#### For citation:

Novikov E. A. Algorithm Variable Order, Step and the Configuration Variables for Solving Stiff Problems. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2013, vol. 13, iss. 3, pp. 35-43. DOI: 10.18500/1816-9791-2013-13-3-35-43

# Algorithm Variable Order, Step and the Configuration Variables for Solving Stiff Problems

An inequality for stability control of a Ceschino’s scheme of second order of accuracy is constructed. A numerical formula of order one is developed that is based on the stages of the this method and its stability interval is extended to 32. On a base of L-stable (2,1)-scheme and a numerical Ceschino’s formula, an algorithm of alternating structure, in which an efficient numerical formula is chosen on an every step by a stability criterion, is constructed. The algorithm is intended for solving stiff and non-stiff problems. There are shown results of calculations, confirming efficiency of this algorithm.

- Hairer E., Wanner G. Solving ordinary differential equations II. Stiff and differential-Algebraic problems. New York, Springer-Verlag, 1996, 601 p.
- Byrne G. D., Hindmarsh A. C. ODE solvers : a review of current and coming attractions. J. of Comput. Physics, 1987, no. 70, pp. 1–62.
- Rosenbrock H. H. Some general implicit processes for the numerical solution of differential equations. Computer, 1963, no. 5, pp. 329–330.
- Novikov V. A., Novikov E. A., Yumatova L. A. Freezing of a matrix of Jacobi in the Rosenbrock method of the second order of accuracy. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 1987, vol. 27, no. 3, pp. 385–390 (in Russian).
- Novikov E. A. Construction of algorithm for the integrating stiff differential equations on nonuniform schemes. Soviet Math. Dokl., 1984, vol. 30, no. 2, pp. 358– 361.
- Novikov E. A. Algorithm of Integrating Stiff Problems Using the Explicit and Implicit Methods. Izv. Sarat. Univ. N.S. Ser. Math. Mech. Inform., 2012, vol. 12, no. 4, pp.19–27 (in Russian).
- Novikov V. A., Novikov E. A. Increase of efficiency of algorithms of integration of the ordinary differential equations at the expense of stability control. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 1985, vol. 25, no. 7, pp. 1023–1030 (in Russian).
- Novikov E. A. Explicit methods for stiff systems. Novosibirsk, Nauka, 1997, 197 p. (in Russian).
- Novikov E. A., Shornikov Yu. V. Computer modeling of stiff hybrid systems. Novosibirsk, publishing house NGTU, 2012, 450 p. (in Russian).
- Novikov E. A., Shitov Yu. A., Shokin Yu. I. Onestep iteration-free methods of solving stiff systems. Soviet Math. Dokl., 1989, vol. 38, no. 1, pp. 212–216.
- Novikov A. E., Novikov E. A. Numerical integration of stiff systems with low accuracy. Mathematical Models and Computer Simulations, 2010, vol. 2, no. 4, pp. 443– 452. DOI: 10.1134/S2070048210040046.
- Ceschino F., Kuntzman J. Numerical solution of initial value problems. New Jersey: Prentice-Hall, Englewood Clis, 1966, 287 p. 13. Hindmarsh A. C. ODEPACK, a systematized collection of ODE solvers. Lawrence Livermore National La- boratory, 1982, preprint UCRL–88007.

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