Global error estimation of linear multistep methods through the Runge-Kutta methods

Document Type : Research Article

Author

• J. Farzi

Department of Mathematics, Sahand University of Technology, P.O. Box 51335-1996, Tabriz, Iran.

10.22067/ijnao.v6i2.35736

Abstract

In this paper, we study the global truncation error of the linear multistep methods (LMM) in terms of local truncation error of the corresponding Runge-Kutta schemes. The key idea is the representation of LMM with a corresponding Runge-Kutta method. For this, we need to consider the multiple step of a linear multistep method as a single step in the corresponding Runge-Kutta method. Therefore, the global error estimation of a LMM through the Runge-Kutta method will be provided. In this estimation, we do not take into account the effects of roundoff errors. The numerical illustrations show the accuracy and efficiency of the given estimation.

Keywords

• Linear multistep methods
• Runge-Kutta methods
• Local trun- cation error
• Global error
• Error estimation

20.1001.1.24236977.2016.6.2.7.1

---

References

1. Ascher, U. M. and Petzold, L. R. Computer methods for ordinary differential equations and differential-algebraic equations, SIAM, Philadelphia, 2008.

2. Butcher, J. C. Numerical methods for ordinary differential equations, 2nd Edition, Wiley, 2008.

3. Cao, Y. and Petzold, L. A posteriori error estimation and global error control for ordinary differential equations by the adjoint method, SIAM J. Sci. Comput. 26 (2004), 359-374.

4. Gottlieb, S., Ketcheson, D. I. and Shu, C. W. High order strong stability preserving time discretizations, J. Sci. Comput. 38 (2009) 251-289.

5. Hadjimichael, Y., Ketcheson, D., Loczi, L. and Nemeth, A. Strong stability preserving explicit linear multistep methods with variable step size, Submitted.

6. Henrici, P. Discrete variable methods in ordinary differential equations, Wiley, New York, 1962.

7. Henrici, P. Error propagation for difference methods, Wiley, New York, 1963.

8. Iserles, A. A First course in the numerical analysis of differential equations, Cambridge University Press, 1996.

9. Lambert, J. D., Numerical methods for ordinarry differential systems: The initial value problem, Wiley, 1993.

10. Press, W.H., Teukolsky, S.A. and Vetterling, W.T., Flannery, BP. Numerical recipes: The art of scientific computing, 3rd ed., New York: Cambridge University Press, 2007.

11. Ruuth, S. J. and Hundsdorfer, W. High-order linear multistep methods with general monotonicity and boundedness properties, Journal of Computational Physics, 209 (2005) 226-248.

12. Suli, E., and Mayers, D. An Introduction to numerical analysis, Cambridge University Press, 2003.