Abstract
An explicit embedded method of the Dormand-Prince type designed for integrating systems of ordinary differential equations of special form is examined. A family of economical fifth-order numerical schemes for integrating systems of structurally separated ordinary differential equations is constructed.
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References
I. V. Olemskoy, “Modification of the Algorithm for Detecting Structural Singularities,” Vestn. St.-Petersburg Univ., Ser. 10 2, 46–54 (2006).
M. F. Subbotin, Introduction to Theoretical Astronomy (Nauka, Moscow, 1968) [in Russian].
S. A. Kutuzov, I. V. Olemskoy, L. P. Osipkov, and V. N. Starkov, Mathematical Methods for Research of Space Systems (St.-Petersburg Gos. Univ., St. Petersburg, 2003).
Yu. M. Kiselev and M. V. Orlov, “Numerical Algorithms for Linear Time-Optimal Control,” Zh. Vychisl. Mat. Mat. Fiz. 31, 1763–1771 (1991).
G. V. Shevchenko, “Numerical Algorithm for a Linear Time-Optimal Control Problem,” Zh. Vychisl. Mat. Mat. Fiz. 42, 1166–1178 (2002) [Comput. Math. Math. Phys. 42, 1123–1134 (2002)].
J. D. Lawson, The Physics of Charged-Particle Beams (Clarendon, Oxford, 1977; Mir, Moscow, 1980).
D. A. Ovsyannikov and N. V. Egorov, Mathematical Modeling of Electron and Ion Beam Forming Systems (St.-Petersburg Gos. Univ., St. Petersburg, 1998) [in Russian].
I. V. Olemskoy, “Numerical Method for Integration of Systems of Ordinary Differential Equations,” in Mathematical Methods for Analyzing Controlled Processes (Leningrad, 1986), pp. 157–160 [in Russian].
I. V. Olemskoy, “Economical Fourth-Order Numerical Integration Scheme for Systems of Special Form,” Proceedings of XXX Scientific Conference on Control Processes and Stability (Nauchno-Issled. Inst. Khimii, St.-Petersburg Gos. Univ., St. Petersburg, 1999), pp. 134–143.
I. V. Olemskoy, “Structural Approach to the Design of Explicit One-Stage Methods,” Zh. Vychisl. Mat. Mat. Fiz. 43, 961–974 (2003) [Comput. Math. Math. Phys. 43, 918–931 (2003)].
I. V. Olemskoy, “Fifth-Order Four-Stage Method for Numerical Integration of Special Systems,” Zh. Vychisl. Mat. Mat. Fiz. 42, 1179–1190 (2002) [Comput. Math. Math. Phys. 42, 1135–1145 (2002)].
J. R. Dormand and P. J. Prince, “New Runge-Kutta Algorithms for Numerical Simulation in Dynamical Astronomy,” Celestial Mech. 18, 223–232 (1978).
J. R. Dormand and P. J. Prince, “A Family of Embedded Runge-Kutta Formulas,” J. Comput. Appl. Math. 6, 19–26 (1980).
J. R. Dormand, M. E. A. El-Mikkawy, and P. J. Prince, “Families of Runge-Kutta-Nystrom Formulas,” J. Numer. Anal. 7, 235–250 (1987).
I. V. Olemskoy, “A Fifth-Order Five-Stage Embedded Method of the Dormand-Prince Type,” Zh. Vychisl. Mat. Mat. Fiz. 45, 1181–1191 (2005) [Comput. Math. Math. Phys. 45, 1140–1150 (2005)].
E. Hairer, S. P. Nörsett, and G. Wanner, Solving Ordinary Differential Equations. I: Nonstiff Problems (Springer-Verlag, Berlin, 1987; Mir, Moscow, 1990).
A. B. Arushanyan and S. F. Zaletkin, Numerical Solution of Ordinary Differential Equations in Fortran (Mosk. Gos. Univ., Moscow, 1990) [in Russian].
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Original Russian Text © A.S. Eremin, I.V. Olemskoy, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 3, pp. 434–448.
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Eremin, A.S., Olemskoy, I.V. An embedded method for integrating systems of structurally separated ordinary differential equations. Comput. Math. and Math. Phys. 50, 414–427 (2010). https://doi.org/10.1134/S0965542510030048
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DOI: https://doi.org/10.1134/S0965542510030048