THE TAU-METHOD AS AN ANALYTIC TOOL IN THE DISCUSSION OF EQUIVALENCE RESULTS ACROSS NUMERICAL-METHODS

A Tau Method approximate solution of a given differential equation def ined on a compact [a, b] is obtained by adding to the right hand side of the equation a specific minimal polynomial perturbation term H-n(x) , which plays the role of a representation of zero in [a,b] by element s of a given subspace of polynomials. Neither discretization nor ortho gonality are involved in this process of approximation. However, there are interesting relations between the Tau Method and approximation me thods based on the former techniques. In this paper we use equivalence results for collocation and the Tau Method, contributed recently by t he authors together with classical results in the literature, to ident ify precisely the perturbation term H(x) which would generate a Tau Me thod approximate solution, identical to that generated by some specifi c discrete methods over a given mesh II is an element of [a, b]. Final ly, we discuss a technique which solves the inverse problem, that is, to find a discrete perturbed Runge-Kutta scheme which would simulate a prescribed Tau Method. We have chosen, as an example, a Tau Method wh ich 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.