DOI QR코드

DOI QR Code

AN IMPROVED IMPLICIT EULER METHOD FOR SOLVING INITIAL VALUE PROBLEMS

  • YUN, BEONG IN (DEPARTMENT OF MATHEMATICS, KUNSAN NATIONAL UNIVERSITY)
  • 투고 : 2022.07.06
  • 심사 : 2022.08.06
  • 발행 : 2022.09.25

초록

To solve the initial value problem we present a new single-step implicit method based on the Euler method. We prove that the proposed method has convergence order 2. In practice, numerical results of the proposed method for some selected examples show an error tendency similar to the second-order Taylor method. It can also be found that this method is useful for stiff initial value problems, even when a small number of nodes are used. In addition, we extend the proposed method by using weighted averages with a parameter and show that its convergence order becomes 2 for the parameter near $\frac{1}{2}$. Moreover, it can be seen that the extended method with properly selected values of the parameter improves the approximation error more significantly.

키워드

과제정보

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) [No. 2021R1F1A1047343].

참고문헌

  1. R. Kress, Numerical Analysis, Graduate Texts in Mathematics 181, Springer Verlag, New York, 1998.
  2. K.A. Atkinson, An Introduction to Numerical Analysis, 2nd edition, John Wiley & Sons, New York, 1989.
  3. R.L. Burden and J.D. Faires, Numerical Analysis, 6th Edition, Springer, New York, 1997.
  4. J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, New York, 2003.
  5. A.M. Wazwaz, A new method for solving singular initial value problems in the second- order ordinary differ- ential equations, Applied Mathematics and Computations, 28 (2002), 45-57. https://doi.org/10.1016/S0096-3003(01)00021-2
  6. K.H. Mohammad, S.A. Mohammad, A.H. Mohammad, A.S. Mohamma, and B.H. Mohammad, An Implicit Method for Numerical Solution of Second Order Singular Initial Value Problems, The Open Mathematics Journal, 7 (2014), 1-5. https://doi.org/10.2174/1874117701407010001
  7. O. Koch, P. Kofler, and E.B. Weinmuller, The implicit Euler method for the numerical solution of singular initial value problems, Applied Numerical Mathematics, 34 (2000), 231-252. https://doi.org/10.1016/S0168-9274(99)00130-0
  8. D.F. Griffiths and D.J. Higham, Numerical methods for ordinary differential equations, Springer, New York, 2010.
  9. A. Guzel and W. Layton, Time filters increase accuracy of the fully implicit method, BIT Numerical Mathe- matics, 58 (2018), 301-315. https://doi.org/10.1007/s10543-018-0695-z
  10. G.G. Dahlquist, A special stability problem for linear multistep methods, BIT Numerical Mathematics, 3 (1963), 27-43. https://doi.org/10.1007/BF01963532
  11. G.G. Dahlquist, W. Liniger, and O. Nevanlinna, Stability of two-step methods for variable integration steps, SIAM Journal on Numerical Analysis, 20 (1983), 1071-1085. https://doi.org/10.1137/0720076
  12. Wikipedia, Stiff equation, http://en.wikipedia.org/wiki/Stiff_equation#A-stability.