Abstract
The present work is focused on the numerical approximation of the shallow water equations. When studying this problem, one faces at least two important issues, namely the ability of the scheme to preserve the positiveness of the water depth, along with the ability to capture the stationary states.We propose here aGodunov-typemethod that fully satisfies the previous conditions, meaning that the method is in particular able to preserve the steady states with non-zero velocity.
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E. Audusse, F. Bouchut, M.O. Bristeau, R. Klein and B. Perthame. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comp., 25 (2004), 2050–2065.
E. Audusse, C. Chalons and P. Ung. A very simple well-balanced positive and entropy-satisfying scheme for the shallow-water equations. Commun. Math. Sci., 13(5) (2015), 1317–1332.
A. Bermudez and M.E. Vazquez-Cendon. Upwind Methods for Hyperbolic Conservation Laws with Source Terms. Comp. & Fluids, 23 (1994), 1049–1071.
C. Berthon and C. Chalons. A fully well-balanced, positive and entropy-satisfying Godunov-type method for the Shallow-Water Equations, to appear in Math. of Comp. (2015).
C. Berthon and F. Foucher. Efficient wellbalanced hydrostatic upwind schemes for shallowwater equations. J. Comput. Phys., 231 (2012), 4993–5015.
F. Bouchut. Non-linear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics, Birkhauser (2004).
F. Bouchut and T. Morales. A subsonic-well-balanced reconstruction scheme for shallow water flows. Siam J. Numer. Anal., 48(5) (2010), 1733–1758.
M.J. Castro, A. Pardo and C. Parés. Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique. Mathematical Models and Methods in Applied Sciences, 17 (2007), 2065–2113.
C. Chalons, F. Coquel, E. Godlewski, P-A Raviart and N. Seguin. Godunov-type schemes for hyperbolic systems with parameter dependent source. The case of Euler system with friction. Math. Models Methods Appl. Sci., 20(11) (2010).
A. Chinnayya, A.-Y. Le Roux and N. Seguin. A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomenon. International Journal on Finite Volume (electronic), 1(1) (2004), 1–33.
G. Gallice. Solveurs simples positifs et entropiques pour les systèmes hyperboliques avec terme source. C. R. Math. Acad. Sci. Paris, 334(8) (2002), 713–716.
G. Gallice. Positive and entropy stable Godunov-type schemes for gas dynamics and MHD equations in Lagrangian or Eulerian coordinates. Numer. Math., 94(4) (2003), 673–713.
P. García-Navarro, P. Brufau, J. Burguete and J. Murillo. The Shallow-Water equations: An example ofHyperbolic System, inMonografías de laReal Academia de Ciencias de Zaragoza (2008).
L. Gosse. A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput.Math. Appl., 39 (2000), 135–159.
L. Gosse. Computing qualitatively correct approximations of balance laws. Exponential-fit,well-balanced and asymptotic-preserving. SEMASIMAI Springer Series, 2 (2013).
J.M. Greenberg and A.Y. Leroux. A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal., 33 (1996), 1–16.
A. Harten, P. Lax and B. Van Leer. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev., 25 (1983), 35–61.
S. Jin. A steady-state capturing method for hyperbolic systems with geometrical source terms. Math. Model. Numer. Anal., 35 (2001), 631–645.
B. Perthame and C. Simeoni. A kinetic scheme for the Saint-Venant system with source term. Calcolo, Springer-Verlag, 38 (2001), 201–231.
Y. Xing. Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium. J. Comput. Phys., 257 (2014), 536–553.
Y. Xing and C.-W. Shu. High order well-balanced finite volume WENO schemes and discontinuousGalerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys., 214 (2006), 567–598.
Y. Xing, C.-W. Shu and S. Noelle. On the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations. J. Sci. Comput., 48 (2011), 339–349.
Y. Xing, X. Zhang and C.-W. Shu. Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour., 33 (2010), 1476–1493.
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Berthon, C., Chalons, C., Cornet, S. et al. Fully well-balanced, positive and simple approximate Riemann solver for shallow water equations. Bull Braz Math Soc, New Series 47, 117–130 (2016). https://doi.org/10.1007/s00574-016-0126-1
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DOI: https://doi.org/10.1007/s00574-016-0126-1
Keywords
- Shallow-water equations
- steady states
- finite volume schemes
- wellbalanced property
- positive preserving scheme