Abstract
Within the large variety of methods of analysis for transient phenomena in nuclear reactors, this article is limited to the discussion of computational methods for the analysis of operational transients, off-normal transients, and hypothetical accidents in power reactors. Though any realistic analysis of such transients requires a coupled treatment of thermodynamics, fluid dynamics and neutronic phenomena, all details of the modeling and computations of thermodynamic and fluid dynamic quantities are omitted in the following presentation.
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References
Yasinsky, J. B and Henry, A. F., “Some Numerical Experiments Concerning Space-Time Kinetics Behavior,” Nucl. Sci. Eng., 22, pp. 171–181, 1965.
Yasinsky, J. B., “On the Use of Point Kinetics for the Analysis of Rod-Ejection Accidents,” Nucl. Sci. Eng., 39, pp. 241–256, 1970.
Kessler, G., “Space-Dependent Dynamic Behavior of the Fast Reactors Using the Time-Discontinuous Synthesis Method,” Nucl. Sci. Eng., 41, pp. 115–148, 1970.
Jackson, J. F., and Kastenberg, W. E., “Space-Time Effects in Fast Reactor Dynamics,” Nucl. Sci. Eng., 42, pp. 278–294, 1970.
Salah, S., Rossi, G. E., and Geets, J. M., “Consequences of Asymmetric Cold Water Addition to a PWR Core from an Inactive Loop,” Trans. Am. Nucl. Soc., 14, Page 756, 1971.
Salah, S., Rossi, C. E., and Geets, J. M., “Three-Dimensional Kinetic Analysis of an Asymmetric Boron Dilution in a PWR Core,” Trans. Am. Nucl. Soc., 15, Page 831, 1972.
Lamarsh, J. R., Introduction to Nuclear Reactor Theory, Addison-Wesley, Reading, Mass., 1966.
Meghreblian, R. V. and Holmes, D. K., Reactor Analysis, McGraw-Hill, New York, 1960.
Henry, A. F., Nuclear Reactor Analysis, MIT Press, Cambridge, Mass., 1975.
Mikhlin, S. G., and Smolitsky, K. L., Approximate Methods for Solution of Differential and Integral Equations, Elsevier, New York, 1967.
Aziz, A. K., and Babuska, I., The Mathematical Foundation of the Finite Element Method with Application to Partial Differential Equations, Academic Press, New York, 1972.
Strang, G., and Fix, G. J., An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, New Jersey, 1973.
Kang, C. M., and Hansen, K. F., “Finite Element Methods for Reactor Analysis,” Nucl. Sci. Eng., 51, pp. 456–495, 1973.
Babuska, I., “Error-Bounds for Finite Element Methods,” Numer. Math, 16, pp. 322–333, 1971.
Schultz, M. H., “L2 Error Bounds for the Rayleigh-Ritz-Galerkin Method,” SIAM J. Numer. Anal., 8, pp. 737–748, 1971.
Nitsche, J. A., “Convergence of Nonconforming Methods,” in Mathematical Aspects of Finite Elements in Partial Differential Equations,” C. deBoor, Ed., Academic Press, New York, 1974.
Birkhofer, A., and Werner, W., “Efficiency of Various Methods for the Analysis of Space-Time Kinetics,” Proc. Conf. Mathematical Models and Computational Techniques for Analysis of Nuclear Systems, CONF-730414, Vol. 2, pp. IX-31–41, 1973.
Birkhofer, A., Langenbuch, S., and Werner, W., “Coarse-Mesh Method for Space-Time Kinetics,” Trans. Am. Nucl. Soc., 18, Page 153, 1974.
Langenbuch, S., Maurer, W., and Werner, W., “Simulation of Transients with Space-Dependent Feedback by Coarse Mesh Flux Expansion Method,” MRR 145, Proc. of Joint NEACRP/CSNI Specialists’ Meeting on New Development in Three-Dimensional Neutron Kinetics, pp. 173–188, 1975.
Schäfer, A., Über das Konvergenzverhalten eines Galerkin-Petrov-Verfahrens Thesis, TU Munchen, 1976.
Finneraan, H., “A Consistent Nodal Method for the Analysis of Space-Time Effects in Large LWR’s,” MRR 145, Proc. of Joint NEACRP/CSNI Specialists’ Meeting on New Developments in Three-Dimensional Neutron Kinetics, pp. 145–172, 1975.
Selengut, D. S., “Variational Analysis of a Multidimensional System,” Page 89, HW-59126, Hanford Laboratory, 1959
Dougherty, D. E., and Shen, C. N., “The Space-Time Neutron Kinetics Equations Obtained by the Semidirect Variational Method,” Nucl. Sci. Eng., 13, pp. 141–152, 1962.
Kaplan, S., Marlowe, O. J., and Bewick, J., “Application of Synthesis Techniques to Problems Involving Time-Dependence,” Nucl. Sci. Eng., 18, pp. 163–176, 1964.
