Geodesic
An algorithm of variable structure based on three-stage explicit-implicit methods
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 433-442.

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An explicit three-stage Runge–Kutta type scheme and L-stable Rosenbrock method are derived, both schemes of order 3. A numerical formula of order 1 is developed on the base of the stages of the explicit third order method. The stability interval of the first order formula is extended up to 18. The integration algorithm of variable order and step is constructed on the base of these three schemes. For each integration step the most efficient numerical scheme is chosen using an inequality for stability control. Numerical results confirming efficiency of the algorithm are given.
Keywords: stiff problem, one-step method, accuracy and stability control
Mots-clés : algorithm of variable structure. 
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A. E. Novikov; E. A. Novikov; M. V. Rybkov. An algorithm of variable structure based on three-stage explicit-implicit methods. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 433-442. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a105/

[1] E. Hairer, S. P. Norsett, G. Wanner, Solving ordinary differential equations I. Nonstiff Problems, Springer-Verlag, Berlin, 1987 | MR | Zbl

[2] E. Hairer, G. Wanner, Solving ordinary differential equations II. Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 1991 | MR | Zbl

[3] E. A. Novikov, Explicit methods for stiff systems, Nauka, Novosibirsk, 1997 (in russian) | MR | Zbl

[4] E. A. Novikov, Yu. V. Shornikov, Computer modelling stiff hybrid systems, NSTU press, Novosibirsk, 2013 (in russian)

[5] H. H. Rosenbrock, “Some general implicit processes for the numerical solution of differential equations”, Computer, 5 (1963), 329–330 | MR | Zbl

[6] V. A. Novikov, E. A. Novikov, L. A. Yumatova, “Freezing the Jacobi matrix in the second order Rosenbrock type method”, Computational Mathematics and Mathematical Physics, 27:3 (1987), 385–390 (in russian) | MR

[7] E. A. Novikov, A. L. Dvinskiy, “Freezing Jacobi Matrix in Rosenbrock-Type Methods”, Computational technologies, 10 (2005), 109–115 (in russian) | Zbl

[8] E. A. Novikov, “Developing the integration algorithm for stiff systems of differential equations on the base of variable structure scheme”, Proceedings of Academy of Sciences of USSR, 278:2 (1984), 272–275 (in russian) | MR | Zbl

[9] A. E. Novikov, E. A. Novikov, “Numerical integration of stiff systems with low accuracy”, Mathematical Models and Computer Simulations, 2:4 (2010), 443–452 (in russian) | MR | Zbl

[10] E. A. Novikov, “An algorithm of alternating order, step, and variable structure for solving stiff problems”, Izvestiya of Saratov University. New series. Series Mathematics. Mechanics. Informatics, 13:3 (2013), 34–41 (in russian) | Zbl

[11] V. A. Novikov, E. A. Novikov, “Stability control of explicit one-step methods for integration of ODEs”, Proceedings of Academy of Sciences of USSR, 277:5 (1984), 1058–1062 (in russian) | MR | Zbl

[12] E. A. Novikov, “An algorithm of alternating order and step based on the explicit three-stage method of the Runge–Kutta type”, Izvestiya of Saratov University. New series. Series Mathematics. Mechanics. Informatics, 11:3/1 (2011), 46–53 (in russian)

[13] G. Dahlquist, “A special stability problem for linear multistep methods”, BIT, 3 (1963), 23–43 | MR

[14] G. V. Demidov, L. A. Yumatova, Study of accuracy of implicit one-step methods, The preprint of the Computer Centre of the USSR AS Siberian branch, No 25, 1976 (in russian)

[15] E. A. Novikov, “Applying the third order Rosenbrock type methods to solving the Ring modulator”, Control systems and Information technologies, 2015, no. 2(60), 20–23 (in russian)

[16] E. A. Novikov, “(2,1)-Method for solving stiff nonautonomous problems”, Automation and Remote Control, 73:1 (2012), 191–197 | MR | Zbl

[17] W. H. Enright, T. E. Hull, Lindberg B., “Comparing numerical methods for stiff systems of O.D.E:s”, BIT, 15 (1975), 10–48 | Zbl

[18] F. Mazzia, C. Magherini, Test Set for Initial Value Problem Solvers, Report Department of Mathematics, University of Bari, Italy, No 4:2008, 2008