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A well-balanced van Leer-type numerical scheme for shallow water equations with variable topography

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Abstract

A well-balanced van Leer-type numerical scheme for the shallow water equations with variable topography is presented. The model involves a nonconservative term, which often makes standard schemes difficult to approximate solutions in certain regions. The construction of our scheme is based on exact solutions in computational form of local Riemann problems. Numerical tests are conducted, where comparisons between this van Leer-type scheme and a Godunov-type scheme are provided. Data for the tests are taken in both the subcritical region as well as supercritical region. Especially, tests for resonant cases where the exact solutions contain coinciding waves are also investigated. All numerical tests show that each of these two methods can give a good accuracy, while the van Leer -type scheme gives a better accuracy than the Godunov-type scheme. Furthermore, it is shown that the van Leer-type scheme is also well-balanced in the sense that it can capture exactly stationary contact discontinuity waves.

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Authors and Affiliations

  1. Nguyen Huu Cau High School, 07 Nguyen Anh Thu str., Trung Chanh Ward, Hoc Mon District, Ho Chi Minh City, Vietnam

    Dao Huy Cuong

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

    Mai Duc Thanh

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

    Dao Huy Cuong

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

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Communicated by: Leslie Greengard

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Cuong, D.H., Thanh, M.D. A well-balanced van Leer-type numerical scheme for shallow water equations with variable topography. Adv Comput Math 43, 1197–1225 (2017). https://doi.org/10.1007/s10444-017-9521-4

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  • DOI: https://doi.org/10.1007/s10444-017-9521-4

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