The continuous extension of the B-spline linear multistep methods for BVPs on non-uniform meshes

B-spline methods are Linear Multistep Methods based on B-splines which have good stability properties [F. Mazzia, A. Sestini, D. Trigiante, B-spline multistep methods and their continuous extensions, SIAM J. Numer Anal. 44 (5) (2006) 1954-1973] when used as Boundary Value Methods [L. Brugnano, D. Trigiante, Convergence and stability of boundary value methods for ordinary differential equations, J. Comput. Appl. Math. 66 (1-2) (1996) 97-109; L. Brugnano, D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon and Breach Science Publishers, Amsterdam, 1998]. In addition, they have an important feature: if k is the number of steps, it is always possible to associate to the numerical solution a Ck spline of degree k + 1 collocating the differential equation at the mesh points. In this paper we introduce an efficient algorithm to compute this continuous extension in the general case of a non-uniform mesh and we prove that the spline shares the convergence order with the numerical solution. Some numerical results for boundary value problems are presented in order to show that the use of the information given by the continuous extension in the mesh selection strategy and in the Newton iteration makes more robust and efficient a Matlab code for the solution of BVPs.