A new finite volume method for flux-gradient and source-term balancing in shallow water equations

Computer Methods in Applied Mechanics and Engineering

Volumn 199, Issue 49-52, 2010, Pages 3324-3335

  Benkhaldoun, Fayssal ^a   Elmahi, Imad ^b   Seaïd, Mohammed ^c  

^a UNIVERSITÉ PARIS 13   (France)

^b Mohammed First University   (Morocco)

^c DURHAM UNIVERSITY   (United Kingdom)

Author keywords

C property; Finite volume method; Shallow water equations; Source terms; Unstructured grids

Indexed keywords

C-PROPERTY; FINITE VOLUME; SHALLOW WATER EQUATIONS; SOURCE TERMS; UNSTRUCTURED GRID;

FINITE VOLUME METHOD;

EID: 78649630858     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2010.07.003     Document Type: Article

References (27)

1. Alcrudo F., Benkhaldoun F. Exact solutions to the Riemann problem of the shallow water equations with a bottom step. Comput. Fluids 2001, 30:643-671.
2. Alcrudo F., Garcia-Navarro P. A high resolution Godunov-type scheme in finite volumes for the 2D shallow water equation. Int. J. Numer. Methods Fluids 1993, 16:489-505.
3. Audusse E., Bouchut F., Bristeau M.O., Klein R., Perthame B. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 2004, 25:2050-2065.
4. Babuska I., Strouboulis T., Updhyay C.S., Gangaraj S.K. A posteriori estimation and adaptive control of the pollution error in the h-version of finite-element method. Int. J. Numer. Methods Engrg. 1995, 38:4207-4235.
5. Benkhaldoun F., Elmahi I., Seaïd M. Well-balanced finite volume schemes for pollutant transport by shallow water equations on unstructured meshes. J. Comput. Phys. 2007, 226:180-203.
6. Benkhaldoun F., Quivy L. A non homogeous Riemann solver for shallow water and two phase flows. Flow Turbulence Combust. 2006, 76:391-402.
7. Bermúdez A., Vázquez M.E. Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 1994, 23:1049-1071.
8. Brufau P., Vázquez-Cendón M.E., García-Navarro P. A numerical model for the flooding and drying of irregular domains. Int. J. Numer. Methods Fluids 2002, 39:247-275.
9. Elmahi I., Benkhaldoun F., Borghi R., Raghay S. Ignition of fuel issuing from a porous cylinder located adjacent to a heated wall: a numerical study. Combust. Theor. Model. 2004, 8:789-809.
10. Garcia-Navarro P., Vazquez-Cendon M.E. On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 2000, 29:951-979.
11. Hubbard M.E., Garcia-Navarro P. Flux Difference splitting and the balancing of source terms and flux gradients. J. Comput. Phys. 2000, 165:89-125.
12. Kurganov A., Levy D. Central-upwind schemes for the Saint-Venant system. Math. Model. Numer. Anal. 2002, 36:397-425.
13. J. LeVeque Randall, Numerical methods for conservation laws, Lectures in Mathematics, ETH Zürich, 1992.
14. Leveque R.J. Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 1998, 146:346-365.
15. Kawahara M., Umetsu T. Finite element method for moving boundary problems in river flow. Int. J. Numer. Methods Fluids 1986, 6:365-386.
16. Noelle S., Pankratz N., Puppo G., Natvig J. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 2006, 213:474-499.
17. Perthame B., Simeoni C. A kinetic scheme for the Saint-Venant system with a source term. CALCOLO 2001, 38:201-231.
18. Rhebergen S., Bokhovea O., van der Vegta J.J.W. Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comput. Phys. 2008, 227:1887-1922.
19. Roe P.L. Approximate riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 1981, 43:357-372.
20. Sahmim S., Benkhaldoun F., Alcrudo F. A sign matrix based scheme for quasi-hyperbolic non-homogeneous PDEs with an analysis of the convergence stagnation problem. J. Comput. Phys. 2007, 226:753-1783.
21. Seaïd M. Non-oscillatory relaxation methods for the shallow water equations in one and two space dimensions. Int. J. Numer. Methods Fluids 2004, 46:457-484.
22. Tassi P.A., Rhebergen S., Vionnet C.A., Bokhove O. A discontinuous Galerkin finite element model for river bed evolution under shallow flows. Comput. Methods Appl. Mech. Engrg. 2008, 197:2930-2947.
23. Toro E.F. Shock-Capturing Methods for Free-Surface Shallow Waters 2001, Wiley, Chichester.
24. Vázquez M.E. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 1999, 148:497-526.
25. Vukovic S., Sopta L. ENO and WENO schemes with the exact conservation property for one-dimensional shallow-water equations. J. Comput. Phys. 2002, 179:593-621.
26. Xing Y., Shu C. High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys. 2006, 214:567-598.
27. Zhou J.G., Causon D.M., Mingham C.G., Ingram D.M. The surface gradient method for the treatment of source terms in the shallow water equations. J. Comput. Phys. 2001, 20:1-25.