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On the Numerical Solution of Nonconservative Hyperbolic Systems of Equations

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Abstract

Issues related to the lack of convergence in the application of formally path-conservative difference schemes for solving nonconservative hyperbolic systems of equations are numerically investigated. This problem is central in constructing well-posed difference schemes for solving this class of problems. The basic concepts of the theory of nonconservative hyperbolic systems of equations and the corresponding problems of constructing difference schemes for their solution are outlined. A variant of the HLL method is proposed that allows using an arbitrary explicitly specified path. For a model system of Burgers equations, the shock adiabates and paths corresponding to the viscous regularization of a system of a given form are explicitly calculated. The reasons for the lack of convergence of numerical solutions of exact ones in the case of incorrect application of the corresponding algorithms are analyzed. It is shown that, at least in the particular case considered, a variant of the HLL method that is formally conservative along the way gives the correct solution of the problem.

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Funding

The results in Secs. 4 and 5 were obtained by R.R. Polekhina with the support of the Ministry of Science and Higher Education of the Russian Federation as part of the implementation of the program of the Moscow Center for Fundamental and Applied Mathematics under agreement no. 075-15-2022-283. The results of Secs. 1–3 were obtained by M.V. Alekseev and E.B. Savenkov with the support of the Russian Science Foundation, project no. 22-11-00203. https://rscf.ru/en/project/22-11-00203/.

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Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Moscow, 125047, Russia

    R. R. Polekhina, M. V. Alekseev & E. B. Savenkov

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  1. R. R. Polekhina
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  2. M. V. Alekseev
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  3. E. B. Savenkov
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Correspondence to R. R. Polekhina, M. V. Alekseev or E. B. Savenkov.

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Translated by V. Potapchouck

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Polekhina, R.R., Alekseev, M.V. & Savenkov, E.B. On the Numerical Solution of Nonconservative Hyperbolic Systems of Equations. Diff Equat 59, 970–984 (2023). https://doi.org/10.1134/S0012266123070108

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  • DOI: https://doi.org/10.1134/S0012266123070108

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