Variable structure algorithm with the rosenbrock method of a third-order approximation applied

About the authors:

Evgeny A. Novikov, Head Researcher, Institute of Computational Modeling, Siberian Branch of Russian Academy of Science, Dr. Phys. and Math. Sci., Professor
Alexander A. Zakharov, Dr. Sci (Tech.), Professor, Secure Smart City Information Technologies Department, University of Tyumen; a.a.zakharov@utmn.ru

Abstract:

An L-stable three-step method of the Rosenbrock type of a third-order approximation has been developed to solve stiff problems. To control the accuracy of calculations, we have written the inequation based on analogous global error estimation. The estimation is performed according to the previous calculations. It allows choosing the integration step size without extensive computation. The study results in the inequation to control the accuracy of calculations, and leads to the variable structure integration algorithm.

References:

1. Hairer, E., Norsett, S.P., Wanner, G. Solving ordinary differential equations. Stiff and differential-algebraic problems. Berlin: Springer-Verlag, 1987. 528 p.

2. Hairer, E., Wanner, G. Solving ordinary differential equations. Non stiff problems. Berlin: Springer-Verlag, 1991. 601 p.

3. Dekker, K., Verwer, J.G. Stability of Runge–Kutta methods for stiff nonlinear differential equations. North-Holland: Walter de Gruyter GmbH & Co. KG, 1984. 321 p.

4. Novikov, E.A., Shornikov, Yu.V. Computer simulation of stiff hybrid systems. Novosibirsk: NSTU Publisher, 2012. 451 p. (in Russian).

5. Benkovich, E.C., Kolesov, Yu.B., Senichenkov, Yu.B. Practical simulation of dynamic systems. Saint Petersburg: BXV-Petersburg, 2002. 464 p. (in Russian).

6. Kolesov, Yu.B., Senichenkov, Yu.B. Simulation systems. Object-oriented approach. Saint Petersburg: BXV-Petersburg, 2006. 192 p. (in Russian).

7. Kolesov, Yu.B. Object-oriented modeling of complex dynamic systems. Saint Petersburg: SPbGPU Publisher, 2004. 239 p. (in Russian).

8. Demidov, G.V., Yumatova, L.A. Investigation of some approximations in connection with the A-stable semi-explicit method // Numerical methods for continuum mechanics. 1977. Vol. 8. № 3. Pp. 68–79. (in Russian).

9. Novikov, A.E., Novikov, E.A. Numerical Integration of Stiff Systems with Low Accuracy // Mathematical Models and Computer Simulations. 2010. Vol. 2. № 4. Pp. 443–452. (in Russian).

10. Novikov, E.A. Explicit methods for stiff systems. Novosibirsk: Nauka, 1997. 197 p.

(in Russian).

11. Novikov, E.A., Zakharov, A.A. Explicit Runge–Kutta methods: algorithms to control the accuracy of the calculations // Tyumen State University Herald. Physics and Mathematics. Tyumen, 2010. № 6. Pp. 101–107. (in Russian).

12. Novikov, E.A., Zakharov, A.A. Algorithm on variable order based Cheskino method steps // Tyumen State University Herald. Physics and Mathematics. Tyumen, 2013. № 7. Pp. 116–123. (in Russian).