Volume 47, Issue 3
A Survey of High Order Schemes for the Shallow Water Equations

Yulong Xing & Chi-Wang Shu

J. Math. Study, 47 (2014), pp. 221-249.

Published online: 2014-09

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  • 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.

  • Keywords

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

  • AMS Subject Headings

65N06, 65N08, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

xingy@math.utk.edu (Yulong Xing)

shu@dam.brown.edu (Chi-Wang Shu)

  • BibTex
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@Article{JMS-47-221, author = {Xing , Yulong and Shu , Chi-Wang }, title = {A Survey of High Order Schemes for the Shallow Water Equations}, journal = {Journal of Mathematical Study}, year = {2014}, volume = {47}, number = {3}, pages = {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.}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v47n3.14.01}, url = {http://global-sci.org/intro/article_detail/jms/9956.html} }
TY - JOUR T1 - A Survey of High Order Schemes for the Shallow Water Equations AU - Xing , Yulong AU - Shu , Chi-Wang JO - Journal of Mathematical Study VL - 3 SP - 221 EP - 249 PY - 2014 DA - 2014/09 SN - 47 DO - http://doi.org/10.4208/jms.v47n3.14.01 UR - https://global-sci.org/intro/article_detail/jms/9956.html KW - Hyperbolic balance laws KW - WENO scheme KW - discontinuous Galerkin method KW - high order accuracy KW - source term KW - conservation laws KW - shallow water equation AB - 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.
Yulong Xing & Chi-Wang Shu. (2019). A Survey of High Order Schemes for the Shallow Water Equations. Journal of Mathematical Study. 47 (3). 221-249. doi:10.4208/jms.v47n3.14.01
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