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A Posteriori Error Estimates for Numerical Solutions to Hyperbolic Conservation Laws

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Abstract

This paper is concerned with a posteriori error bounds for a wide class of numerical schemes, for \(n\times n\) hyperbolic conservation laws in one space dimension. These estimates are achieved by a “post-processing algorithm”, checking that the numerical solution retains small total variation, and computing its oscillation on suitable subdomains. The results apply, in particular, to solutions obtained by the Godunov or the Lax–Friedrichs scheme, backward Euler approximations, and the method of periodic smoothing. Some numerical implementations are presented.

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Acknowledgements

This research was partially supported by NSF with Grant DMS-2006884, “Singularities and error bounds for hyperbolic equations”.

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Authors and Affiliations

  1. Department of Mathematics, Penn State University, University Park, PA, 16802, USA

    Alberto Bressan, Maria Teresa Chiri & Wen Shen

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  1. Alberto Bressan
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  2. Maria Teresa Chiri
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  3. Wen Shen
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Correspondence to Wen Shen.

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Communicated by Constantine Dafermos

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Bressan, A., Chiri, M.T. & Shen, W. A Posteriori Error Estimates for Numerical Solutions to Hyperbolic Conservation Laws. Arch Rational Mech Anal 241, 357–402 (2021). https://doi.org/10.1007/s00205-021-01653-4

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