Higher-order accurate least-squares methods for first-order initial value problems

Abstract

In this paper, various least-squares procedures to solve first-order initial value problems are studied. The accuracy and stability properties are investigated by applying the methods to a linear first-order ordinary differential equation. In relating the least-squares procedures to the weighted residual method, the weighting functions can be identified as the residuals obtained by substituting the trial functions into the governing equation. By using a different set of functions to construct the residual weighting functions, a more general ‘pseudo’-least-squares method is proposed here. Instead of having the weighting functions specified explicitly and the characteristics of the resultant algorithms investigated, the weighting parameter method is adopted. The required residual weighting functions can be reconstructed from the selected weighting parameters. The weighting parameters corresponding to the A-stable generalized Padé approximations are presented in this paper. The order of accuracy is 4n−1 in general if n unknown variables are used in approximating the solutions. It is found that a direct application of the least-squares procedures to multi-degree-of-freedom systems may result in loss of accuracy. By studying the uncoupling conditions for the multi-degree-of-freedom systems, modified forms are suggested for various least-squares methods to maintain the accuracy and stability properties. A two-degree-of-freedom system is used to illustrate the accuracy of the standard, pseudo- and modified-least-squares procedures. Copyright © 1999 John Wiley & Sons, Ltd.