BIT Numerical Mathematics

, Volume 52, Issue 3, pp 725–740

# Superconvergent interpolants for collocation methods applied to Volterra integro-differential equations with delay

Article

## Abstract

Standard software based on the collocation method for differential equations delivers a continuous approximation (called the collocation solution) which augments the high order discrete approximate solution that is provided at mesh points. This continuous approximation is less accurate than the discrete approximation. For ‘non-standard’ Volterra integro-differential equations with constant delay, that often arise in modeling predator-prey systems in Ecology, the collocation solution is C0 continuous. The accuracy is O(hs+1) at off-mesh points and O(h2s) at mesh points where s is the number of Gauss points used per subinterval and h refers to the stepsize. We will show how to construct C1 interpolants with an accuracy at off-mesh points and mesh points of the same order (2s). This implies that even for coarse mesh selections we achieve an accurate and smooth approximate solution. Specific schemes are presented for s=2, 3, and numerical results demonstrate the effectiveness of the new interpolants.

### Keywords

Delay Volterra integro-differential equations Piecewise polynomial collocation Bootstrapping Order conditions

### Mathematics Subject Classification (2000)

65R20 65L60 65L06

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## Copyright information

© Springer Science + Business Media B.V. 2012

## Authors and Affiliations

1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada