On a class of spline-collocation methods for solving second-order initial-value problems

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.