Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T20:19:39.111Z Has data issue: false hasContentIssue false
  • English
  • Français

Momentum Conservative Schemes for Shallow Water Flows

Published online by Cambridge University Press:  28 May 2015

S. R. Pudjaprasetya  and
I. Magdalena
S. R. Pudjaprasetya*
Affiliation:
Industrial and Financial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia
I. Magdalena*
Affiliation:
Industrial and Financial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia
*
Corresponding author. Email Address: [email protected]
Corresponding author. Email Address: [email protected]
Get access

Abstract

We discuss the implementation of the finite volume method on a staggered grid to solve the full shallow water equations with a conservative approximation for the advection term. Stelling & Duinmeijer [15] noted that the advection approximation may be energy-head or momentum conservative, and if suitable which of these to implement depends upon the particular flow being considered. The momentum conservative scheme pursued here is shown to be suitable for 1D problems such as transcritical flow with a shock and dam break over a rectangular bed, and we also found that our simulation of dam break over a dry sloping bed is in good agreement with the exact solution. Further, the results obtained using the generalised momentum conservative approximation for 2D shallow water equations to simulate wave run up on a conical island are in good agreement with benchmark experimental data.

Keywords

65M0876M1235L65Finite volume methodstaggered gridconservative schemeshallow water equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 2 , May 2014 , pp. 152 - 165
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arakawa, A. and Lamb, V.R., A potential enstrophy and energy conserving scheme for the shallow water equations. Monthly Weather Review 109, 18–36 (1981).Google Scholar
[2]Bouchut, F., Nonlinear Stability of Finite Volume Method for Hyperbolic Conservation Laws and Well-Balanced Schemes four Sources. Birkhauser Verlag, Germany, 2004.Google Scholar
[3]Briggs, M.J., Synolakis, C.E, Harkins, G.S. and Green, D.R., Laboratoryexperiments oftsunami runup on a circular island, Pure and Applied Geophysics 144, 569–593 (1995).Google Scholar
[4]Casulli, V., Semi-implicit finite difference method for three-dimensional shallow water flow, Int. J. Numer. Meth. Fluids 15, 169–148 (1992).Google Scholar
[5]Chanson, H., A simple solution of the laminar dam break wave, J. Appl. Fluid Mech. 1, 1–10 (2008).Google Scholar
[6]Chen, Q., Kirby, J.T., Dalrymple, R.A., Kennedy, A.B. and Chawla, A., Boussinesq modeling of wave transformation, breaking, and runup. II: 2D, J. Waterway, Port, Coastal, and Ocean Engineering, 126), 48–56 (2008).Google Scholar
[7]Doyen, D. and Gunawan, P.H., Explicit Staggered Schemes for the Shallow Water Equations with Topography, in preparation.Google Scholar
[8]Ham, F.E., Lien, F.S. and Strong, A.B., A fully conservative second order finite difference scheme for incompressible flow on nonuniform grids, J. Comp. Phys. 177, 117–133 (2002).CrossRefGoogle Scholar
[9]Jochen, K., Ocean Modelling for Beginners. Springer-Heidelberg, Berlin Heidelberg, 2009.Google Scholar
[10]Kurganov, A. and Levy, D., Central-upwind schemes for the Saint-Venant system, Math. Model. Numer. Anal. 36, 397–425 (2002).Google Scholar
[11]LeVeque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasisteady wave- propagation algorithm, J. Comp. Phys. 146, 346–365 (1998).CrossRefGoogle Scholar
[12]Liu, P., Cho, Y.S., Briggs, M.J., Kanoglu, U. and Synolakis, C.E., Runup ofsolitary wave on a circular island, J. Fluid Mech. 302, 259–285 (1995).Google Scholar
[13]Morinishi, Y., Lund, T.S., Vasilyev, O.V and Moin, P., Fully conservative higher order finite difference schemes for incompressible flow, J. Comp. Phys. 143, 90–124 (1998).Google Scholar
[14]Noelle, S., Pankratz, N., Puppo, G. and Natvig, J. R., Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comp. Phys. 13, 474–499 (2006).CrossRefGoogle Scholar
[15]Stelling, G.S. and Duinmeijer, S.P.A., A staggered conservative scheme for everyFroude number in rapidly varied shallow water flows, Int. J. Numer. Meth. Fluids 43, 1329–1354 (2003).CrossRefGoogle Scholar
[16]Stelling, G.S. and Zijlema, M., An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation, Int. J. Numer. Meth. Fluids 43, 1–23 (2003).Google Scholar
[17]Tang, H. Z., Tang, T., and Xu, K., A gas-kinetic scheme for shallow-water equations with source terms, Z. Angew. Math. Phys. 55, 365–382 (2004).Google Scholar
[18]Toro, E.F., Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, 3rd Edition. Springer Verlag, Berlin, (2009).Google Scholar
[19]Vasilyev, O., High order finite difference schemes on non-uniform meshes with good conservation properties, J. Comp. Phys, 157, 746–761 (2000).CrossRefGoogle Scholar
[20]Wei, Y, Mao, X.Z. and Cheung, K.F., Well-balanced finite-volume model for long-wave runup, J. Waterway, Port, Coastal, and Ocean Engineering 132, 114–124 (2006).Google Scholar
[21]Xing, Y. and Shu, C.W., A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class ofhyperbolic systems with source terms, Commun. Comput. Phys. 1, 100–134 (2006).Google Scholar
[22]Zijlema, M. and Stelling, G.S., Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure, Coastal Eng. 55, 780–790 (2008).CrossRefGoogle Scholar