A well-balanced high-order scheme on van Leer-type for the shallow water equations with temperature gradient and variable bottom topography | Advances in Computational Mathematics
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A well-balanced high-order scheme on van Leer-type for the shallow water equations with temperature gradient and variable bottom topography

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Abstract

A well-balanced high-order scheme for shallow water equations with variable topography and temperature gradient is constructed. This scheme is of van Leer-type and is based on exact Riemann solvers. The scheme is shown to be able to capture almost exactly the stationary smooth solutions as well as stationary elementary discontinuities. Numerical tests show that the scheme gives a much better accuracy than the Godunov-type scheme and can work well even in the resonant regime. Wave interaction problems are also tested where the scheme possesses a good accuracy. It turns out that the superbee limiter can provide us with more accurate approximations than van Leer’s limiter.

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Acknowledgments

The author would like to thank the reviewers for their very constructive comments and fruitful discussions. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number “101.02-2019.306”.

Funding

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number “101.02-2019.306.”

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Science, 227 Nguyen Van Cu str., District 5, Ho Chi Minh City, Vietnam

    Nguyen Xuan Thanh

  2. Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam

    Mai Duc Thanh

  3. Department of Mathematics, Ho Chi Minh City University of Pedagogy, 280 An Duong Vuong str., District 5, Ho Chi Minh City, Vietnam

    Dao Huy Cuong

  4. Vietnam National University, Ho Chi Minh City, Vietnam

    Nguyen Xuan Thanh & Mai Duc Thanh

Authors
  1. Nguyen Xuan Thanh
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  2. Mai Duc Thanh
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  3. Dao Huy Cuong
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Correspondence to Mai Duc Thanh.

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Communicated by: Jan Hesthaven

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Thanh, N.X., Thanh, M.D. & Cuong, D.H. A well-balanced high-order scheme on van Leer-type for the shallow water equations with temperature gradient and variable bottom topography. Adv Comput Math 47, 13 (2021). https://doi.org/10.1007/s10444-020-09832-9

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  • DOI: https://doi.org/10.1007/s10444-020-09832-9

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