On some 4-Point Spline Collocation Methods for Solving Ordinary Initial Value Problems

This paper discusses collocation schemes based on seventh C 3 -splines with four collocation points s_{i-1+\alpha}=x_{i-1}+\alpha h , x_{i-1+\beta}=x_{i-1}+\beta h , x_{i-1+\theta}=x_{i-1}+\theta h and x_i=x_{i-1}+h in each subinterval [ x i @ 1 , x i ], i = 1(1) N for solving initial value problems in ordinary differential equations including stiff equations. Here 0 < f < g < è < 1 are arbitrarily given. It is shown that the methods are convergent and the order of convergence is seven if: \eqalign {&\beta(\theta -\theta^2)+\beta^2(\theta^2-\theta)+\alpha^2[\theta^2-\theta +\beta^2(1+4\theta -5\theta^2)+\theta(4\theta^2-4\theta -1)]\cr &\qquad \qquad \qquad +\alpha[\theta -\theta^2+\beta (1+4\theta -4\theta^2)+\beta^2(4\theta^2+4\theta -1)]\le 0 and they are unstable if f , g , è < 0.7279115. Moreover, the absolute stability properties of the methods are considered. It shows that with 0.888035 h f < g < è < 1 the methods are A-stable while they are not if f h 0.5; on other hand, the sizes of regions of absolute stability increase when f , g , è M 1 m.