Yasinsky, J. B., “The Solution of the Space-Time Neutron Group Diffusion Equations by a Time Discontinuous Synthesis Method,” Nucl. Sci. Eng., 29, pp. 381–391, 1967.
Stacey, W. M. Jr., “Variational Functionals for Space-Time Neutronics,” Nucl. Sci. Eng., 30, pp. 448–463, 1967.
Yasinsky, J. B., and Kaplan S., “Synthesis of Three-Dimensional Flux Shapes Using Discontinuous Sets of Trial Functions,” Nucl. Sci. Eng., 28, pp. 426–440, 1967.
Stacey, W. M. Jr., “A Variational Multichannel Space-Time Synthesis Method for Nonseparable Reactor Transients,” Nucl. Sci. Eng., 34, pp. 45–56. 1968.
Stacey, W. M. Jr., Space-Time Nuclear Reactor Kinetics, Academic Press, New York, 1969.
Yasinsky, J. B., and Henry, A. F., “Some Numerical Experiments Concerning Space-Time Reactor Kinetics Behavior,” Nucl, Sci. Eng., 22, pp. 171–181, 1965.
Boresen, S., “A Simplified, Coarse-Mesh, Three-Dimensional Diffusion Scheme for Calculating the Gross Power Distribution in a Boiling Water Reactor,” Nucl. Sci. Eng., 44, pp. 37–43, 1971.
Boresen, S., “Characteristics and Performance of the 3D LWR Simulator PRESTO,” Trans. Am. Nucl. Soc., 15, Page 956, 1972.
Delp, D. L., Fischer, D. L., Harriman, J. M., and Stedwell, M. J., “FLARE — A Three-Dimensional Boiling Water Reactor Simulator,” GEAP-4598, General Electric, 1964.
Goldstein, L., Nakache, F., and Veras, A., “Calculation of Fuel-Cycle Burnup and Power Distribution of Dresden-I Reactor with the TRILUX Fuel Management Program,” Trans. Am. Nucl. Soc., 10, Page 300, 1967.
Deppe, L. O., Hansen, K. F., “Applications of the Finite Element Method to Two-Dimensional Diffusion Problems,” Nucl. Sci. Eng., 54, pp. 456–465, 1974.
Gear, C. W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, Englewood Cliff, New Jersey, 1971.
Lapidus, L., and Seinfeld, J. H., Numerical Solution of Ordinary Differential Equations, Academic Press, New York, 1971.
Enright, W. H., “Second Derivative Multistep Methods for Stiff Ordinary Differential Equations,” SIAM J. Anal., 11, pp. 321–331, 1974.
Hofer, E., “A Partially Implicit Method for Large Stiff Systems of ODEs with Only Few Equations Introducing Small Time-Constants,” SIAM J. Numer. Anal., 13–5, 1976.
Richtmeyer, R. D., and Morton, K. W., Difference Methods for Initial Value Problems, Interscience Publishers, New York, 1967.
Birkhofer, A., and Werner, W., “Eine Methode zur Berechnung der raum- und zeit-abhängigen Leistungsverteilung in Kernreaktoren,” Atomkernenergie, 15, pp. 97–102, 1970.
Wight, A. L., Hansen, K. F., Ferguson, D. R., “Application of Alternating-Direction Implicit Methods to Space-Dependent Kinetics Equations,” Nucl. Sci. Eng., 44, pp. 239–251, 1971.
Ferguson, D. R., and Hansen, H. F., “Solution of the Space-Dependent Reactor Kinetics Equations in Three Dimensions,” Nucl. Sci. Eng., 51, pp. 189–205, 1973.
MEKIN: MIT-EPRI Nuclear Reactor Core Kinetics Code, 1975.
Gordon, P., “Nonsymmetric Difference Equations,” J. Soc. Indust. Appl. Math., 13, pp. 667–673, 1965.
Peaceman, D. W., and Rachford, H. H. Jr., “The Numerical Solution of Parabolic and Elliptic Differential Equations,” J. Soc. Indust. Appl. Math., 3, pp. 42–65, 1955.
Douglas, J., and Gunn, J. E., “A General Formulation of Alternating Direction Methods,” Num. Math., 6, pp. 428–453, 1965.
Janenko, N. N., “Die Zwischenschrittmethode zur Lösung mehrdimensionaler Probleme der mathematischen Physik,” Lecture Notes in Mathematics, 109, Springer-Verlag, Heidelberg, 1969.
Birkhofer, A., and Werner, W., “Fully Implicit Matrix Decomposition Method for Space-Time Kinetics,” Trans. Am. Nucl. Soc., 15, pp. 789–790, 1972.
Langenbuch, S., and Werner, W., “Implicit Matrix Decomposition Scheme for Coarse-Mesh Methods”, Trans. Am. Nucl. Soc., 21, Page 224, 1975.
Varga, R. S., Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1962.
