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Beyond Superconvergence of Collocation Methods for Volterra Integral Equations of the First Kind

Paul P. B. Eggermont

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Abstract

We discuss superconvergence aspects of collocation methods for Volterra integral equations of the first kind. If piecewise polynomials of degree ≤ p are used, then convergence of order p + 2 is best possible. We show here that one may perform some postprocessing on the collocation solution to obtain convergence of order p + 3. This possibility arises because of the oscillating error in the collocation solution. The relevance of superconvergence techniques to a third order Runge-Kutta method is also discussed.

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What this paper is about

We discuss superconvergence aspects of collocation methods for Volterra integral equations of the first kind. If piecewise polynomials of degree ≤ p are used, then convergence of order p + 2 is best possible. We show here that one may perform some postprocessing on the collocation solution to obtain convergence of order p + 3. This possibility arises because of the oscillating error in the collocation solution. The relevance of superconvergence techniques to a third order Runge-Kutta method is also discussed.

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Available abstract

We discuss superconvergence aspects of collocation methods for Volterra integral equations of the first kind. If piecewise polynomials of degree ≤ p are used, then convergence of order p + 2 is best possible. We show here that one may perform some postprocessing on the collocation solution to obtain convergence of order p + 3. This possibility arises because of the oscillating error in the collocation solution. The relevance of superconvergence techniques to a third order Runge-Kutta method is also discussed.

Key concepts: Superconvergence, Collocation (remote sensing), Piecewise, Mathematics, Collocation method, Convergence (economics), Orthogonal collocation, Volterra integral equation

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