Two-point boundary value problems, Green's functions, and product integration

When the Green's function for a two-point boundary value problem can be found, the solution for any forcing term reduces to a quadrature. Here we investigate using this as the basis for numerical schemes for boundary value problems and parabolic initial-boundary value problems. Application of Gaussian quadrature is disappointing, only converging slowly due to the discontinuous derivative of the Green's functions; however, rapid convergence is recovered by the use of product integration; that is, the prior accurate evaluation of Green's integral for a set of basis functions and the subsequent evaluation for arbitrary forcing functions by a matrix-vector multiplication. The method requires more operations than finite differencing for the same number of nodes, but converges far more rapidly. For small numbers of nodes, the performance is similar to that for orthogonal collocation; however, at high resolution, collocation accuracy deteriorates due to ill-conditioning whereas the present method is stable