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The Estimation of Truncation Error by \(\tau \)-Estimation for Chebyshev Spectral Collocation Method

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Abstract

In this paper we show how to accurately estimate the local truncation error of the Chebyshev spectral collocation method using \(\tau \)-estimation. This method compares the residuals on a sequence of approximations with different polynomial orders. First, we focus the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the truncation error. Then, we show the validity of the analysis for the incompressible Navier–Stokes equations. First on the Kovasznay flow, where an analytical solution is known, and finally in the lid driven cavity (LDC). We demonstrate that this approach yields a highly accurate estimation of the truncation error if the precision of the approximations increases with the polynomial order.

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Acknowledgments

This research is supported by the European project ANADE (PINT-GA-2011-289428). Furthermore, the authors would like to thank Professor David A. Kopriva for his support, and for many invaluable discussions.

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  1. E.T.S.I. Aeronáuticos (Universidad Politécnica de Madrid), Ciudad Universitaria, 28040 , Madrid, Spain

    G. Rubio, F. Fraysse, J. de Vicente & E. Valero

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  1. G. Rubio
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  2. F. Fraysse
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  3. J. de Vicente
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  4. E. Valero
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Correspondence to G. Rubio.

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Rubio, G., Fraysse, F., de Vicente, J. et al. The Estimation of Truncation Error by \(\tau \)-Estimation for Chebyshev Spectral Collocation Method. J Sci Comput 57, 146–173 (2013). https://doi.org/10.1007/s10915-013-9698-8

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  • DOI: https://doi.org/10.1007/s10915-013-9698-8

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