Wachspress, E. L., Iterative Solution of Elliptic Systems, Prentice-Hall, Englewood Cliffs, New Jsersy, 1966.
Nakamura, S., “A Variational Rebalancing Method for Linear Iterative Convergence Schemes for Neutron Diffusion and Transport Equations,” Nucl. Sci. Eng., 39, pp. 278–283, 1970.
Nakamura, S., “Coarse Mesh Acceleration of Iterative Solution of Neutron Diffusion Equation,” Nucl. Sci. Eng., 43, pp. 116–120, 1971.
Fröhlich, R., “A Theoretical Foundation of Coarse Mesh Variational Techniques,” Report CNM-R-2, Vol, 1, Page 219, (CONF-670501), 1967.
Anderson, M. M., Buckner, M. R., Carswell, J. H., Dodds, H. L., Gregory, M. V., Honeck, H. C., Routt, K. R., and Stewart, J. W., “Three-Dimensional Coupled Neutronic and Engineering Calculations of Savannah River Reactors,” Proc. Conf. Computational Methods in Nuclear Engineering, CONF-750413, Vol. II, VI, pp. 123–141, 1975.
Hestenes, M. R., and Stiefel, E., “Methods of Conjugate Gradients for Solving Linear Systems,” Nat. Bur. Standards, J. of Res., 49, pp. 409–436, 1952.
Langenbuch, S., and Werner, W., “Eine Methode zur Verbesserung der Zeitintegration in 3d Neutronenkinetik-Rechnungen durch eine Form der Periodenfaktorisierung,” Proc. Reaktortagung, 1976.
Garland, W. J., Vlachopoulos, J., Harms, A. A., “A Summation-Exponent Analysis for Space-Dependent Reactor Transients,” Trans. Am. Nucl. Soc., 18, Page 322, 1974.
Devought, J, and Mund, E., “A-Stable Algorithms for Neutron Kinetics,” MRR 145, Proc. of the Joint NEACRP/CSNI Specialists 1 Meeting on New Developments in Three-Dimensional Neutron Kinetics, pp. 21–71, 1975.
Yasinsky, J. B., “Combined Space-Time Synthesis with Axially Discontinuous Trial Functions,” USAEC Report, WAPD-TM-736, Westinghouse Electric Corp., Bettis Atomic Power Laboratory, 1967.
Yasinsky, J. B., “Numerical Studies of Combined Space-Time Synthesis,” Nucl. Sci. Eng., 34, pp. 158–168, 1968.
Henry, A. F., “Review of Computational Methods for Space-Dependent Kinetics,” Dynamics of Nuclear Systems, University of Arizona Press, Tuscon, Ariz., 1972.
Ott, K., and Madell, J. T., “Quasistatic Treatment of Spatial Phenomena in Reactor Dynamics,” Nucl. Sci. Eng., 26, pp. 563–565, 1966.
Ott, K., and Meneley, D. A., “Accuracy of the Quasistatic Treatment of Spatial Reactor Kinetics,” Nucl. Sci. Eng., 36, pp. 402–411, 1969.
Wagner, M. R., Finnemann, H., Lee, R. R., Meneley, D. A., Michelsen, B., Misfeldt, I., Vondy, D. R., Werner, W., “Multidimensional LWR Benchmark Problems,” Trans. Am. Nucl. Soc., 23, 1976.
Werner, W., Finnemann, H., Langenbuch, S., “Two- and Three-Dimensional BWR Kinetics Benchmark Problem,” Trans. Am. Nucl. Soc., 23, 1976.
“Nuclear Reactor Core Analysis Code: VENTURE,” Oak Ridge National Lab., 1976.
Cadwell, W. R., PDQ-7 Reference Manual, WAPD-TM-678, January 1967.
Misfeldt, I., “Solution of the Multigroup Neutron Diffusion Equations by the Finite Element Method,” RIS-M-1809, Danish AEC, Research Establishment RIS, Denmark, July 1975.
Buckel, G., “Vorschlag für ein Benchmark Problem in xyz- und Dreiecks-z-Geometrie,” INR Notiz, 335, 1975.
Buckel, G., Approximation der stationären, dreidimensionalen Mehrgruppen-Neutronen-Diffusionsgleichung durch ein Syntheseverfahren mit dem Karlsruher Synthese-Programm KASY, KfK-1349, 1971.
Dodds, H. L. Jr., Honeck, H. C, Hostetier, D. E., “Coarse-Mesh-Method for Two-Diraensional Mixed-Lattice Diffusion Theory Calculations,” Trans. Am. Nucl. Soc., 21, Page 223, 1975.
Schmidt, F. A. R., IKE Stuttgart, FRG, Private Communication.
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Werner, W. (1977). Kinetics of Nuclear System Solution Methods for the Space-Time Dependent Neutron Diffusion Equation. In: Henley, E.J., Lewins, J., Becker, M. (eds) Advances in Nuclear Science and Technology. Advances in Nuclear Science and Technology, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-9913-1_4
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