Abstract
A numerical method for solving the two-dimensional shallow water equations was presented. This method was based on the third-order genuinely multidimensional semi-discrete central scheme for spatial discretization and the optimal third-order Strong Stability Preserving (SSP) Runge-Kutta method for time integration. The third-order compact Central Weighted Essentially Non-Oscillatory (CWENO) reconstruction was adopted to guarantee the non-oscillatory behavior of the presented scheme and improve the resolution. Two kinds of source terms were considered in this work. They were evaluated using different approaches. The resulting scheme does not require Riemann solvers or characteristic decomposition, hence it retains all the attractive features of central schemes such as simplicity and high resolution. To evaluate the performance of the presented scheme, several numerical examples were tested. The results demonstrate that our method is efficient, stable and robust.
Similar content being viewed by others
References
WANG Jia-song, NI Han-gen. A high-resolution finite-volume method for solving shallow water equation [J]. Journal of Hydrodynamics, Ser. B, 2000, 12(1): 35–41.
WANG Jia-song, HE You-sheng. Finite volume TVD algorithm for dam-break flows in open channels [J]. Journal of Hydrodynamics, Ser. B, 2003, 15(3): 28–34.
PAN Cun-hong, LIN Bing-yao, MAO Xian-zhong. A Godunov-type scheme for 2-D shallow water flow with bottom topography [J]. Journal of Hydrodynamics, Ser. A, 2003, 18(1): 16–23. (in Chinese)
XU Kun, PAN Cun-hong. Kinetic flux vector scheme for the 1-D shallow water equation with source terms [J]. Journal of Hydrodynamics, Ser. A, 2002, 17(2): 140–147. (in Chinese)
WEI Wen-li, LIU Yu-ling, WANG De-yi et al. The MacCormack scheme and the boundary-fitted coordinate system for numerical simulation of super critical free surface flows[J]. Journal of Hydrodynamics, Ser. B, 2002, 14(2): 122–126.
KURGANOV A., TADMOR E. New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations [J]. Journal of Computational Physics, 2000, 160(1): 241–282.
GOTTARDI G., VENUTELLI M. Central scheme for two-dimensional dam-break flow simulation [J]. Advances in Water Resources, 2004, 27(3): 259–268.
RUSSO G. Central schemes for balance laws [A]. Eighth International conference on Hyperbolic problems: Theory, Numerical, Applications[C]. Magdeburg, Germany, 2000, 821–829.
KURGANOV A., LEVY D. Central-upwind schemes for the Saint-Venant system [J]. Math. Model. Numer. Anal., 2002, 36(3): 397–425.
KURGANOV A., PETROVA G. A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems [J]. Numer. Math., 2001, 88(4): 683–729.
LEVY D., PUPPO G., RUSSO G. Compact central WENO schemes for multidimensional conservation laws [J]. SIAM J. Sci. Comput., 2000, 22(2): 656–672.
GOTTLIEB S., SHU C. W., TADMOR E. Strong stability preserving high order time discretization methods [J]. SIAM Rev., 2001, 43(1): 89–112.
LEVEQUE R. J. Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm[J]. Journal of Computational Physics, 1998, 146(1): 346–365.
LISKA R., WENDROFF B. Two-dimensional shallow water equations by composite schemes [J]. Int. J. Numer. Meth. Fluids, 1999, 30(4): 461–479.
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (Grant No: 60134010).
Biography: CHEN Jian-zhong (1976-), Male, Ph. D. Student
Rights and permissions
About this article
Cite this article
Chen, Jz., Shi, Zk. Solution of 2D Shallow Water Equations by Genuinely Multidimensional Semi-Discrete Central Scheme. J Hydrodyn 18, 436–442 (2006). https://doi.org/10.1016/S1001-6058(06)60117-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1016/S1001-6058(06)60117-0