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A SOLUTION OF THE CONSERVATION LAW FORM OF THE SERRE EQUATIONS

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Hyperbolic equations and systems Incompressible inviscid fluids

Published online by Cambridge University Press:  28 March 2016

C. ZOPPOU ,
S. G. ROBERTS  and
J. PITT
C. ZOPPOU*
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, ACT 2001, Australia email Christopher.Zoppou@anu.edu.au, Stephen.Roberts@anu.edu.au, Jordan.Pitt@anu.edu.au
S. G. ROBERTS
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, ACT 2001, Australia email Christopher.Zoppou@anu.edu.au, Stephen.Roberts@anu.edu.au, Jordan.Pitt@anu.edu.au
J. PITT
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, ACT 2001, Australia email Christopher.Zoppou@anu.edu.au, Stephen.Roberts@anu.edu.au, Jordan.Pitt@anu.edu.au
*
✉Christopher.Zoppou@anu.edu.au
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Abstract

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The nonlinear and weakly dispersive Serre equations contain higher-order dispersive terms. These include mixed spatial and temporal derivative flux terms which are difficult to handle numerically. These terms can be replaced by an alternative combination of equivalent temporal and spatial terms, so that the Serre equations can be written in conservation law form. The water depth and new conserved quantities are evolved using a second-order finite-volume scheme. The remaining primitive variable, the depth-averaged horizontal velocity, is obtained by solving a second-order elliptic equation using simple finite differences. Using an analytical solution and simulating the dam-break problem, the proposed scheme is shown to be accurate, simple to implement and stable for a range of problems, including flows with steep gradients. It is only slightly more computationally expensive than solving the shallow water wave equations.

Keywords

dispersive wavesconservation lawsSerre equationsfinite-volume method

MSC classification

Primary: 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Secondary: 35L65: Conservation laws 65M08: Finite volume methods
Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 4 , April 2016 , pp. 385 - 394
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2016 Australian Mathematical Society

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