Second derivative General Linear Method in Nordsieck form

Robert I. Okuonghae; Monday Ndidi Oziegbe Ikhile

doi:10.33993/jnaat481-1140

Authors

Robert I. Okuonghae Department of Mathematics, University of Benin, Nigeria https://orcid.org/0000-0003-3776-1172
Monday Ndidi Oziegbe Ikhile Department of Mathematic; Faculty of Physical Science, University of Benin, Benin City, Nigeria https://orcid.org/0000-0002-2929-6576

DOI:

https://doi.org/10.33993/jnaat481-1140

Keywords:

General linear methods (GLM), Nordsieck vectors/methods, second derivative GLM, RK-starters, variable stepsize

Abstract views: 268

Abstract

This paper considers the construction of second derivative general linear methods (SD-GLM) from hybrid LMM and their transformation to Nordsieck
GLM.
How the Runge-Kutta starters for the methods can be derived are given.
The representation of the methods in Nordsieck form has the advantage of easy implementation in variable stepsize.

Downloads

Download data is not yet available.

Author Biography

Monday Ndidi Oziegbe Ikhile, Department of Mathematic; Faculty of Physical Science, University of Benin, Benin City, Nigeria

1. Professor M.N.O. Ikhile, Ph.D (Department of Mathematics)

2. Professor R. I. Okuonghae, Ph.D (Department of Mathematics)

References

A. Nordsieck, On Numerical Integration of Ordinary Differential Equations, Mathematics of Computation, 16, 1962, pp. 22-49. https://doi.org/10.2307/2003809 DOI: https://doi.org/10.1090/S0025-5718-1962-0136519-5

J. C. Butcher, G. Hojjati, Second derivative methods with RK stability. Numerical Algorithms, 40, 2005, pp. 415-429. https://doi.org/10.1007/s11075-005-0413-1 DOI: https://doi.org/10.1007/s11075-005-0413-1

J. C. Butcher, Numerical Methods for Ordinary Differential Equations. Second Edition., J. Wiley, Chichester, 2016. https://doi.org/10.1002/9781119121534 DOI: https://doi.org/10.1002/9781119121534

J. C. Butcher, A. E. O'Sullivan, Nordsieck methods with an off-step point. Numerical Algorithms, 31, (2002), pp. 87-101. https://doi.org/10.1023/a:1021104222126 DOI: https://doi.org/10.1023/A:1021104222126

J. C. Butcher and N. Rattenbury, ARK methods for stiff problems. Appl. Num. Maths., 53, (2005) 165-181. https://doi.org/10.1016/j.apnum.2004.09.033 DOI: https://doi.org/10.1016/j.apnum.2004.09.033

J. C. Butcher and N. Moir, Experiments with a new fifth order method}. Numerical Algorithms, 33, (2003), pp. 137-151. https://doi.org/10.1023/a:1025503719518 DOI: https://doi.org/10.1023/A:1025503719518

G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3, (1963), pp. 27-43. https://doi.org/10.1007/bf01963532 DOI: https://doi.org/10.1007/BF01963532

C. W. Gear, Runge-Kutta starters for multistep methods, ACM Trans. Math. Software 6, (1980), 263-279. https://doi.org/10.1145/355900.355901 DOI: https://doi.org/10.1145/355900.355901

W. B. Gragg and H. J. Stetter, Generalized multistep predictor-corrector methods., J. Assoc. Comput. Mach., 11, (1964), pp. 188-209. https://doi.org/10.1145/321217.321223 DOI: https://doi.org/10.1145/321217.321223

E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary Differential Equations II. Non-stiff Problems, Springer-Verlag, Berlin, 1980. https://doi.org/10.1007/bf02252917

E. Hairer, S. P. Norsett} and G. Wanner, Multistep-multistage multi-derivative methods for Ordinary Differential Equations., Computing, 11, (1973), pp. 114-123. https://doi.org/10.1007/bf02252917 DOI: https://doi.org/10.1007/BF02252917

S. J. Y, Huang, Implementation of General linear methods for stiff ordinary differential equations. Ph.D Thesis. Dept. of Mathematics, The University of Auckland, 2005.

R. I. Okuonghae, S. O. Ogunleye} and M. N. O. Ikhile, Some explicit general linear methods for IVPs in ODEs, J. of Algorithms and Comp. Technology, 7, No. 1 (2013), pp. 41-63. https://doi.org/10.1260/1748-3018.7.1.41 DOI: https://doi.org/10.1260/1748-3018.7.1.41

R. I. Okuonghae, Variable order explicit second derivative general linear methods, Comp. Applied Maths, 33, (2014), pp. 243-255. https://doi.org/10.1007/s40314-013-0058-y DOI: https://doi.org/10.1007/s40314-013-0058-y

R. I. Okuonghae and M. N. O. Ikhile, On the construction of high order A(alpha)-stable hybrid linear multistep methods for stiff IVPs and ODEs. Journal of Numerical Analysis and Application., 15, No. 3, (2012), pp. 231-241. https://doi.org/10.1134/s1995423912030056 DOI: https://doi.org/10.1134/S1995423912030056

R. I. Okuonghae and M. N. O. Ikhile, Second derivative general linear methods.,Numerical Algorithms. 67, 3, (2014), pp. 637-654. https://doi.org/10.1007/s11075-013-9814-8 DOI: https://doi.org/10.1007/s11075-013-9814-8

W. M. Wright, Explicit general linear methods with inherent Runge-Kutta stability.,Numerical Algorithms. 31, (2002), pp. 381-399. https://doi.org/10.1023/A:1021195804379 DOI: https://doi.org/10.1023/A:1021195804379

E. A. Karam, G. Hojjati and A. Abdi, Sequential second derivative general linear methods for stiff systems. Bulletin of the Iranian Mathematical Society; 40, 1, (2014), 83-100.

A. Movahedinejad, G. Hojjati and A. Abdi, Second derivative general linear methods with inherent Runge-Kutta stability. Numerical Algorithms 73, 2, (2016), 371-389.https://doi.org/10.1007/s11075-016-0099-6. DOI: https://doi.org/10.1007/s11075-016-0099-6

R. I. Okuonghae and M.N.O. Ikhile, Second derivative general linear methods with nearly ARK stability. J. Numer. Math., Vol. 22, No. 2, (2014), pp. 165-176. https://doi.org/10.1515/jnma-2014-0007. DOI: https://doi.org/10.1515/jnma-2014-0007