Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-12T21:58:18.033Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

Basco, D. R., “Computation of rapidly varied, unsteady, free-surface flow”, Water-Resources Investigations Report 83-4284, US Geological Survey, 1987; http://pubs.usgs.gov/wri/1983/4284/report.pdf.Google Scholar
Carter, J. D. and Cienfuegos, R., “Solitary and cnoidal wave solutions of the Serre equations and their stability”, Eur. J. Mech. B 30 (2011) 259–268; doi:10.1016/j.euromechflu.2010.12.002.Google Scholar
Chazel, F., Lannes, D. and Marche, F., “Numerical simulation of strongly non-linear and dispersive waves using a Green–Naghdi model”, J. Sci. Comput. 48 (2011) 105–116; doi:10.1007/s10915-010-9395-9.Google Scholar
El, G. A., Grimshaw, R. H. J. and Smyth, N. F., “Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory”, Physica D 237 (2008) 2423–2435; doi:10.1016/j.physd.2008.03.031.Google Scholar
Green, A. E. and Naghdi, P. M., “A derivation of equations for wave propagation in water of variable depth”, J. Fluid Mech. 78 (1976) 237–246; doi:10.1017/S0022112076002425.CrossRefGoogle Scholar
Harten, A., “High resolution schemes for hyperbolic conservation laws”, J. Comput. Phys. 49 (1983) 357–393; http://arrow.utias.utoronto.ca/∼groth/aer1319/Handouts/Additional_Reading_Material/JCP-1983-harten.pdf.Google Scholar
Kurganov, A., Noelle, S. and Petrova, G., “Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations”, J. Sci. Comput. 23 (2002) 707–740; doi:10.1137/S1064827500373413.Google Scholar
Li, M., Guyenne, P., Li, F. and Xu, L., “High order well-balanced CDG-FE methods for shallow water waves by a Green–Naghdi model”, J. Comput. Phys. 257 (2014) 169–192; doi:10.1016/j.jcp.2013.09.050.Google Scholar
Macdonald, C. B., Gottlieb, S. and Ruuth, S. J., “A numerical study of diagonally split Runga–Kutta methods for PDEs with discontinuities”, J. Sci. Comput. 6 (2008) 89–112; doi:10.1007/s10915-007-9180-6.CrossRefGoogle Scholar
Madsen, P. A. and Sørensen, O. R., “A new form of Boussinesq equations with improved linear dispersion characteristics. Part II. A slowly-varying bathymetry”, Coast. Eng. 18 (1992) 183–204; doi:10.1016/0378-3839(92)90019-Q.Google Scholar
Panda, N., Dawson, C., Zhang, Y., Kennedy, A. B., Westerink, J. J. and Donahue, A. S., “Discontinuous Galerkin methods for solving Boussinesq–Green–Naghdi equations in resolving non-linear and dispersive surface water waves”, J. Comput. Phys. 273 (2014) 572–588; doi:10.1016/j.jcp.2014.05.035.Google Scholar
Seabra-Santos, F. J., Renouard, D. P. and Temperville, A. M., “Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle”, J. Fluid Mech. 176 (1987) 117–134; doi:10.1017/S0022112087000594.Google Scholar
Serre, F., “Contribution à l’étude des écoulements permanents et variables dans les canaux”, La Houille Blanche 6 (1953) 830–872; doi:10.1051/lhb/1953058.Google Scholar
Shu, C. W. and Osher, S., “Efficient implementation of essentially non-oscillatory shock-capturing schemes”, J. Comput. Phys. 77 (1988) 439–471; doi:10.1016/0021-9991(88)90177-5.CrossRefGoogle Scholar
Su, C. H. and Gardner, C. S., “Korteweg–de Vries equation and generalisations. III. Derivation of the Korteweg–de Vries equation and Burgers equation”, J. Math. Phys. 10 (1969) 536–539; doi:10.1063/1.1664873.Google Scholar
van Leer, B., “Towards the ultimate conservative difference scheme, V. A second-order sequel to Godunov’s method”, J. Comput. Phys. 32 (1979) 101–136; doi:10.1016/0021-9991(79)90145-1.Google Scholar
Zoppou, C. and Roberts, S., “Catastrophic collapse of water supply reservoirs in urban areas”, J. Hydraul. Eng. 125 (1999) 11–34; doi:10.1061/(ASCE)0733-9429(1999)125:7(686).CrossRefGoogle Scholar
Zou, Z. L., “Higher order Boussinesq equations”, Ocean Eng. 26 (1999) 767–792; doi:10.1016/S0029-8018(98)00019-5.Google Scholar