Linear systems of singular linear ordinary differential equations with variable coefficient matrices and non-smooth inhomogeneities arise in several applications, either as models by themselves or as discretizations of certain partial differential equations. Based on previous results on the existence and uniqueness of smooth solutions for this class of systems, based in particular on the properties of the spectrum of the coefficient matrix, in the present paper a convergence analysis of the piecewise polynomial collocation method as a numerical integrator is carried out. The class of collocation methods considered here to approximate the solution of the problem is based on the construction on a grid of size \(h\) of a piecewise continuous polynomial function that reduces in each subinterval \([t_j, t_{j+1} = t_j + h]\) of the grid to a polynomial of degree less or equal to \(k\), the number of equidistantly spaced collocation nodes in each subinterval.
The treatment proceeds by stages: first a class of boundary value problems that can be expressed as a well-posed initial value problem (with all boundary conditions posed at \(t=0\)) is considered; then the collocation scheme is shown to converge for terminal value problems, and finally a general value problem is analyzed. The main result is contained in Theorem 7.2, which states that, under quite general assumptions, the unique solution \(p\) of the collocation scheme verifies \(\|p - y\| \leq \mathrm{const}\cdot h^k\), where \(y\) is the unique solution of the boundary value problem. In other words, the collocation method with \(k\) collocation points retains its classical stage order \(k\) uniformly in \(t\).
The numerical examples provided clearly illustrate this behavior. In particular, if Gaussian points are taken as collocation points, then only order \(k+1\) is exhibited (instead of the typical order \(2k\)).