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A Godunov type scheme and error estimates for scalar conservation laws with Panov-type discontinuous flux

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Abstract

This article concerns a scalar conservation law where the flux is of Panov type and may contain spatial discontinuities. We define a notion of entropy solution and discuss the existence via Godunov type finite volume approximation. We further show that our numerical scheme converges the entropy solution at an optimal rate of \({\mathcal {O}}(\sqrt{\varDelta t}).\) To the best of our knowledge, the error estimates of the numerical scheme are the first of its kind for conservation laws with discontinuous flux where spatial discontinuities can accumulate. We present numerical examples that illustrate the theory.

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Acknowledgements

First and last authors, would like to thank Department of Atomic Energy, Government of India, under Project No. 12-R &D-TFR-5.01-0520. First author would also like to acknowledge Inspire faculty-research grant DST/INSPIRE/04/2016/000237. We thank the anonymous referees for their careful reading of the paper and constructive inputs which improved the scope of the study.

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Authors and Affiliations

  1. Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Post Bag No 6503, Sharadanagar, Bangalore, 560065, India

    Shyam Sundar Ghoshal & Ganesh Vaidya

  2. MiraCosta College, 3333 Manchester Avenue, Cardiff-by-the-Sea, CA, 92007-1516, USA

    John D. Towers

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  1. Shyam Sundar Ghoshal
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  3. Ganesh Vaidya
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Correspondence to Shyam Sundar Ghoshal.

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Ghoshal, S.S., Towers, J.D. & Vaidya, G. A Godunov type scheme and error estimates for scalar conservation laws with Panov-type discontinuous flux. Numer. Math. 151, 601–625 (2022). https://doi.org/10.1007/s00211-022-01297-w

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