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A thorough description of one-dimensional steady open channel flows using the notion of viscosity solution

Author

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  • Mean, Sovanna
  • Unami, Koichi
  • Okamoto, Hisashi
  • Fujihara, Masayuki

Abstract

Determining water surface profiles of steady open channel flows in a one-dimensional bounded domain is one of the well-trodden topics in conventional hydraulic engineering. However, it involves Dirichlet problems of scalar first-order quasilinear ordinary differential equations, which are of mathematical interest. We show that the notion of viscosity solution is useful in thoroughly describing the characteristics of possibly non-smooth and discontinuous solutions to such problems, achieving the conservation of momentum and the entropy condition. Those viscosity solutions are the generalized solutions in the space of bounded measurable functions. Generalized solutions to some Dirichlet problems are not always unique, and a necessary condition for the non-uniqueness is derived. A concrete example illustrates the non-uniqueness of discontinuous viscosity solutions in a channel of a particular cross-sectional shape.

Suggested Citation

  • Mean, Sovanna & Unami, Koichi & Okamoto, Hisashi & Fujihara, Masayuki, 2022. "A thorough description of one-dimensional steady open channel flows using the notion of viscosity solution," Applied Mathematics and Computation, Elsevier, vol. 415(C).
    • Handle: RePEc:eee:apmaco:v:415:y:2022:i:c:s0096300321008122
    DOI: 10.1016/j.amc.2021.126730
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    References listed on IDEAS

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    1. Unami, Koichi & Mohawesh, Osama & Fadhil, Rasha M., 2019. "Time periodic optimal policy for operation of a water storage tank using the dynamic programming approach," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 418-431.
    2. K. Ammari & B. Chentouf, 2010. "Further Results on the Robust Regulation of a One-Dimensional Dam-River System," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 597-606, December.
    3. Glaubitz, Jan, 2019. "Shock capturing by Bernstein polynomials for scalar conservation laws," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    4. B. Chentouf & J. M. Wang, 2007. "Stabilization of a One-Dimensional Dam-River System: Nondissipative and Noncollocated Case," Journal of Optimization Theory and Applications, Springer, vol. 134(2), pages 223-239, August.
    5. Chentouf, Boumediène & Smaoui, Nejib, 2018. "Stability analysis and numerical simulations of a one dimensional open channel hydraulic system," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 498-511.
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    1. Chentouf, Boumediène & Smaoui, Nejib, 2018. "Stability analysis and numerical simulations of a one dimensional open channel hydraulic system," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 498-511.
    2. K. Ammari & B. Chentouf, 2010. "Further Results on the Robust Regulation of a One-Dimensional Dam-River System," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 597-606, December.

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