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Rigorous integration of non-linear ordinary differential equations in chebyshev basis

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Abstract

In this paper, we introduce a new approach to multiple step verified integration of non-linear ordinary differential equations. The approach is based on the technique of a Taylor model integration, however, a novel method is introduced to suppress the wrapping effect over several integration steps. This method is simpler and more robust compared to the known methods. It allows more general inputs, while it does not require rigorous matrix inversion. Moreover, our integration algorithm allows the use of various types of underlying function enclosures. We present rigorous arithmetic operations with function enclosures based on the truncated Chebyshev series. Computational experiments are used to show the wrapping effect suppression of our method and to compare integration algorithm that uses Chebyshev function enclosures with the existing algorithms that use function enclosures based on the truncated Taylor series (Taylor models).

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Authors and Affiliations

  1. Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 2, 182 07, Prague 8, Czech Republic

    Tomáš Dzetkulič

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  1. Tomáš Dzetkulič
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Dzetkulič, T. Rigorous integration of non-linear ordinary differential equations in chebyshev basis. Numer Algor 69, 183–205 (2015). https://doi.org/10.1007/s11075-014-9889-x

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  • DOI: https://doi.org/10.1007/s11075-014-9889-x

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