One parameter family of linear difference equations and the stability problem for the numerical solution of ODEs
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The study of the stability properties of numerical methods leads to considering linear difference equations depending on a complex parameter q. Essentially, the associated characteristic polynomial must have constant type for q ∈ C− . Usually such request is proved with the help of computers. In this paper, by using the fact that the associated polynomials are solutions of a “Legendre-type” difference equation, a complete analysis is carried out for the class of linear multistep methods having the highest possible order. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1. Introduction The problem to approximate the solutions of differential equations by substituting to them “appropriate” difference equations is as old as the differential calculus. Of course the main problem to be solved is the control of the errors between the continuous and the discrete solutions. In the fifties, the fundamental importance of the stability properties of the difference equations on the error propagation was recognized. After that, a lot of efforts has been done in this field, mainly when the methods are applied to dissipative problems. In such a case, the first approximation theorem permits to transform the nonlinear problem into a linear one. Regarding the class of linear multistep methods (LMMs), the propagation of the errors can be studied by means of a linear difference equation which, in the scalar case, depends on a complex parameter q = hλ, where h is the stepsize and λ is the derivative at the critical point of the function defining the differential equation. Obviously, the characteristic polynomial of the derived difference equation also depends on the same parameter q. The order of the equation of the error is, usually, greater than the one of the differential equation. Therefore, an higher number of conditions are needed to get the solution we are interested in. When all the conditions are fixed at the beginning of the interval of integration, it is well-known that the asymptotic stability of the zero solution of the error equation is equivalent to require that the characteristic polynomial is a Schur polynomial, that is all its roots lie in the unit circle, for all q ∈ C− . On the contrary, when the conditions are split between the beginning and the end of the interval of integration, the concept of Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 19276, Pages 1–14 DOI 10.1155/ADE/2006/19276
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Difference equations and the stability problem
stability needs to be generalized. In such a case the notion of well-conditioning is more appropriate. Essentially, such notion requires that under the perturbation δη of the imposed conditions, the perturbation of the solution δy should be bounded as follows: δy ≤ δη,
(1.1)
where is independent on the number of points in the discrete mesh. It is worth to note that when the discrete equation is of higher order with respect to the continuous initial value problem, it is not necessary to approximate the latter with a discrete
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