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Abstract
We propose global collocation methods for second-order initial-value problems y″=f(x, y) and y″=f(x, y, y′). The present methods are based on quintic C2-splines S(x) with three collocation points , j=1, …, 3 in each subinterval [x
i−1, x
i
], i=1, …, N. It is shown that the method (c
1=(5−√5)/10, c
2=(5+√ 5)/10) has a convergence of order six, while in the remaining cases (c
1, c
2∈(0, 1), with c
1≠c
2) the order is five. The absolute stability properties appear that for all c
1, c
2∈[0.8028, 1) with c
1≠c
2, the methods are A-stable independent of the particular choice of the collocation points, while the sixth-order method has a large region of absolute stability. Moreover, the sixth-order method has a phase-lag of order six with actual phase-lag (3/25(8!))v
6, and it possesses (0, 37.5)∪(60, 122.178) as the interval of periodicity and absolute stability. The superiority of the obtained methods is demonstrated by considering periodic stiff problems of practical interest.
Acknowledgements
The author is indebted to Professor S. E. El. Gendi for various valuable suggestions and constructive criticism.