A survey of high order schemes for the shallow water equations

Yulong Xing University of Tennessee Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1610.25038

Journal of Mathematical Study, 47, 221-249, 2014
In this paper, we survey our recent work on designing high order positivity-preserving well-balanced finite difference and finite volume WENO (weighted essentially non-oscillatory) schemes, and discontinuous Galerkin finite element schemes for solving the shallow water equations with a non-flat bottom topography. These schemes are genuinely high order accurate in smooth regions for general solutions, are essentially non-oscillatory for general solutions with discontinuities, and at the same time they preserve exactly the water at rest or the more general moving water steady state solutions. A simple positivity-preserving limiter, valid under suitable CFL condition, has been introduced in one dimension and reformulated to two dimensions with triangular meshes, and we prove that the resulting schemes guarantee the positivity of the water depth.
hyperbolic balance laws; WENO scheme; discontinuous Galerkin method; high order accuracy; source term; conservation laws; shallow water equation
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  title={A survey of high order schemes for the shallow water equations},
  author={Yulong Xing, and Chi-Wang Shu},
  booktitle={Journal of Mathematical Study},
Yulong Xing, and Chi-Wang Shu. A survey of high order schemes for the shallow water equations. 2014. Vol. 47. In Journal of Mathematical Study. pp.221-249. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012041726327353156.
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