High Order Continuous Approximation for the Top Order Methods

The Top Order Methods are a class of linear multistep schemes to be used as Boundary Value Methods and with the feature of having maximal order (2k if k is the number of steps). This often implies that accurate numerical approximations of general BVPs can be produced just using the 3-step TOM. In this work, we consider two different possibilities for defining a continuous approximation of the numerical solution, the standard C-1 cubic spline collocating the differential equation at the knots and a C2k-1 spline of degree 2k. The computation of the B-spline coefficients of this higher degree spline requires the solution of N + 2k banded linear systems of size 4k x 4k. The resulting B-spline function is convergent of order 2k to the exact solution of the continuous BVPs.