Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Exact Riemann solutions to shallow water equations


Authors: Ee Han and Gerald Warnecke
Journal: Quart. Appl. Math. 72 (2014), 407-453
MSC (2010): Primary 76B15, 76H05, 35L60
DOI: https://doi.org/10.1090/S0033-569X-2014-01353-3
Published electronically: June 13, 2014
MathSciNet review: 3237558
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Abstract: We determine completely the Riemann solutions to the shallow water equations with a bottom step, including the dry bed problem. The nonstrict hyperbolicity of this first-order system of partial differential equations leads to resonant waves and nonunique solutions. To address these difficulties we construct the L-M and R-M curves in the state space. For the bottom step elevated from left to right, we classify the L-M curve into five different cases and the R-M curve into two different cases based on the subcritical and supercritical Froude number of the Riemann initial data as well as the jump of the bottom step. The behaviors of all basic cases of the L-M and R-M curves are fully analyzed. We observe that the non-uniqueness of the Riemann solutions is due to bifurcations on the L-M or R-M curves. The possible solutions including classical waves and resonant waves as well as dry bed state are solved in a uniform framework for any given Riemann initial data.


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Additional Information

Ee Han
Affiliation: Institut fuer Analysis und Numerik, Otto-von-Guericke-Universitaet Magdeburg, D-39106 Magdeburg, Germany
Email: eehan@math.uni-bremen.de

Gerald Warnecke
Affiliation: Institut fuer Analysis und Numerik, Otto-von-Guericke-Universitaet Magdeburg, D-39106 Magdeburg, Germany
Email: warnecke@ovgu.de

DOI: https://doi.org/10.1090/S0033-569X-2014-01353-3
Keywords: Shock waves, rarefaction waves, velocity function, stationary waves, resonant waves, Froude number, nonuniqueness solutions
Received by editor(s): June 12, 2012
Published electronically: June 13, 2014
Additional Notes: The first author is supported by Micro-Macro-Interactions in structured Media and Particle Systems
Article copyright: © Copyright 2014 Brown University

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