BS2 methods for semi-linear second order boundary value problems

A new class of Linear Multistep Methods based on B-splines for the numerical solution of semi-linear second order Boundary Value Problems is introduced. The presented schemes are called BS2 methods, because they are connected to the BS (B-spline) methods previously introduced in the literature to deal with first order problems. We show that, when using an even number of steps, schemes with good general behavior are obtained. In particular, the absolute stability of the 2-step and 4-step BS2 methods is shown. Like BS methods, BS2 methods are of particular interest, because it is possible to associate with the discrete solution a spline extension which collocates the differential equation at the mesh points.