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Differential equation software for the computation of error-controlled continuous approximate solutions

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Abstract

In this paper, we survey selected software packages for the numerical solution of boundary value ODEs (BVODEs), time-dependent PDEs in one spatial dimension (1DPDEs), and initial value ODEs (IVODEs). A unifying theme of this paper is our focus on software packages for these problem classes that compute error-controlled, continuous numerical solutions. A continuous numerical solution can be accessed by the user at any point in the domain. We focus on error-control software; this means that the software adapts the computation until it obtains a continuous approximate solution with a corresponding error estimate that satisfies the user tolerance. The second section of the paper will provide an overview of recent work on the development of COLNEWSC, an updated version of the widely used collocation BVODE solver, COLNEW, that returns an error-controlled continuous approximate solution based on the use of a superconvergent interpolant to the underlying collocation solution. The third section of the paper gives a brief review of recent work on the development of a new 1DPDE solver, BACOLIKR, that provides time- and space-dependent event detection for an error-controlled continuous numerical solution. In the fourth section of the paper, we briefly review the state of the art in IVODE software for the computation of error-controlled continuous numerical solutions.

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Acknowledgements

The authors would like to thank the referees for many helpful suggestions.

Funding

The second author is funded by the Natural Sciences and Engineering Research Council of Canada and Saint Mary’s University.

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  1. Saint Mary’s University, Halifax, NS, B3H 3C3, Canada

    Mark Adams & Paul Muir

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Correspondence to Paul Muir.

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Adams, M., Muir, P. Differential equation software for the computation of error-controlled continuous approximate solutions. Numer Algor 96, 1021–1044 (2024). https://doi.org/10.1007/s11075-024-01784-1

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