Numerical solution of the discontinuous-bottom Shallow-water Equations with hydrostatic pressure distribution at the step

Abstract Free-surface flows are usually modelled by means of the Shallow-water Equations: this system of hyperbolic equations exhibits a source term which is proportional to the product of the water depth by the bed slope, and which takes into account the effect of gravity onto fluid mass. Recently, much attention has been paid to the case in which bottom discontinuities are present in the physical domain to be represented: in this case, it is difficult to define the non-conservative product in the distributional sense. Here, the discontinuous-bottom Shallow-water Equations with hydrostatic pressure distribution at the bed step (Bernetti et al., 2006) are discussed in the context of the theory of Dal Maso et al. (1995) [9] ; finally, a first-order numerical scheme is presented, which is consistent for regular solutions, and which is able to capture contact discontinuities at bottom steps. Numerous tests are presented to show the feasibility of the scheme and its ability to converge to the exact solution in the cases of smooth as well as discontinuous bed profiles.

[1]  Manuel J. Castro,et al.  WELL-BALANCED NUMERICAL SCHEMES BASED ON A GENERALIZED HYDROSTATIC RECONSTRUCTION TECHNIQUE , 2007 .

[2]  C. Parés.  Numerical methods for nonconservative hyperbolic systems: a theoretical framework. , 2006 .

[3]  Discontinuous Solutions of the “Shallow Water$rdquo; Equations for Flow Over a Bottom Step , 2002 .

[4]  Giorgio Rosatti,et al.  The Riemann Problem for the one-dimensional, free-surface Shallow Water Equations with a bed step: Theoretical analysis and numerical simulations , 2010, J. Comput. Phys..

[5]  Javier Murillo,et al.  Generalized Roe schemes for 1D two-phase, free-surface flows over a mobile bed , 2008, J. Comput. Phys..

[6]  Eleuterio F. Toro,et al.  Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry , 2008, J. Comput. Phys..

[7]  Hervé Capart,et al.  Riemann wave description of erosional dam-break flows , 2002, Journal of Fluid Mechanics.

[8]  Valerio Caleffi,et al.  Depth–energy and depth–force relationships in open channel flows: Analytical findings , 2008 .

[9]  Carlos Parés,et al.  On the well-balance property of Roe?s method for nonconservative hyperbolic systems , 2004 .

[10]  David L. George,et al.  Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation , 2008, J. Comput. Phys..

[11]  Willi H. Hager,et al.  Hydraulic jumps at positive and negative steps , 1986 .

[12]  Emmanuel Audusse,et al.  A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows , 2004, SIAM J. Sci. Comput..

[13]  Rémi Abgrall,et al.  A comment on the computation of non-conservative products , 2010, J. Comput. Phys..

[14]  Brett F. Sanders,et al.  Non-reflecting boundary flux function for finite volume shallow-water models , 2002 .

[15]  Carlos Parés,et al.  Godunov method for nonconservative hyperbolic systems , 2007 .

[16]  Valerio Caleffi,et al.  Well-Balanced Bottom Discontinuities Treatment for High-Order Shallow Water Equations WENO Scheme , 2009 .

[17]  Gregory A. Lawrence.  Steady Flow over an Obstacle , 1987 .

[18]  Carlos Parés Madroñal,et al.  On the Convergence and Well-Balanced Property of Path-Conservative Numerical Schemes for Systems of Balance Laws , 2011, J. Sci. Comput..

[19]  Luigi Fraccarollo,et al.  A well-balanced approach for flows over mobile-bed with high sediment-transport , 2006, J. Comput. Phys..

[20]  Manuel Jesús Castro Díaz,et al.  Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes , 2008, J. Comput. Phys..

[21]  Fayssal Benkhaldoun,et al.  Exact solutions to the Riemann problem of the shallow water equations with a bottom step , 2001 .

[22]  M. Thanh,et al.  The Riemann problem for the shallow water equations with discontinuous topography , 2007, 0712.3778.

[23]  Luka Sopta,et al.  Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations , 2004 .