Abstract
In this paper we present main points in the process of application of numerical schemes for hyperbolic balance laws to water wave propagation and flooding. The appropriate mathematical models are the one-dimensional open channel flow equations and the two-dimensional shallow water equations. Therefore good simulation results can only be obtained with well-balanced numerical schemes such as the ones developed by Bermùdez and Vázquez, Hubbard and García-Navarro, LeVeque, etc. as well as the ones developed by the authors of this paper. We also propose a modification of the well-balanced Q-scheme for the two-dimensional shallow water equations that solves the wetting and drying problem. Finally, we present numerical results for three simulation tasks: the CADAM dam break experiment, the water wave propagation in the Toce river, and the catastrophic dam break on the Malpasset river.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. S. Bale, R. J. LeVeque, S. Mitran, and J. A. Rossmanith, A wave propagation method for conservation laws and balance laws with spatially varying flux functions, preprint, (2001).
A. Bermùdez, A. Dervieux, J. A. Désideéri and M. E. Vázquez, Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Comput. Methods Appl. Mech. Eng. 155, 49 (1998).
A. Bermùdez and M. E. Váquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids 23(8), 1049 (1994).
M. O. Bristeau and B. Perthame, Transport of pollutant in shallow water using kinetic schemes, ESAIM-Proceedings Vol. 10 — CEMRACS, 9 (1999).
P. Brufau, M. E. Vázquez-Cendón, P. García-Navarro, A numerical model for the flooding and drying of irregular domains, Int. J. for Numerical Methods in Fluids, Vol. 39, 247–275 (2002).
P. Brufau, P. García-Navarro, Unsteady free surface flow simulation over complex topography with multidimensional upwind technique, J. of Computational Physics, Vol. 186, 503–526 (2003).
J. Burguete and P. García-Navarro, Efficient construction of high-resolution TVD conservative schemes for equations with source terms: application to shallow water flows, Int. J. Numer. Meth. Fluids 37 (2001). doi: 10.1002/fld.175.
A. Chinnaya and A.-Y. LeRoux, A new general Riemann solver for the shallow water equations, with friction topography, http://www.math.ntnu.no/conservation/1999/021.html.
N. Crnjaric-Zic, S. Vukovic, and L. Sopta, Extension of ENO and WENO schemes to one-dimensional sediment transport equations, Comput. Fluids, article in press.
G. Dal Maso, P. G. LeFloch and F. Murat, Definition and weak stability of a non-conservative product, J. Math. Pures Appl., 74, 483, (1995).
P. García-Navarro and M. E. Vázquez-Cendón, On numerical treatment of the source terms in the shallow water equations, Comput. Fluids 29, 951 (2000).
J. M. Greenberg and A.-Y. LeRoux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33, 1 (1996).
L. Gosse, A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms, Comput. Math. Appl., 39, 135 (2000).
L. Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms, Math. Models Methods Appl. Sci., 11, 339 (2001).
A. Harten and S. Osher, Uniformly high-order accurate non-oscillatory schemes, I, SIAM J. Numer. Anal. 24, 279 (1987).
A. Harten, B. Engquist, S. Osher, and S. R. Chakravarthy, Uniformly high-order accurate non-oscillatory schemes, III, J. Comput. Phys. 71, 231 (1987).
M. E. Hubbard and P. García-Navarro, Flux difference splitting and the balancing of source terms and flux gradients, J. Comput. Phys. 165, (2000), doi: 10.1006/jcph.2000.6603.
P. Jenny and B. Müller, Rankine-Hugoniot-Riemann solver considering source terms and multidimensional effects, J. Comput. Phys. 145, 575 (1998).
S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms, Math. Model. Num. Anal., 35 631 (2001).
P. G. LeFloch and A. E. Tzavaras, Representation of week limits and definition of non-conservative products, SIAM J. Math. Anal., 30, 1309, (1999).
R. J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J. Comput. Phys. 146, 346 (1998).
X.-D. Liu, S. Osher, and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115, 200, (1994).
B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, CALCOLO, 38(4), 201 (2001).
J. D. Towers, A difference scheme for conservation laws with a discontinuous flux: the nonconvex case, SIAM J. Numer. Anal., 39, 1197 (2001).
M. E. Vázquez-Cendón, Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. Comput. Phys. 148, 497 (1999).
S. Vukovic and L. Sopta, ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations, J. Comput. Phys. 179, 593 (2002), doi: 10.1006/jcph.2002.7076.
S. Vukovic and L. Sopta, Upwind schemes with exact conservation property for one-dimensional open channel flow equations, SIAM J. Scient. Comput., 24(5), 1630 (2003).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer
About this paper
Cite this paper
Sopta, L., Črnjarić-Žic, N., Vuković, S., Holjević, D., Škifić, J., Družeta, S. (2005). Numerical Simulations of Water Wave Propagation and Flooding. In: Drmač, Z., Marušić, M., Tutek, Z. (eds) Proceedings of the Conference on Applied Mathematics and Scientific Computing. Springer, Dordrecht. https://doi.org/10.1007/1-4020-3197-1_22
Download citation
DOI: https://doi.org/10.1007/1-4020-3197-1_22
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-3196-0
Online ISBN: 978-1-4020-3197-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)