Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) no. 3, p. 423-446
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.
@article{M2AN_2011__45_3_423_0,
     author = {Bryson, Steve and Epshteyn, Yekaterina and Kurganov, Alexander and Petrova, Guergana},
     title = {Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {3},
     year = {2011},
     pages = {423-446},
     doi = {10.1051/m2an/2010060},
     zbl = {1267.76068},
     mrnumber = {2804645},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2011__45_3_423_0}
}
Bryson, Steve; Epshteyn, Yekaterina; Kurganov, Alexander; Petrova, Guergana. Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) no. 3, pp. 423-446. doi : 10.1051/m2an/2010060. http://www.numdam.org/item/M2AN_2011__45_3_423_0/

[1] R. Abgrall, On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comput. Phys. 114 (1994) 45-58. | MR 1286187 | Zbl 0822.65062

[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. | MR 1609613 | Zbl 0913.76063

[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. | MR 2086830 | Zbl 1133.65308

[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). | MR 2128209 | Zbl 1086.65091

[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. | MR 2202233 | Zbl 1089.76036

[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. | MR 2414928 | Zbl 1151.65068

[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. | JFM 03.0482.04

[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. | Zbl 0747.65072

[10] T. Gallouët, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32 (2003) 479-513. | MR 1966639 | Zbl 1084.76540

[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. | MR 1821555 | Zbl 0997.76023

[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. | MR 1854647 | Zbl 0967.65098

[13] M.E. Hubbard, Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. J. Comput. Phys. 155 (1999) 54-74. | MR 1716501 | Zbl 0934.65109

[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. | Zbl 1014.76051

[15] S. Jin, A steady-state capturing method for hyperbolic system with geometrical source terms. ESAIM: M2AN 35 (2001) 631-645. | Numdam | MR 1862872 | Zbl 1001.35083

[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. | MR 2196590 | Zbl 1083.35062

[17] D. Kröner, Numerical Schemes for Conservation Laws. Wiley, Chichester (1997). | Zbl 0872.76001

[18] A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397-425. | Numdam | MR 1918938 | Zbl 1137.65398

[19] A. Kurganov and C.-T. Lin, On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys. 2 (2007) 141-163. | MR 2305919 | Zbl 1164.65455

[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. | MR 2128595 | Zbl 1071.65122

[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. | MR 2310637 | Zbl 1226.76008

[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. | MR 1756766 | Zbl 0987.65085

[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. | MR 1919599 | Zbl 1058.76046

[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. | MR 1860961 | Zbl 0998.65091

[25] R.J. Leveque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146 (1998) 346-365. | MR 1650496 | Zbl 0931.76059

[26] R. Leveque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics, Cambridge University Press (2002). | MR 1925043 | Zbl 1010.65040

[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. | MR 1976211 | Zbl 1038.65078

[28] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-463. | MR 1047564 | Zbl 0697.65068

[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. | MR 2207248 | Zbl 1088.76037

[30] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231. | MR 1890353 | Zbl 1008.65066

[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. | MR 1871169

[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. | MR 1910803

[33] P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21 (1984) 995-1011. | MR 760628 | Zbl 0565.65048

[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. | MR 1703646 | Zbl 0939.76063

[35] M. Wierse, A new theoretically motivated higher order upwind scheme on unstructured grids of simplices. Adv. Comput. Math. 7 (1997) 303-335. | MR 1449684 | Zbl 0889.65103

[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. | MR 2144699 | Zbl 1114.76340

[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. | MR 2216604 | Zbl 1115.65096