Eduardo L. Ortiz

Numerical solution of eigenvalue problems for partial differential equations with the Tau-lines method

Abstract We discuss a hybrid approach which uses the Tau Method in combination with the Method of Lines and treat a number of eigenvalue problems defined by partial differential equations with constant and variable coefficients, on rectangular or circular domains and with the eigenvalue parameters entering in the equation or in the boundary conditions. We obtain results of considerable accuracy which compare favourably with those published in the very recent literature and obtained by using advanced formutions of the boundary integral equation method or the multigrid method.

A bidimensional tau-elements method for the numerical solution of nonlinear partial differential equations with an application to burgers' equation

Abstract We discuss a direct formulation of the Tau Method in two dimensions which differs radically from former techniques in that no discretization is introduced in any of the variables. A segmented formulation in terms of Tau elements is discussed and applied to the numerical solution of nonlinear partial differential equations which develop steep fronts in their domain.

Error analysis of the Tau method: dependence of the error on the degree and on the length of the interval of approximation

Abstract In this paper, we discuss the dependence of Tau Method approximations on (i) the degree of approximant n and (ii) the length of the interval of approximation h . We shall show that the Tau Method parameters τ i (i) decay exponentially in terms of n and (ii) for a fixed n they decay in terms of h as ( h /2) n .

Numerical solution of ordinary and partial functional-differential eigenvalue problems with the Tau method

We apply a recent new formulation of the Tau Method to reduce the numerical treatment of eigenvalue problems for ordinary and partialfunctional-differential equations to that of generalized algebraic eigenvalue problems. We find accurate numerical results through the use of a simple algorithm which we discuss in applications to several concrete examples. Extrapolation is used to refine the results already obtained.ZusammenfassungMit Hilfe einer neuen Formulierung der Tau-Methode werden Eigenwertprobleme für gewöhnliche und partielle Funktional-Differentialgleichungen auf verallgemeinerte algebraische Eigenwertprobleme reduziert. Ein einfacher Algorithmus liefert genaue numerische Ergebnisse, die wir an Hand einiger konkreter Beispiele diskutieren. Zur weiteren Verbesserung der Ergebnisse wird Extrapolation verwendet.

A unified approach to the Tau Method and Chebyshev series expansion techniques

Abstract In this paper, we show the full equivalence between the recursive [1] and operational [2] formulations of the Tau Method. We then use such methods as analytic tools in the simulation of other numerical techniques for the approximate solution of differential equations, as Tau Methods with special perturbation terms. In this paper, we consider, as examples of this approach, two classical numerical techniques based on Chebyshev series expansions. We introduce a numerical parameter, the length of a method, to compare different numerical techniques with reference to a given basis. Our results make possible the recursive formulation of a variety of series expansion methods.

A recursive formulation of collocation in terms of canonical polynomials

In this paper we discuss two related but analytically different techniques: the collocation method and Ortizs recursive formulation of the Tau Method. Specifically, we show that it is possible to simulate with the Tau Method collocation approximants for any desired degree. We give a representation for collocation approximants in terms ofshifted canonical polynomials, which are introduced here. We show that in the linear case computing a collocation approximant of orderN by this new approach requiresO(N) arithmetic operations while obtaining the same approximant by the direct approach involvesO(N3). Furthermore, our technique leads to a recursive formulation of collocation. We discuss separately the linear and nonlinear cases and propose a more efficienteconomized approach for the latter.ZusammenfassungWir diskutieren zwei verwandte aber analytisch verschiedene Techniken: Kollokation und die rekursive Formulierung der Tau-Methode von Ortiz. Dabei zeigen wir, dass man mit der Tau-Methode Kollokationsnäherungen beliebigen Grades simulieren kann. Wir stellen Kollokationsnäherungen durch hier eingeführte verschobene kanonische Polynome dar. Wir zeigen, dass die Berechnung einer Kolokationsnäherung der OrdnungN bei diesem Vorgehen im linearen Fall nurO(N) Operationen kostet im Vergleich mitO(N3) Operationen beim direkten Vorgehen. Ausserdem führt unser Vorgehen auf eine rekursive Formulierung der Kollokation. Wir behandeln den linearen und den nichtlinearen Fall und schlagen für letzteren ein effizienteres Vorgehen vor.

Tau method approximation of differential eigenvalue problems where the spectral parameter enters nonlinearly

Abstract We discuss the use of recent new formulations of the Tau method for the numerical approximation of differential eigenvalue problems where the spectral parameter appears nonlinearly. Our approach enables us to translate the differential eigenvalue problem into a generalized algebraic eigenvalue problem, which is formulated by using a standard technique easy to implement in a computer. We consider several examples and report results of high accuracy.

A Tau method based on non-uniform space-time elements for the numerical simulation of solitons

Abstract In this paper we discuss the numerical simulation of soliton solutions of Schrodngers system of nonlinear partial differential equations. The technique used is based on a novel approach to the Tau Method based on space-time elements . The original operator is not discretized in either space or time. The perturbation term, which in the Tau Method is a by-product of the computation, is used to measure the error in the equation, or defect , and therefore to control the accuracy of the computation in an adaptive way. This leads to an accurate and economical numerical procedure which, even for approximations of a very low degree, can simulate surfaces with very sharp gradients in the direction of both, space and time. The Tau Method formulation presented in this paper preserves very accurately the principle of conservation of energy. However, our numerical examples support the idea that the latter is a necessary but not a sufficient condition to guarantee an efficient approximation of soliton solutions of Schrodingers equation by means of a method of numerical simulation.

Error analysis of the Tau method: dependence of the approximation error on the choice of perturbation term

We consider a system of ordinary differential equations with constant coefficients and deduce asymptotic estimates for the Tau Method approximation error vector per step for different choices of the perturbation term Hn(x). The cases considered are Legendre polynomials, Chebyshev polynomials, powers of x and polynomials of the form (x2 − r2)n, −r ⩽ x ⩽ r. The first two are standard choices for the Tau Method, for Chebyshev and Legendre series expansion techniques and also for collocation; the third one realizes the classical power series expansion techniques in the framework of the Tau Method and the last is related to the trial functions used in weighted residuals methods; we shall refer to it as the weighted residuals choice. We show that the resulting Tau Method implementations can be arranged into the following scale of increasing error estimates at the end point x = r: Legendre < Chebyshev ⪡ Power series < Weighted residuals. For the interesting case of Legendre Tau approximations, we offer upper and lower error bounds for the end point of the interval of approximation. In particular, this last estimates solve a conjecture on increased accuracy at the end point of the interval of approximation formulated by Lanczos in 1956. Such conjecture has equivalent forms for other polynomial methods for the numerical solution of differential equations. Although formulated in the convenient framework of the recursive Tau Method (see Ortiz [1]), the results given here apply, without essential modifications, to Chebyshev or Legendre series expansion techniques for differential equations, collocation and spectral methods. We give numerical examples which confirm the sharpness of the lemmas and theorems given in this paper. Finally, we discuss in an example the application of our results to the analysis of singularly perturbed differential equations.

Differential equations with piecewise approximate coefficients: Discrete and continuous estimation for initial and boundary value problems

Abstract We give sharp error estimates for both function and derivative when the coefficients and right hand side of a given initial or boundary value problem for ordinary differential equations are replaced by local approximations. These estimates are given for partition points and also continuously on subintervals. Numerical examples demonstrate the accuracy of our estimates.