Smooth Spline Collocations for BVPs

A class of knot spline collocation methods related to B-splines is here considered for the numerical solution of general Boundary Value Problems. When correctly used, the k--th method of this class produces a k+1 degree spline function with smoothness C^k and whose approximation order for smooth solutions is O(h^p), p >=k+1. An important feature of these collocation methods is that the numerical solution, i.e. the set of spline values at the knots can be determined just using related Linear Multistep methods, here denoted BS methods. The k-step BS method has order p and must be used as a Boundary Value Method combined with suitable additional left and right auxiliary methods. Considering that in many case it can be sufficient to approximate the BVP solution at the mesh points, we first determine the numerical solution. Successively, the collocation spline coefficients in the B--spline basis can be cheaply computed as the solution of an Hermite interpolation problem at the knots by solving local linear systems of size 2k+2.In the uniform case the coefficients of the k-step BS method are explicitly known because they are just the values and the derivative values at its active knots of the uniform B-spline of degree k+1 and integer knots. In the non uniform case their determination requires the solution of related local linear systems. For the k-step BS method, these systems have size 2k+2 and they can be efficiently and accurately solved using a symmetric block factorization which can also profitably reused to compute the spline coefficients.