Numerical Efficiency in Monte Carlo Simulations—Case Study of a River Thermodynamic Model
Publication: Journal of Environmental Engineering
Volume 130, Issue 4
Abstract
Trade-offs between precision of numerical solutions to deterministic models of the environment, and the number of model realizations achievable within a framework of Monte Carlo simulation, are investigated and discussed. A case study of a model of river thermodynamics is employed. It is shown that the tractability of Monte Carlo simulation relies on adaptation of the numerical solution time-step, giving results with a guaranteed error in the time domain as well as near-optimum speed of calibration under any chosen accuracy criteria. Time-step control is implemented using two adaptive Runge–Kutta methods: a second order scheme with first order error estimator, and an embedded fourth-fifth order scheme. In the case study, where the effects of sparse and imprecise data dominate the overall modeling error, both the schemes appear adequate. However, the higher order scheme is concluded to be generally more reliable and efficient, and has wide potential to improve the value of applying the Monte Carlo method to environmental simulation. The problem of reconciling spatial error with the specified temporal error is discussed.
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References
Ashton, G. (1979). “Modeling of ice in rivers.” Modeling of rivers, H. W. Shen, ed., Wiley, New York.
Ashton, G. (1986). “Thermal regime of lakes and rivers.” River and lake ice engineering, G. Ashton, ed., Water Resources Publications, Littleton, Colo.
Berezin, I. S., and Zdidkov, N. P. (1965). Computing methods, Permagon, Oxford.
Beven, K. J., and Binley, A. M.(1992). “The future of distributed models: Model calibration and predictive uncertainty.” Hydrolog. Process., 6, 279–298.
Bras, R. L. (1990). Hydrology: An introduction to hydrologic science, Addison-Wesley, Reading, Mass.
Carver, M. B. (1976). “The choice of algorithms in automated method of lines solution of partial differential equations.” Numerical methods for differential systems, L. Lapidus and W. E. Schiesser, eds., Academic, New York, 243–265.
Cash, J. R., and Karp, A. H.(1990). “A variable order {Runge-Kutta} method for initial value problems with rapidly varying right-hand sides.” ACM Trans. Math. Softw., 16, 201–222.
Chapra, S. C. (1997). Surface water-quality modeling, McGraw-Hill, New York.
Chapra, S. C., and Canale, R. P. (1998). Numerical recipes for engineers, 3rd Ed., McGraw-Hill, New York, 710–718.
Franks, S., and Beven, K.(1997). “Bayesian estimation of uncertainty in land-surface-atmosphere flux predictions.” J. Geophys. Res., 102, 23991–23999.
Gustafsson, K. (1993). “Control of error and convergence in ODE solvers.” PhD thesis, Lund Institute of Technology, Sweden.
Hairer, E., Nørsett, S. P., and Wanner, G. (1993). Solving ordinary differential equations I, 2nd Ed., Springer, New York, 164–202.
Hondzo, M., and Stefan, H. G.(1994). “River-bed heat conduction prediction.” Water Resour. Res., 30(5), 1503–1513.
Lal, A. M., and Shen, H. W.(1991). “Mathematical model for river ice processes.” J. Hydraul. Eng., 117(7), 851–867.
MacKay, M. D., Beckman, R. J., and Conover, W. J.(1979). “A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.” Technometrics, 21(2), 239–245.
McIntyre, N. R., Wheater, H. S., and Lees, M. J.(2002). “Estimation and propagation of parametric uncertainty in environmental models.” J. Hydroinformatics, 4(3), 177–198.
Reichart, P., and Omlin, M.(1996). “On the usefulness of over-parameterised ecological models.” Ecol. Modell., 95, 289–299.
Shen, H. T., and Chiang, L. A.(1984). “Simulation of growth and decay of river ice cover.” J. Hydraul. Eng., 110(7), 958–971.
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Information
Published In
Journal of Environmental Engineering
Volume 130 • Issue 4 • April 2004
Pages: 456 - 464
Copyright
Copyright © 2004 American Society of Civil Engineers.
History
Received: Apr 5, 2002
Accepted: Apr 29, 2003
Published online: Mar 15, 2004
Published in print: Apr 2004
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