On application of solution continuation method with respect to the best exponential argument in solving stiff boundary value problems

Cover Page

Cite item

Full Text

Abstract

The problematic of solving stiff boundary value problems permeates numerous scientific and engineering disciplines, demanding novel approaches to surpass the limitations of traditional numerical techniques. This research delves into the implementation of the solution continuation method with respect to the best exponential argument, to address these stiff problems characterized by rapidly evolving integral curves. The investigation was conducted by comparing the efficiency and stability of this novel method against the conventional shooting method, which has been a cornerstone in addressing such problems but struggles with the erratic growth of integral curves. The results indicate a marked elevation in computational efficiency when the problem is transformed using the exponential best argument. This method is particularly pronounced in scenarios where integral curves exhibit exponential growth speed. The main takeaway from this study is the instrumental role of the regularization parameter. Its judicious selection based on the unique attributes of the problem can dictate the efficiency of the solution. In summary, this research not only offers an innovative method to solve stiff boundary value problems but also underscores the nuances involved in method selection, potentially paving the way for further refinements and applications in diverse domains.

About the authors

Ekaterina D. Tsapko

Joint Stock Company “Interregional Energy Service Company ‘Energoefficiency Technologies’ ”

Author for correspondence.
Email: zapkokaty@gmail.com
ORCID iD: 0000-0002-4215-3510

Support engineer for Visiology platform

12 Semashko St., bldg. 8, Nizhnii Novgorod, 603155, Russian Federation

Sergey S. Leonov

RUDN University; Moscow Aviation Institute

Email: powerandglory@yandex.ru
ORCID iD: 0000-0001-6077-0435

Candidate of Physical and Mathematical Sciences, Assistant Professor of Nikolsky Mathematical Institute of Peoples’ Friendship University of Russia named after Patrice Lumumba; Assistant Professor of Department of Mechatronics and Theoretical Mechanics of Moscow Aviation Institute

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 4 Volokolamskoe shosse, Moscow, 125993, Russian Federation

Evgenii B. Kuznetsov

Moscow Aviation Institute

Email: kuznetsov@mai.com
ORCID iD: 0000-0002-9452-6577

Doctor of Physical and Mathematical Sciences, Professor of Department of Mechatronics and Theoretical Mechanics

4 Volokolamskoe shosse, Moscow, 125993, Russian Federation

References

  1. E. Hairer and G. Wanner, Solving ordinary differential equations, II: stiff and differential-algebraic problems. Berlin: Springer-Verlag, 1996.
  2. U. M. Ascher and L. R. Petzold, Computer methods for ordinary differential equations and differential-algebraic equations. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1998. DOI: 10. 1137/1.9781611971392.
  3. K. Dekker and J. G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations. North-Holland, Amsterdam, New York, Oxford, 1984.
  4. V. I. Shalashilin and E. B. Kuznetsov, Parametric continuation and optimal parametrization in applied mathematics and mechanics. Dordrecht, Boston, London: Kluwer Academic Publishers, 2003.
  5. E. B. Kuznetsov, S. S. Leonov, and E. D. Tsapko, “Applying the best parameterization method and its modifications for numerical solving of some classes of singularly perturbed problems,” vol. 274, 2022, pp. 311- 330. doi: 10.1007/978-981-16-8926-0_21.
  6. E. B. Kuznetsov, S. S. Leonov, and E. D. Tsapko, “A new numerical approach for solving initial value problems with exponential growth integral curves,” in IOP Conference Series: Materials Science and Engineering, AMMAI’2020, IOP Publishing, 2020. doi: 10.1088/1757899X/927/1/012032.
  7. K. W. Chang and F. A. Howes, Nonlinear singular perturbation phenomena: theory and application. New York: Springer, 1984.
  8. R. Temam, Navier-Stokes Equations: theory and numerical analysis. AMS Chelsea Publishing, 1997.
  9. H. Schlichting and K. Gersten, Boundary-layer theory. Springer, 2016. [10] P. Wesseling, Principles of computational fluid dynamics. Springer, 2009.
  10. G. A. Dahlquist, “A special stability problem for linear multistep methods,” BIT Numerical Mathematics, vol. 3, no. 1, pp. 27-43, 1963. doi: 10.1007/BF01963532.
  11. E. B. Kuznetsov, S. S. Leonov, and E. D. Tsapko, “Estimating the domain of absolute stability of a numerical scheme based on the method of solution continuation with respect to a parameter for solving stiff initial value problems,” Computational Mathematics and Mathematical Physics, vol. 63, no. 4, pp. 557-572, 2023. doi: 10.1134/S0965542523040115.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download