The Tau Method as an analytic tool in the discussion of equivalence results across numerical methods

Abstract

A Tau Method approximate solution of a given differential equation defined on a compact [a, b] is obtained by adding to the right hand side of the equation a specific minimal polynomial perturbation termH _n(x), which plays the role of a representation of zero in [a,b] by elements of a given subspace of polynomials. Neither discretization nor orthogonality are involved in this process of approximation. However, there are interesting relations between the Tau Method and approximation methods based on the former techniques. In this paper we use equivalence results for collocation and the Tau Method, contributed recently by the authors together with classical results in the literature, to identify precisely the perturbation termH(x) which would generate a Tau Method approximate solution, identical to that generated by some specific discrete methods over a given mesh Π ∈ [a, b]. Finally, we discuss a technique which solves the inverse problem, that is, to find adiscrete perturbed Runge-Kutta scheme which would simulate a prescribed Tau Method. We have chosen, as an example, a Tau Method which recovers the same approximation as an orthogonal expansion method. In this way we close the diagram defined by finite difference methods, collocation schemes, spectral techniques and the Tau Method through a systematic use of the latter as an analytical tool.