Math @ Duke

Publications [#304481] of Anita T. Layton
Papers Published
 Layton, AT; Minion, ML, Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations,
BIT Numerical Mathematics, vol. 45 no. 2
(2005),
pp. 341373 [doi]
(last updated on 2018/08/20)
Abstract: This paper concerns a class of deferred correction methods recently developed for initial value ordinary differential equations; such methods are based on a Picard integral form of the correction equation. These methods divide a given timestep [t n ,t n+1] into substeps, and use function values computed at these substeps to approximate the Picard integral by means of a numerical quadrature. The main purpose of this paper is to present a detailed analysis of the implications of the location of quadrature nodes on the accuracy and stability of the overall method. Comparisons between GaussLegendre, GaussLobatto, GaussRadau, and uniformly spaced points are presented. Also, for a given set of quadrature nodes, quadrature rules may be formulated that include or exclude function values computed at the lefthand endpoint t n . Quadrature rules that do not depend on the lefthand endpoint (which are referred to as righthand quadrature rules) are shown to lead to L(α)stable implicit methods with α≈π/2. The semiimplicit analog of this property is also discussed. Numerical results suggest that the use of uniform quadrature nodes, as opposed to nodes based on Gaussian quadratures, does not significantly affect the stability or accuracy of these methods for orders less than ten. In contrast, a study of the reduction of order for stiff equations shows that when uniform quadrature nodes are used in conjunction with a righthand quadrature rule, the form and extent of orderreduction changes considerably. Specifically, a reduction of order to script O sign(ε2) is observed for uniform nodes as opposed to script O sign(εΔt) for nonuniform nodes, where Δt denotes the time step and ε a stiffness parameter such that ε→0 corresponds to the problem becoming increasingly stiff. © Springer 2005.


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