Abstract
Boundary conditions implementation in previously proposed bicompact schemes is studied. These schemes are constructed by the method of lines for a linear transport equation. These schemes are conservative, monotonic, and economical and can be solved by running calculation method. Methods are proposed for the boundary conditions implementation in bicompact schemes that ensure their high accuracy by using A- and L-stable diagonally implicit Runge-Kutta schemes with the third-order approximation for the time integration of the transport equation.
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References
A. V. Shil’kov, S. V. Shil’kova, E. N. Aristova, and V. Ya. Gol’din, “Efficient Precise Calculations of Atmospheric Radiation Based on ATRAD System,” Dokl. Math. 60(3), 469 (1999).
E. N. Aristova and V. Ya. Gol’din. Computation of Anisotropy Scattering of Solar Radiation in Atmosphere (Monoenergetic Case),” J. Quant. Spectrosc. Radiat. Transfer 67 (2), 139 (2000).
E. N. Aristova, D. F. Baydin, and V. Ya. Gol’din, “Two Variants of a Economical Method for Solving the Transport Equation in r-z Geometry on the Basis of Transition to Vladimirov’s Variables,” Mat. Model. 18(7), 43 (2006) [in Russian].
E. N. Aristova and V. Ya. Gol’din. “The Neutron Transport Equation Solving Coupled with the Quasi-Diffusion Equations Solving in r-z Geometry,” Mat. Model. 18 (11), 61 (2006) [in Russian].
D. F. Baydin, E. N. Aristova, and V. Ya. Gol’din, “Comparison of the Efficiency of the Transport Equation Calculation Methods in Characteristics Variables,” Transp. Theor. Stat. Phys. 37(02–04), 286 (2008).
V. Ya. Gol’din, G. A. Pestryakova, Yu. V. Troshchiev, and E. N. Aristova, “Investigation of Self-Adjustable Neutron-Nuclear Mode of the Second Kind in the Fast Reactor,” Mat. Model. 12(4), 33 (2000) [in Russian].
E. N. Aristova, “Simulation of Radiation Transport in a Channel on the Basis of the Quasi-Diffusion Method,” Transp. Theor. Stat. Phys. 37(05–07), 483 (2008).
E. N. Aristova, Modeling of the Radiation-Substance Interaction. Applying of Quasi-Diffusion Method (Lambert Academic Publishing, Saarbrucken, Germany, 2011).
E. N. Aristova, Modeling Energy Transfer Processes of a Laser Pulse with a Significant Role of Plasma Intrinsic Emmission on the Basis of the LATRANT complex,” in: Problems of Computational Mathematics, Mathematical Modeling and Informatics (M3 Press, Moscow, 2006) [in Russian].
L. P. Bass, O. V. Nikolaeva, V. S. Kuznetsov et al., “Optical Radiation Propagation Modeling in the Phantom of Biological Tissue by the Supercomputer MBC1000/M,” Mat. Model. 18(1), 29 (2006) [in Russian].
V. S. Kuznetsov, O. V. Nikolaeva I. P. Bass, et al., “Modeling of Ultrashort Light Pulse Propagation in a Strongly Scattering Medium.” Math. Models Computer Simul. 2(1), 22 (2010).
B. V. Rogov and M. N. Mikhailovskaya, “Monotone Bicompact Schemes for a Linear Advection Equation,” Dokl. Math. 83(1), 121 (2011).
B. V. Rogov and M. N. Mikhailovskaya, “Monotonic Bicompact Schemes for a Linear Transport Equation,” Math. Models Computer Simul. 4(1), 2 (2012).
A. A. Samarskii, Theory of Difference Schemes (Nauka, Moscow, 1989) [in Russian].
K. Dekker and J. G. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam-New York, 1984)
N. N. Kalitkin, Numerical Methods (Nauka, Moscow, 1978) [in Russian].
B. V. Rogov and M. N. Mikhailovskaya, “Fourth-Order Accurate Bicompact Schemes for Hyperbolic Equations,” Dokl. Math. 81 (1), 146 (2010).
N. N. Kalitkin and P. V. Koryakin, “Bicompact Schemes and Layered Media,” Dokl. Math. 77(2), 320 (2008).
U.G. Pirumov, Numerical Methods (Drofa, Moscow, 2007) [in Russian].
N. N. Kalitkin, “Numerical Methods for Solving Stiff Systems,” Mat. Model. 7(5), 8 (1995) [in Russian].
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Berlin, Springer, 1987).
N. N. Kalitkin, A. B. Al’shin, E. A. Al’shina, and B. V. Rogov, Calculations on Quasi-Uniform Grids (Fizmatlit, Moscow, 2005) [in Russian].
B. V. Rogov and M. N. Mikhailovskaya, “On the Convergence of Compact Difference Schemes,” Math. Models Computer Simul. 1(1), 91 (2009).
E. A. Al’shina, E. M. Zaks, and N. N. Kalitkin, “Optimal Parameters for Explicit Runge-Kutta Schemes of Lower Orders” Mat. Model. 18(2), 61 (2006) [in Russian].
E. A. Al’shina, E. M. Zaks, and N. N. Kalitkin, “Optimal First to Sixth Order Runge-Kutta schemes,” Comput. Math. Math. Phys. 48(3), 395 (2008).
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Original Russian Text © E.N. Aristova, B.V. Rogov, 2012, published in Matematicheskoe Modelirovanie, 2012, Vol. 24, No. 10, pp. 3–14.
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Aristova, E.N., Rogov, B.V. Boundary conditions implementation in bicompact schemes for the linear transport equation. Math Models Comput Simul 5, 199–207 (2013). https://doi.org/10.1134/S2070048213030022
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DOI: https://doi.org/10.1134/S2070048213030022