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A Survey of High Order Schemes for the Shallow Water Equations

A Survey of High Order Schemes for the Shallow Water Equations
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Year:    2014

Author:    Yulong Xing, Chi-Wang Shu

Journal of Mathematical Study, Vol. 47 (2014), Iss. 3 : pp. 221–249

Abstract

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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jms.v47n3.14.01

Journal of Mathematical Study, Vol. 47 (2014), Iss. 3 : pp. 221–249

Published online:    2014-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    29

Keywords:    Hyperbolic balance laws WENO scheme discontinuous Galerkin method high order accuracy source term conservation laws shallow water equation.

Author Details

Yulong Xing

Chi-Wang Shu

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  31. A Well-Balanced Active Flux Method for the Shallow Water Equations with Wetting and Drying

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  32. A Class of Positive Semi-discrete Lagrangian–Eulerian Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

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