A new hydrostatic reconstruction scheme based on subcell reconstructions

Guoxian Chen Computational Science Hubei Key Laboratory, School of Mathematics and Statistics, Wuhan University Sebastian Noelle Institute for Geometry and Practical Mathematics, RWTH Aachen University

Analysis of PDEs Fluid Dynamics and Shock Waves Numerical Analysis and Scientific Computing mathscidoc:2203.03002

SIAM J. NUMER. ANAL., 55, (2), 758-784, 2017.4
A key difficulty in the analysis and numerical approximation of the shallow water equations is the nonconservative product of measures due to the gravitational force acting on a sloped bottom. Solutions may be nonunique, and numerical schemes are not only consistent discretizations of the shallow water equations, but they also determine how to model the physics. Our derivation is based on a subcell reconstruction using infinitesimal singular layers at the cell boundaries, as inspired by S. Noelle, Y. Xing, and C.-W. Shu [J. Comput. Phys., 226 (2007), pp. 29-58]. One key step is to separate the singular measures. Another aspect is the reconstruction of the solution variables in the singular layers. We study three reconstructions. The first leads to the well-known scheme of Audusse et al., [SIAM J. Sci. Comput., 25 (2004), pp. 2050-2065], which introduces the hydrostatic reconstruction. The second is a modification proposed in [T. Morales de Luna, M. J. Castro D´ıaz, and C. Par´es, Appl. Math. Comput., 219 (2013), pp. 9012-9032], which analyzes whether a wave has enough energy to overcome a step. The third is our new scheme, which borrows its structure from the wet-dry front. For a number of cases discussed in recent years, where water runs down a hill, Audusse’s scheme converges slowly or fails. Morales’ scheme gives a visible improvement. Both schemes are clearly outperformed by our new scheme.
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@inproceedings{guoxian2017a,
  title={A NEW HYDROSTATIC RECONSTRUCTION SCHEME BASED ON SUBCELL RECONSTRUCTIONS},
  author={Guoxian Chen, and Sebastian Noelle},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310100531031760911},
  booktitle={SIAM J. NUMER. ANAL.},
  volume={55},
  number={2},
  pages={758-784},
  year={2017},
}
Guoxian Chen, and Sebastian Noelle. A NEW HYDROSTATIC RECONSTRUCTION SCHEME BASED ON SUBCELL RECONSTRUCTIONS. 2017. Vol. 55. In SIAM J. NUMER. ANAL.. pp.758-784. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310100531031760911.
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