Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system

ESAIM: Mathematical Modelling and Numerical Analysis

Volumn 45, Issue 3, 2011, Pages 423-446

  Bryson, Steve ^a   Epshteyn, Yekaterina ^b   Kurganov, Alexander ^c   Petrova, Guergana ^d  

^a NASA AMES RESEARCH CENTER   (United States)

^b CARNEGIE MELLON UNIVERSITY   (United States)

^c Tulane University   (United States)

^d TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Hyperbolic systems of conservation and balance laws; Saint Venant system of shallow water equations; semi discrete central upwind schemes

Indexed keywords

TOPOGRAPHY;

BALANCE LAW; BOTTOM TOPOGRAPHY; CENTRAL-UPWIND SCHEME; HIGH RESOLUTION; POSITIVITY PRESERVING; SAINT-VENANT SYSTEM; SHALLOW WATER EQUATIONS; TRIANGULAR GRID;

EQUATIONS OF MOTION;

EID: 79952553523     PISSN: 28227840     EISSN: 28047214     Source Type: Journal    
DOI: 10.1051/m2an/2010060     Document Type: Article

References (37)

1. R. Abgrall, On essentially non-oscillatory schemes on unstructured meshes: Analysis and implementation. J. Comput. Phys. 114 (1994) 45-58. (Pubitemid 124013471)
2. N. Andrianov, Testing numerical schemes for the shallow water equations. Preprint available at http://www-ian.math. uni-magdeburg.de/home/andriano/ CONSTRUCT/testing.ps.gz (2004).
3. P. Arminjon, M.-C. Viallon and A. Madrane, A finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for conservation laws on unstructured grids. Int. J. Comput. Fluid Dyn. 9 (1997) 1-22.
4. 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. Comput. 25 (2004) 2050-2065.
5. F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Math Series, Birkhäuser Verlag, Basel (2004).
6. S. Bryson and D. Levy, Balanced central schemes for the shallow water equations on unstructured grids. SIAM J. Sci. Comput. 27 (2005) 532-552.
7. I. Christov and B. Popov, New nonoscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws. J. Comput. Phys. 227 (2008) 5736-5757.
8. A.J.C. de Saint-Venant, Théorie du mouvement non-permanent des eaux, avec application aux crues des rivière et à l'introduction des marées dans leur lit. C. R. Acad. Sci. Paris 73 (1871) 147-154.
9. L.J. Durlofsky, B. Engquist and S. Osher, Triangle based adaptive stencils for the solution of hyperbolic conservation laws. J. Comput. Phys. 98 (1992) 64-73.
10. T. Galloüet, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32 (2003) 479-513. (Pubitemid 36178533)
11. J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89-102. (Pubitemid 33771939)
12. S. Gottlieb, C.-W. Shu and E. Tadmor, High order time discretization methods with the strong stability property. SIAM Rev. 43 (2001) 89-112. (Pubitemid 32406893)
13. M.E. Hubbard, Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. J. Comput. Phys. 155 (1999) 54-74.
14. M.E. Hubbard, On the accuracy of one-dimensional models of steady converging/diverging open channel flows. Int. J. Numer. Methods Fluids 35 (2001) 785-808. (Pubitemid 32292404)
15. S. Jin, A steady-state capturing method for hyperbolic system with geometrical source terms. ESAIM: M2AN 35 (2001) 631-645. (Pubitemid 33570241)
16. S. Jin and X. Wen, Two interface-type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations. SIAM J. Sci. Comput. 26 (2005) 2079-2101. (Pubitemid 41641363)
17. D. Kröner, Numerical Schemes for Conservation Laws. Wiley, Chichester (1997).
18. A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397-425.
19. A. Kurganov and C.-T. Lin, On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys. 2 (2007) 141-163.
20. A. Kurganov and G. Petrova, Central-upwind schemes on triangular grids for hyperbolic systems of conservation laws. Numer. Methods Partial Diff. Equ. 21 (2005) 536-552. (Pubitemid 40668505)
21. A. Kurganov and G. Petrova, A second-order well-balanced positivity preserving scheme for the Saint-Venant system. Commun. Math. Sci. 5 (2007) 133-160.
22. A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 214-282.
23. A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer. Methods Partial Diff. Equ. 18 (2002) 584-608.
24. A. Kurganov, S. Noelle and G. Petrova, Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton- Jacobi equations. SIAM J. Sci. Comput. 23 (2001) 707-740.
25. R.J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wavepropagation algorithm. J. Comput. Phys. 146 (1998) 346-365. (Pubitemid 128347128)
26. R. LeVeque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics, Cambridge University Press (2002).
27. K.-A. Lie and S. Noelle, On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws. SIAM J. Sci. Comput. 24 (2003) 1157-1174.
28. H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-463.
29. S. Noelle, N. Pankratz, G. Puppo and J. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213 (2006) 474-499. (Pubitemid 43240121)
30. B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231. (Pubitemid 33743526)
31. G. Russo, Central schemes for balance laws, in Hyperbolic problems: Theory, numerics, applications II, Internat. Ser. Numer. Math. 141, Birkhäuser, Basel (2001) 821-829.
32. G. Russo, Central schemes for conservation laws with application to shallow water equations, in Trends and applications of mathematics to mechanics: STAMM 2002, S. Rionero and G. Romano Eds., Springer-Verlag Italia SRL (2005) 225-246.
33. P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21 (1984) 995-1011. (Pubitemid 14664098)
34. B. van Leer, Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101-136.
35. M. Wierse, A new theoretically motivated higher order upwind scheme on unstructured grids of simplices. Adv. Comput. Math. 7 (1997) 303-335.
36. Y. Xing and C.-W. Shu, High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208 (2005) 206-227. (Pubitemid 40714221)
37. Y. Xing and C.-W. Shu, A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. Commun. Comput. Phys. 1 (2006) 100-134.