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

Published online by Cambridge University Press:  11 October 2010

Steve Bryson ,
Yekaterina Epshteyn ,
Alexander Kurganov  and
Guergana Petrova
Steve Bryson
Affiliation:
NASA Ames Research Center, Moffett Field, CA 94035, USA. Stephen.T.Bryson@nasa.gov
Yekaterina Epshteyn
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA. rina10@andrew.cmu.edu
Alexander Kurganov
Affiliation:
Mathematics Department, Tulane University, New Orleans, LA 70118, USA. kurganov@math.tulane.edu
Guergana Petrova
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA. gpetrova@math.tamu.edu
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Abstract

We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove that the scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be applied to models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of numerical examples.

Keywords

Hyperbolic systems of conservation and balance lawssemi-discrete central-upwind schemesSaint-Venant system of shallow water equations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 3 , May 2011 , pp. 423 - 446
Copyright
© EDP Sciences, SMAI, 2010

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References

Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comput. Phys. 114 (1994) 45–58. CrossRef
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).
Arminjon, P., Viallon, M.-C. and Madrane, A., 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. CrossRef
Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R. and Perthame, B., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050–2065. CrossRef
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).
Bryson, S. and Levy, D., Balanced central schemes for the shallow water equations on unstructured grids. SIAM J. Sci. Comput. 27 (2005) 532–552. CrossRef
Christov, I. and Popov, B., New nonoscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws. J. Comput. Phys. 227 (2008) 5736–5757. CrossRef
de Saint-Venant, A.J.C., 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.
Durlofsky, L.J., Engquist, B. and Osher, S., Triangle based adaptive stencils for the solution of hyperbolic conservation laws. J. Comput. Phys. 98 (1992) 64–73. CrossRef
Gallouët, T., Hérard, J.-M. and Seguin, N., Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32 (2003) 479–513. CrossRef
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.
Gottlieb, S., Shu, C.-W. and Tadmor, E., High order time discretization methods with the strong stability property. SIAM Rev. 43 (2001) 89–112. CrossRef
Hubbard, M.E., Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. J. Comput. Phys. 155 (1999) 54–74. CrossRef
Hubbard, M.E., On the accuracy of one-dimensional models of steady converging/diverging open channel flows. Int. J. Numer. Methods Fluids 35 (2001) 785–808. 3.0.CO;2-H>CrossRef
Jin, S., A steady-state capturing method for hyperbolic system with geometrical source terms. ESAIM: M2AN 35 (2001) 631–645. CrossRef
Jin, S. and Wen, X., Two interface-type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations. SIAM J. Sci. Comput. 26 (2005) 2079–2101. CrossRef
D. Kröner, Numerical Schemes for Conservation Laws. Wiley, Chichester (1997).
Kurganov, A. and Levy, D., Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397–425. CrossRef
Kurganov, A. and Lin, C.-T., On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys. 2 (2007) 141–163.
Kurganov, A. and Petrova, G., Central-upwind schemes on triangular grids for hyperbolic systems of conservation laws. Numer. Methods Partial Diff. Equ. 21 (2005) 536–552. CrossRef
Kurganov, A. and Petrova, G., A second-order well-balanced positivity preserving scheme for the Saint-Venant system. Commun. Math. Sci. 5 (2007) 133–160. CrossRef
Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 214–282.
Kurganov, A. and Tadmor, E., Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer. Methods Partial Diff. Equ. 18 (2002) 584–608. CrossRef
Kurganov, A., Noelle, S. and Petrova, G., Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23 (2001) 707–740. CrossRef
LeVeque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146 (1998) 346–365. CrossRef
R. LeVeque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics, Cambridge University Press (2002).
Lie, K.-A. and Noelle, S., 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. CrossRef
Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–463. CrossRef
Noelle, S., Pankratz, N., Puppo, G. and Natvig, J., Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213 (2006) 474–499. CrossRef
Perthame, B. and Simeoni, C., A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201–231. CrossRef
G. Russo, Central schemes for balance laws, in Hyperbolic problems: theory, numerics, applications II, Internat. Ser. Numer. Math. 141, Birkhäuser, Basel (2001) 821–829.
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.
Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21 (1984) 995–1011. CrossRef
van Leer, B., Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101–136. CrossRef
Wierse, M., A new theoretically motivated higher order upwind scheme on unstructured grids of simplices. Adv. Comput. Math. 7 (1997) 303–335. CrossRef
Xing, Y. and Shu, C.-W., High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208 (2005) 206–227. CrossRef
Xing, Y. and Shu, C.-W., 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.