On the C-property and Generalized C-property of Residual Distribution for the Shallow Water Equations

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.

[1]  Rémi Abgrall,et al.  Residual distribution schemes: Current status and future trends , 2006 .

[2]  Mario Ricchiuto,et al.  Stabilized residual distribution for shallow water simulations , 2009, J. Comput. Phys..

[3]  Alfredo Bermúdez,et al.  Upwind methods for hyperbolic conservation laws with source terms , 1994 .

[4]  Yulong Xing,et al.  High order well-balanced schemes , 2010 .

[5]  Rémi Abgrall,et al.  Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes , 2007, J. Comput. Phys..

[6]  Rémi Abgrall,et al.  Toward the ultimate conservative scheme: following the quest , 2001 .

[7]  Yulong Xing,et al.  High-order well-balanced finite volume WENO schemes for shallow water equation with moving water , 2007, J. Comput. Phys..

[8]  D. A. Dunavant.  High degree efficient symmetrical Gaussian quadrature rules for the triangle , 1985 .

[9]  Herman Deconinck,et al.  Residual distribution for general time-dependent conservation laws , 2005 .

[10]  Herman Deconinck,et al.  Residual Distribution Schemes: Foundations and Analysis , 2007 .

[11]  Pilar García-Navarro,et al.  Unsteady free surface flow simulation over complex topography with a multidimensional upwind technique , 2003 .

[12]  Herman Deconinck,et al.  A Conservative Formulation of the Multidimensional Upwind Residual Distribution Schemes for General Nonlinear Conservation Laws , 2002 .

[13]  Rémi Abgrall,et al.  High Order Fluctuation Schemes on Triangular Meshes , 2003, J. Sci. Comput..

[14]  Thomas J. R. Hughes,et al.  Encyclopedia of computational mechanics , 2004 .

[15]  Rémi Abgrall,et al.  A simple construction of very high order non-oscillatory compact schemes on unstructured meshes , 2009 .

[16]  G. Petrova,et al.  A SECOND-ORDER WELL-BALANCED POSITIVITY PRESERVING CENTRAL-UPWIND SCHEME FOR THE SAINT-VENANT SYSTEM ∗ , 2007 .

[17]  Emmanuel Audusse,et al.  A well-balanced positivity preserving second-order scheme for shallow water flows on unstructured meshes , 2005 .

[18]  Yulong Xing,et al.  High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms , 2006, J. Comput. Phys..

[19]  Yulong Xing,et al.  High order finite difference WENO schemes with the exact conservation property for the shallow water equations , 2005 .

[20]  Chi-Wang Shu,et al.  High-order well-balanced schemes and applications to non-equilibrium flow , 2009, J. Comput. Phys..