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

Abstract

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+\ heta}=x_{i-1}+\ heta 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(\ heta -\ heta^2)+\\beta^2(\ heta^2-\ heta)+\\alpha^2[\ heta^2-\ heta +\\beta^2(1+4\ heta -5\ heta^2)+\ heta(4\ heta^2-4\ heta -1)]\\cr &\\qquad \\qquad \\qquad +\\alpha[\ heta -\ heta^2+\\beta (1+4\ heta -4\ heta^2)+\\beta^2(4\ heta^2+4\ heta -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.