We develop well-balanced finite-volume central schemes on overlapping cells for the Saint-Venant shallow water system and its variants. The main challenge in deriving the schemes is related to the fact that the Saint-Venant system contains a geometric source term due to nonflat bottom topography and therefore a delicate balance between the flux gradients and source terms has to be preserved. We propose a constant subtraction technique, which helps one to ensure a well-balanced property of the schemes, while maintaining arbitrary high-order of accuracy. Hierarchical reconstruction limiting procedure is applied to eliminate spurious oscillations without using characteristic decomposition. Extensive one- and two-dimensional numerical simulations are conducted to verify the well-balanced property, high-order of accuracy, and non-oscillatory high-resolution for both smooth and nonsmooth solutions.
Bibliographical noteFunding Information:
Suo Yang and Yingjie Liu: Research supported in part by NSF Grant DMS-1115671.
Alexander Kurganov: Research supported in part by NSF Grant DMS-1216957 and ONR Grant N00014-12-1-0833.
© 2014, Springer Science+Business Media New York.
- Balance laws
- Central scheme
- Central schemes on overlapping cells
- Saint-Venant system
- Shallow water
- Well-balanced scheme