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On backward Euler approximations for systems of conservation laws

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Abstract

We study approximate solutions to a hyperbolic system of conservation laws, constructed by a backward Euler scheme, where time is discretized while space is still described by a continuous variable \(x\in {\mathbb R}\). We prove the global existence and uniqueness of these approximate solutions, and the invariance of suitable subdomains. Furthermore, given a left and a right state \(u_l, u_r\) connected by an entropy-admissible shock, we construct a traveling wave profile for the backward Euler scheme connecting these two asymptotic states in two main cases. Namely: (1) a scalar conservation law, where the jump \(u_l-u_r\) can be arbitrarily large, and (2) a strictly hyperbolic system, assuming that the jump \(u_l-u_r\) occurs in a genuinely nonlinear family and is sufficiently small.

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Acknowledgements

The authors would like to thank A.Bressan for suggesting the problem and for the extremely useful discussions. This research was partially supported by NSF with grant DMS-2006884.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Queen’s University, Kingston, ON, K7L3N6, Canada

    Maria Teresa Chiri

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

    Minyan Zhang

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  1. Maria Teresa Chiri
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  2. Minyan Zhang
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The authors equally contributed to the research presented in the paper.

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Correspondence to Maria Teresa Chiri.

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Chiri, M.T., Zhang, M. On backward Euler approximations for systems of conservation laws. Nonlinear Differ. Equ. Appl. 31, 37 (2024). https://doi.org/10.1007/s00030-023-00920-5

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