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Comparative Analysis of the Accuracy of Three Different Schemes in the Calculation of Shock Waves

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Abstract

We perform a comparative analysis of the accuracy of the weighted essentially nonoscillatory (WENO), compact high-order weak approximation (CWA), and central-upwind (CU) schemes used to compute discontinuous solutions containing shocks propagating with variable velocity. We demonstrate that the the accuracy of the formally high-order WENO and CU schemes, which are constructed using nonlinear flux correction mechanisms, reduces to approximately first order integral convergence on intervals in which one of the endpoints is in the region of influence of a shock wave. At the same time, the CWA scheme, which is designed to be high-order in the weak sense and does not rely on any nonlinear flux corrections, retains approximately the second order of integral convergence even in the regions of influence of shock waves. As a result, in these areas, the accuracy of the WENO and CU schemes is significantly lower than the accuracy of the CWA scheme. We provide a theoretical justification of these numerical results.

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Funding

This study was financially supported by the Russian Foundation for Basic Research and the National Natural Science Foundation of China (NSFC) as part of scientific project no. 21-51-53012 and was partially supported by NSFC grants nos. 11771201 and 1201101343, as well as by the Guangdong Provincial Foundation, Laboratory of Computing and Materials Science (no. 2019B030301001).

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Authors and Affiliations

  1. Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia

    O. A. Kovyrkina & V. V. Ostapenko

  2. Novosibirsk State University, Novosibirsk, Russia

    O. A. Kovyrkina & V. V. Ostapenko

  3. Department of Mathematics, SUSTech International Center for Mathematics and Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology, Shenzhen, China

    A. A. Kurganov

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  1. O. A. Kovyrkina
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Correspondence to O. A. Kovyrkina, A. A. Kurganov or V. V. Ostapenko.

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Kovyrkina, O.A., Kurganov, A.A. & Ostapenko, V.V. Comparative Analysis of the Accuracy of Three Different Schemes in the Calculation of Shock Waves. Math Models Comput Simul 15, 401–414 (2023). https://doi.org/10.1134/S2070048223030092

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