Abstract
In this paper we consider the discretization of the Shallow Water equations by means of Residual Distribution (RD) schemes. We review the conditions allowing the exact preservation of some exact steady solutions. These conditions are shown to be related both to the type of spatial approximation and to the quadrature used to evaluate the cell residual. Numerical examples are shown to validate the theory.
Similar content being viewed by others
References
Abgrall, R.: Toward the ultimate conservative scheme: following the quest. J. Comput. Phys. 167(2), 277–315 (2001)
Abgrall, R.: Residual distribution schemes: current status and future trends. Comput. Fluids 35(7), 641–669 (2006)
Abgrall, R., Roe, P.L.: High order fluctuation schemes on triangular meshes. J. Sci. Comput. 19(3), 3–36 (2003)
Abgrall, R., Larat, A., Ricchiuto, M., Tavé, C.: Simplified stabilisation procedures for residual distribution schemes. Comput. Fluids 38(7), 1314–1323 (2009)
Audusse, E., Bristeau, M.-O.: A well-balanced positivity preserving second order scheme for shallow water flows on unstructured meshes. J. Comput. Phys. 206(1), 311–333 (2005)
Bermúdez, A., Vásquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071 (1994)
Brufau, P., García-Navarro, P.: Unsteady free surface flow simulation over complex topography with a multidimensional upwind technique. J. Comput. Phys. 186(2), 503–526 (2003)
Csík, Á., Ricchiuto, M., Deconinck, H.: A conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws. J. Comput. Phys 179(2), 286–312 (2002)
Deconinck, H., Ricchiuto, M.: Residual distribution schemes: foundation and analysis. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics. Wiley, New York (2007). doi:10.1002/0470091355.ecm054
Dunavant, D.A.: High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int. J. Numer. Methods Eng. 21, 1129–1148 (1985)
Kurganov, A., Petrova, G.: A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5(1), 133–160 (2007)
Ricchiuto, M., Bollermann, A.: Stabilized residual distribution for shallow water simulations. J. Comput. Phys. 228(4), 1071–1115 (2009)
Ricchiuto, M., Abgrall, R., Deconinck, H.: Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes. J. Comput. Phys. 222, 287–331 (2007)
Ricchiuto, M., Csík, Á., Deconinck, H.: Residual distribution for general time dependent conservation laws. J. Comput. Phys. 209(1), 249–289 (2005)
Xing, Y., Shu, C.-W.: High-order finite difference WENO schemes with the exact conservation property for the shallow-water equations. J. Comput. Phys. 208(1), 206–227 (2005)
Xing, Y., Shu, C.-W.: High-order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys. 214(2), 567–598 (2006)
Xing, Y., Noelle, S., Shu, C.-W.: High-order well-balanced schemes. In: Puppo, G., Russo, G. (eds.) Numerical Methods for Relaxation Systems and Balance Equations, Quaderni di Matematica. Dipartimento di Matematica, Seconda Universita di Napoli, Italy (to appear)
Xing, Y., Noelle, S., Shu, C.-W.: High-order well-balanced finite volume WENO schemes for shallow water equations with moving water. J. Comput. Phys. 226(1), 29–58 (2007)
Yee, H.C., Wang, W., Shu, C.-W., Sjögreen, B.: High-order well-balanced schemes and applications to non-equilibrium flow. J. Comput. Phys. 228(1), 6682–6702 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ricchiuto, M. On the C-property and Generalized C-property of Residual Distribution for the Shallow Water Equations. J Sci Comput 48, 304–318 (2011). https://doi.org/10.1007/s10915-010-9369-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-010-9369-y