NUMERICAL MODELLING OF THE RING MODULATOR BY THE METHOD FOR IMPLICIT SYSTEMS SOLUTION

Novikov Evgeniy Aleksandrovich, Doctor of physical and mathematical sciences, chief researcher, Institute of computational modeling of the Siberian Branch of the Russian Academy of Sciences (building 44, 50 Academgorodok campus, Krasnoyarsk, Russia), novikov@icm.krasn.ru

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Background. At schematic designing of radioelectronic circuits and other important applications there occurs a necessity to solve the Cauchy problem for stiff systems of ordinary differential equations, unsolved for derivatives. The known methods are mainly aimed at solving explicit problems. Even in a basic case, reduction of the implicit system to an explicit form is associated with solution of a linear system of algebraic equations at each step. The matrix at derivatives usually is poor conditioned and often degenerate, and the problem in an explicit form is stiff. For its solution one needs applying the L-stability methods, which also require decomposition of the matrix. Efficiency of calculations can be increased by contemporary solving the system and meeting requirements of L-stability for a numerical scheme applying the same matrix.
Materials and methods. The decision function and its derivative were calculated approximately. For calculation accuracy control inequalities were applied. The first inequality provided accuracy of calculations and the second one was used for accuracy control of solution derivative calculations.
Results. The author developed an algorithm based on the L-stability method for solution of implicit problems. The method differs from the classic Rosenbrock schemes by approximate calculation of a solution derivative. The author constructed inequalities for stability control and gave the results of modelling the Ring Modulator.
Conclusion. The algorithm is designed for solving the Cauchy problem for implicit stiff systems of ordinary differential equations. The calculation results confirm the effectiveness of the constructed algorithm.

implicit system, Rosenbrock methods, accuracy control, ring modulator.

1. Hairer E., Wanner G. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problem. Berlin: Springer-Verlag, 1996, 614 p.
2. Brayton R. K., Gustavson F. G., Hachtel G. D. Proceedings of IEEE, 1972, pp. 98–108.
3. Boyarintsev Yu. E., Danilov V. A., Loginov A. A., Chistyakov V. F. Chislennye metody resheniya singulyarnykh system [Numerical methods for solving singular systems]. Novosibirsk: Nauka, 1989, 223 p.
4. Gear C. W. SIAM J. Sci. Stat. Comput. 1988, vol. 9, no. 1, pp. 39–47.
5. Novikov E. A., Yumatova L. A. Soviet Math. Dokl. 1988, vol. 36, no. 1, pp. 138–141.
6. Levykin A. I., Novikov E. A. Doklady RAN [Reports of the Russian Academy of Sciences]. 1996, vol. 348, no. 4, pp. 442–445.
7. Dekker K., Verwer J. G. Stability of Runge-Kutta methods for stiff nonlinear differential equations. North-Holland, 1984, 332 p.
8. Novikov E. A., Shornikov Yu. V. Komp'yuternoe modelirovanie zhestkikh gibridnykh sistem [Computer modeling of stiff hybrid systems]. Novosibirsk: Nauka, 2013, 451 p.
9. Rosenbrock H. H. Computer. 1963, no. 5, pp. 329–330.
10. Artem'ev S. S., Demidov G. V., Novikov E. A. Minimizatsiya ovrazhnykh funktsiy chislennym metodom dlya resheniya zhestkikh sistem uravneniy [Minimization of ravine functions by the numerical method for solving stiff systems of equations]. Preprint № 74. Novosibirsk: VTs SO AN SSSR, 1980, 13 p.
11. Novikov E. A. Yavnye metody dlya zhestkikh sistem [Explicit methods for stiff systems]. Novosibirsk: Nauka, 1997, 194 p.
12. Novikov E. A., Dvinskiy A. L. Vychislitel'nye tekhnologii [Computing technologies]. 2003, vol. 8, no. 3, pp. 272–278.
13. Mazzia F., Iavernaro F. Test Set for Initial Value Problem Solver. University of Bari: Department of Mathematics, 2003, 295 p.