DOI: https://doi.org/10.26089/NumMet.v16r342

A variable structure algorithm using the (3,2)-scheme and the Fehlberg method

Authors

  • E.A. Novikov

Keywords:

stiff systems
k)-schemes
Fehlberg method
Runge–Kutta methods
accuracy and stability control
variable structure algorithm
ordinary differential equations
numerical methods

Abstract

A third-order (3,2)-method allowing freezing the Jacobi matrix is constructed. Its main and intermediate numerical schemes are L-stable. An accuracy control inequality is obtained using an embedded method of second order. A stability control inequality for the explicit three-stage Runge-Kutta-Fehlberg method of third order is proposed. A variable structure algorithm is formulated. An explicit or L-stable method is chosen according to the stability criterion at each step. Numerical results are discussed.


Published

2015-08-09

Issue

Section

Section 1. Numerical methods and applications

Author Biography

E.A. Novikov


References

  1. E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems (Springer, Berlin, 1996; Mir, Moscow, 1999).
  2. E. Hairer, S. P. Nörsett, and G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems (Springer, Berlin, 1987; Mir, Moscow, 1990).
  3. E. A. Novikov and Yu. V. Shornikov, Computer Simulation of Stiff Hybrid Systems (Novosibirsk Tech. Univ., Novosibirsk, 2012) [in Russian].
  4. N. D. Demidenko, V. A. Kulagin, and Yu. I. Shokin, Modeling and Computational Technologies of Distributed Systems (Nauka, Novosibirsk, 2012) [in Russian].
  5. H. H. Rosenbrock, “Some General Implicit Processes for the Numerical Solution of Differential Equations,” Comput. J. 5 (4), 329-330 (1963).
  6. V. A. Novikov, E. A. Novikov, and L. A. Yumatova, “Freezing of the Jacobi Matrix in the Second Order Rosenbrock Method,” Zh. Vychisl. Mat. Mat. Fiz. 27 (3), 385-390 (1987) [USSR Comput. Math. Math. Phys. 27 (2), 41-45 (1987)].
  7. E. A. Novikov and A. L. Dvinskii, “Jacoby Matrix Freezing for Rosenbrock-Type Methods,” Vychisl. Tekhnol. 10, 108-114 (2005).
  8. A. E. Novikov and E. A. Novikov, “Numerical Integration of Stiff Systems with Low Accuracy,” Mat. Model. 22 (1), 46-56 (2010) [Math. Models Comput. Simul. 2 (4), 443-452 (2010)].
  9. E. A. Novikov, “Construction of an Algorithm for the Integrating Stiff Differential Equations on Nonuniform Schemes,” Dokl. Akad. Nauk SSSR 278 (2), 272-275 (1984) [Sov. Math. Dokl. 30 (2), 358-361 (1984)].
  10. V. A. Novikov and E. A. Novikov, “Control of the Stability of Explicit One-Step Methods of Integrating Ordinary Differential Equations,” Dokl. Akad. Nauk SSSR 277 (5), 1058-1062 (1984) [Sov. Math. Dokl. 30 (1), 211-215 (1984)].
  11. E. A. Novikov, Explicit Methods for Stiff Systems (Nauka, Novosibirsk, 1997) [in Russian].
  12. E. A. Novikov, Yu. A. Shitov, and Yu. I. Shokin, “One-Step Noniteration Method for Solving Stiff Systems,” Dokl. Akad. Nauk SSSR 301 (6), 1310-1314 (1988).
  13. G. G. Dahlquist, “A Special Stability Problem for Linear Multistep Methods,” BIT Numer. Math. 3 (1), 27-43 (1963).
  14. G. V. Demidov and L. A. Yumatova, The Investigation of Precision of Implicit One-Step Methods , Preprint No. 25 (Comput. Center of Siberian Branch of USSR Academy of Sciences, Novosibirsk, 1976).
  15. E. Fehlberg, Low Order Classical Runge-Kutta Formulas with Step Size Control and Their Application to Some Heat Transfer Problems , NASA Technical Report R 315 (NASA, Huntsville, 1969).
  16. G. D. Byrne and A. C. Hindmarsh, “Stiff ODE Solvers: A Review of Current and Coming Attractions,” J. Comput. Phys. 70 (1), 1-62 (1987).
  17. W. H. Enright, T. E. Hull, and B. Lindberg, “Comparing Numerical Methods for Stiff Systems of ODE’s,” BIT Numer. Math. 15 (1), 10-48 (1975).
  18. F. Mazzia and C. Magherini, Test Set for Initial Value Problem Solvers , Technical Report 4/2008 (University of Bari, Bari, 2008).