On some 4-point spline collocation methods for solving second-order initial value problems

Abstract

This paper discusses collocation schemes based on seventh C 3 -splines with four collocation points x i−1+α=x i−1+αh,x i−1+β=x i−1+βh,x i−1+θ=x i−1+θh and x i=x i−1+h in each subinterval [x i−1,x i], i=1(1)N , for solving second-order initial value problems in ordinary differential equations including stiff equations. Here 0<α<β<θ<1 are arbitrarily given. It is shown that the methods are convergent and the order of convergence is seven if: 1−α−β−θ+αβ+αθ+βθ−2αβθ⩽0 and they are unstable if α,β,θ<0.5. Moreover, the absolute stability properties of the methods are considered. It shows that with 0.888035⩽α<β<θ<1 the methods possess unbounded regions of absolute stability, on other hand, the sizes of regions of absolute stability increase when α,β and θ are close to 1 − .