Abstract
A finite analytic (not finite-difference) scheme is developed in the characteristic value for the solution of even- and odd-parity kinetic equations of neutron and photon transport with algebraic and centered forms of the scattering integral in one-dimensional problems with the symmetry of a plane layer, cylinder, and sphere.
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A. V. Shilkov, “Even- and odd-parity kinetic equations. I: Algebraic and centered forms of the scattering integral,” Math. Models Comput. Simul. 6(5), 465–480 (2014).
A. N. Tikhonov and A. A. Samarskii, “Homogeneous high-order difference schemes on nonuniform grids,” Zh. Vychisl. Mat. Mat. Fiz. 1(3), 425–440 (1961).
A. A. Samarskii, Introduction to the Theory of Difference Equations (Nauka, Moscow, 1971).
F. V. Atkinson, Discrete and Continuous Boundary Problems (Academic Press, New York/London, 1964).
A. M. Il’in, “A difference scheme for a differential equation with a small parameter at the highest derivative,” Mat. Zam. 6(2), 237–248 (1969).
K. V. Emel’yanov, “A difference scheme for three-dimensional elliptic equation with a small parameter at the highest derivatives,” Tr. Inst. Mat. Mekh Ural. Nauchn. Tsentr., Akad. Nauk SSSR No. 11, 30–42 (1973) [in Russian].
C. J. Chen, H. Naseri-Nashet, and K. S. Ho, “Finite Analytic Numerical Solution of Heat Transfer in Two Dimensional Cavity Flow,” Journal of Numerical Heat Transfer 4, 179–197 (1981).
C. J. Chen and H. C. Chen, “Finite analytic numerical method for unsteady two-dimensional Navier-Stokes equations,” J. Comp. Phys. 53(2), 209–226 (1984).
W. Lin, Y. Haik, R. Bernatz, and C. J. Chen, “Finite analytic method and its applications: a review,” Dyn. Atmos. Ocean. 27(1), 17–33 (1998).
C. J. Chen, R. Bernatz, K. D. Carlson, and W. Lin, Finite Analytic Method in Flows and Heat Transfer (Taylor and Francis, New York, 2000).
J. P. Pontaza, H. C. Chen, and J. N. Reddy, “A local-analytic-based discretization procedure for the numerical solution of incompressible flows,” Int. J. Numer. Methods Fluids 49(6), 657–699 (2005).
Z. F. Liu and X. H. Wang, “Finite analytic numerical method for two-dimensional fluid flow in heterogeneous porous media,” J. of Comp. Phys. 235(2), 286–301 (2013).
V. Ya. Gol’din, N. N. Kalitkin, and T. V. Shishova, “Nonlinear finite-difference schemes for hyperbolic equations,” Zh. Vychisl. Mat. Mat. Fiz. 5(5) (1965).
E. N. Aristova, Modelling the Interaction of Radiation with Matter (Lambert, Academic Publishing Saabrucken, 2011).
V. S. Vladimirov, “Numerical solution of the kinetic equation for a sphere,” in Computational Mathematics (Akad. Nauk SSSR, Moscow, 1958) [in Russian].
V. S. Vladimirov, “Mathematics and the creation of the first samples of atomic weapons,” Zh. At. Strat., 5(42), 27–29 (2009).
A. A. Abramov, “A variant of the sweep method,” Zh. Vychisl. Mat. Mat. Fiz. 1(2), 349–351 (1961).
G. I. Marchuk, Methods for the Calculation of Nuclear Reactors (Atomizdat, Moscow, 1961) [in Russian].
B. N. Chetverushkin, Mathematical Modeling of the Dynamics of a Radiating Gas (Nauka, Moscow, 1985) [in Russian].
R. D. Richtmyer and K.W. Morton, Difference Methods for Initial-Value Problems (Wiley, New York, London, 1967).
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Original Russian Text © A.V. Shilkov, 2014, published in Matematicheskoe Modelirovanie, 2014, Vol. 26, No. 7, pp. 33–53.
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Shilkov, A.V. Even- and odd-parity kinetic equations of particle transport. 2: A finite analytic characteristic scheme for one-dimensional problems. Math Models Comput Simul 7, 36–50 (2015). https://doi.org/10.1134/S2070048215010093
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DOI: https://doi.org/10.1134/S2070048215010093