519.63 Development and testing for methods of solving stiff ordinary differential equations

Galanin M. P. (Bauman Moscow State Technical University/Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Khodzhaeva S. R. (Bauman Moscow State Technical University)

THE PAPER IS AIMED AT RESEARCH OF THE (M,K)-METHOD, CROS, FINITE SUPERELEMENT METHOD AND 4-STAGE EXPLICIT RUNGE–KUTTA METHOD FOR SOLVING STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. ANALYSIS OF TESTS RESULTS SHOWED THAT THE BEST CHOICE FOR SYSTEMS WITH HIGH STIFFNESS IS CROS. THE FINITE SUPERELEMENT METHOD IS THE «PRECISE» METHOD FOR SOLVING LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS, THE BEST SUPPORTING METHOD FOR ITS IMPLEMENTATION IS (4,2)-METHOD. THE VARIATION OF THE FINITE SUPERELEMENT METHOD HAS BEEN BUILT AND TESTED FOR SOLVING NONLINEAR PROBLEMS, THIS METHOD PROVED TO BE UNSUITABLE FOR PROBLEMS WITH HIGH STIFFNESS.


doi: 10.18698/2309-3684-2014-4-95119


The paper is aimed at research of the (m,k)-method, CROS, finite superelement method and 4-stage explicit Runge–Kutta method for solving stiff systems of ordinary differential equations. Analysis of tests results showed that the best choice for systems with high stiffness is CROS. The finite superelement method is the «precise» method for solving linear systems of ordinary differential equations, the best supporting method for its implementation is (4,2)-method. The variation of the finite superelement method has been built and tested for solving nonlinear problems, this method proved to be unsuitable for problems with high stiffness.


[1] Galanin M.P., Savenkov E.B. Methods of numerical analysis of mathematical models. Moscow, BMSTU Publ., 2010, 591 p.
[2] Kalitkin N.N. Numerical methods. Moscow, Nauka Publ., 1978, 512 p.
[3] Arushanyan O.B., Zaletkin S.F., Kalitkin N.N. Vychislitel’nye Metody i Programmirovanie. Numerical Methods and Programming, 2002, vol. 3, pp. 11−19.
[4] Novikov E.A. Vychislitel‘nye tekhnologii. Computational Technologies, 2007, vol. 12, no. 5, pp. 103−115.
[5] Kalitkin N.N. Semi-explicit schemes for the problems of high rigidity. Encyclopedia of low emperature plasma, series B, vol. VII-1, part. 1. Moscow, Yanus-К Publ., 2008, pp. 153−171.
[6] Al’shin A.B., Al’shina E.A., Kalitkin N.N., Koryagina A.B. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki. Computational Mathematics and Mathematical Physics, 2006, vol. 46, no. 8, pp. 1392−1414.
[7] Al’shin A.B., Al’shina E.A., Limonov A.G. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki. Computational Mathematics and Mathematical Physics, 2009, vol. 49, no. 2, pp. 270−287.
[8] Shirkov P.D. Matematicheskoe Modelirovanie. Mathematical Models and Computer Simulations, 1992, vol. 4, no. 8, pp. 47−57.
[9] Kalitkin N.N., Panchenko S.L. Matematicheskoe Modelirovanie. Mathematical Models and Computer Simulations, 1999, vol. 11, no. 6, pp. 52−81.
[10] Rosenbrock H. H. Some general implicit processes for the numerical solution of differential equations. Comput J., 1963, vol. 5, no. 4, рр. 329−330.
[11] Fedorenko R.P. An Introduction to Computational Physics. Moscow, MIPT Publ., 1994, 528 p.
[12] Galanin M.P., Milyutin D.C., Savenkov E.B. The development, research and application of the finite super element method for solving biharmonic equation. Keldysh Institute of Applied Mathematics RAS, Preprint, 2005, no. 59, 26 p.
[13] Galanin M.P., Savenkov E.B. The method of finite superelement for the problem of high-speed skin layer. Keldysh Institute of Applied Mathematics RAS, Preprint, 2004, no. 3, 32 p.
[14] Galanin M.P., Savenkov E.B. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki. Computational Mathematics and Mathematical Physics, 2003, vol. 43, no. 5, pp. 713−729.
[15] Samarsky A.A., Gulin A.V. Chi. Numerical methods. Moscow, Nauka Publ., 1989, 432 p.
[16] Hairer E., Norsett S., Wanner J. Solving ordinary differential equations. Part 2. Stiff and Differential-Algebraic Problems. New York, Springer, 1991.
[17] Galanin M.P., Khodzhaeva S.R. The methods of solving stiff ordinary differential equations. The results of test calculations. Keldysh Institute of Applied Mathematics RAS, Preprint, 2013, no. 98, 29 p.


Galanin M., Khodzhaeva S. Development and testing for methods of solving stiff ordinary differential equations. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 95-119



Download article

Колличество скачиваний: 155