scholar
- Young, D. 1971. Iterative Solution of Large Linear Systems. New York: Academic Press.Google Scholar
- Zang, T. A., C. Streett, and M. Y. Hussanini. 1989. Spectral methods for CFD. ICASE Report No. 89–13, February 17, 1989. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center.Google Scholar
- Zienkiewicz, O. C. 1977. The Finite Element Method. 3
^{rd}Ed. New York: McGraw-Hill Book Company.Google Scholar