We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semi-discrete central schemes for systems of hyperbolic conservation laws to balance laws. Special attention is paid to the discretization of the source term such as to preserve stationary steady-state solutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water. This important feature allows one to compute solutions for problems that include dry areas.
@article{M2AN_2002__36_3_397_0,
author = {Kurganov, Alexander and Levy, Doron},
title = {Central-upwind schemes for the Saint-Venant system},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {36},
year = {2002},
pages = {397-425},
doi = {10.1051/m2an:2002019},
mrnumber = {1918938},
zbl = {1137.65398},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2002__36_3_397_0}
}
Kurganov, Alexander; Levy, Doron. Central-upwind schemes for the Saint-Venant system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) pp. 397-425. doi : 10.1051/m2an:2002019. http://gdmltest.u-ga.fr/item/M2AN_2002__36_3_397_0/
[1] , Fourth Order Chebyshev Methods with Recurrence Relation. SIAM J. Sci. Comput. 23 (2002) 2041-2054. | Zbl 1009.65048
[2] and, Second Order Chebyshev Methods Based on Orthogonal Polynomials. Numer. Math. 90 (2001) 1-18. | Zbl 0997.65094
[3] and, Généralisation du schéma de Nessyahu-Tadmor pour une équation hyperbolique à deux dimensions d'espace. C. R. Acad. Sci. Paris Sér. I Math. t. 320 (1995) 85-88. | Zbl 0831.65091
[4] , and, 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. | Zbl 0913.76063
[5] , and, Kinetic Schemes for Saint-Venant Equations With Source Terms on Unstructured Grids. INRIA Report RR-3989 (2000).
[6] and, Upwind Methods for Hyperbolic Conservation Laws With Source Terms. Comput. & Fluids 23 (1994) 1049-1071. | Zbl 0816.76052
[7] , and, High Order Central Schemes for Hyperbolic Systems of Conservation Laws. SIAM J. Sci. Comput. 21 (1999) 294-322. | Zbl 0940.65093
[8] , and, A Sequel to a Rough Godunov Scheme. Application to Real Gas Flows. Comput. & Fluids 29-7 (2000) 813-847. | Zbl 0961.76048
[9] , and, High Order Time Discretization Methods with the Strong Stability Property. SIAM Rev. 43 (2001) 89-112. | Zbl 0967.65098
[10] and, Systems of Conservation Equations with a Convex Extension. Proc. Nat. Acad. Sci. USA 68 (1971) 1686-1688. | Zbl 0229.35061
[11] , and, Some Approximate Godunov Schemes to Compute Shallow-Water Equations with Topography. Computers and Fluids (to appear). | MR 1966639 | Zbl 1084.76540
[12] and, Derivation of Viscous Saint-Venant System for Laminar Shallow Water; Numerical Validation. Discrete Contin. Dynam. Systems Ser. B 1 (2001) 89-102. | Zbl 0997.76023
[13] , A Well-Balanced Scheme Using Non-Conservative Products Designed for Hyperbolic Systems of Conservation Laws With Source Terms. Math. Models Methods Appl. Sci. 11 (2001) 339-365. | Zbl 1018.65108
[14] ,, and, Uniformly High Order Accurate Essentially Non-Oscillatory Schemes III. J. Comput. Phys. 71 (1987) 231-303. | Zbl 0652.65067
[15] and, Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws. SIAM J. Sci. Comput. 19 (1998) 1892-1917. | Zbl 0914.65095
[16] , A Steady-state Capturing Method for Hyperbolic System with Geometrical Source Terms. ESAIM: M2AN 35 (2001) 631-645. | Numdam | Zbl 1001.35083
[17] and, A Third-Order Semi-Discrete Scheme for Conservation Laws and Convection-Diffusion Equations. SIAM J. Sci. Comput. 22 (2000) 1461-1488. | Zbl 0979.65077
[18] , and, Semi-Discrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton-Jacobi Equations. SIAM J. Sci. Comput. 23 (2001) 707-740. | Zbl 0998.65091
[19] and, A Third-Order Semi-Discrete Genuinely Multidimensional Central Scheme for Hyperbolic Conservation Laws and Related Problems. Numer. Math. 88 (2001) 683-729. | Zbl 0987.65090
[20] and, Central Schemes and Contact Discontinuities. ESAIM: M2AN 34 (2000) 1259-1275. | Numdam | Zbl 0972.65055
[21] and, New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations. J. Comput. Phys. 160 (2000) 214-282. | Zbl 0987.65085
[22] , Towards the Ultimate Conservative Difference Scheme, V. A Second Order Sequel to Godunov's Method. J. Comput. Phys. 32 (1979) 101-136. | Zbl 0939.76063
[23] , Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods: The Quasi-Steady Wave-Propagation Algorithm. J. Comput. Phys. 146 (1998) 346-365. | Zbl 0931.76059
[24] and, Wave Propagation Methods for Conservation Laws with Source Terms, Hyperbolic Problems: Theory, Numerics, Applications, Vol. II, Zürich (1998). Birkhäuser, Basel, Internat. Ser. Numer. Math. 130 (1999) 609-618. | Zbl 0927.35062
[25] , and, Central WENO Schemes for Hyperbolic Systems of Conservation Laws. ESAIM: M2AN 33 (1999) 547-571. | Numdam | Zbl 0938.65110
[26] , and, Compact Central WENO Schemes for Multidimensional Conservation Laws. SIAM J. Sci. Comput. 22 (2000) 656-672. | Zbl 0967.65089
[27] , and, Central Schemes for Systems of Balance Laws, Hyperbolic Problems: Theory, Numerics, Applications, Vol. II, Zürich (1998). Birkhäuser, Basel, Internat. Ser. Numer. Math. 130 (1999) 651-660. | Zbl 0926.35081
[28] and, Nonoscillatory High Order Accurate Self Similar Maximum Principle Satisfying Shock Capturing Schemes. I. SIAM J. Numer. Anal. 33 (1996) 760-779. | Zbl 0859.65091
[29] , and, Weighted Essentially Non-Oscillatory Schemes. J. Comput. Phys. 115 (1994) 200-212. | Zbl 0811.65076
[30] and, Third Order Nonoscillatory Central Scheme for Hyperbolic Conservation Laws. Numer. Math. 79 (1998) 397-425. | Zbl 0906.65093
[31] , High Order Explicit Methods for Parabolic Equations. BIT 38 (1998) 372-390. | Zbl 0909.65060
[32] and, Non-Oscillatory Central Differencing for Hyperbolic Conservation Laws. J. Comput. Phys. 87 (1990) 408-463. | Zbl 0697.65068
[33] , A Comparison of Third and Second Order Accurate Finite Volume Schemes for the Two-Dimensional Compressible Euler Equations, Hyperbolic Problems: Theory, Numerics, Applications, Vol. I, Zürich (1998). Birkhäuser, Basel, Internat. Ser. Numer. Math. 129 (1999) 757-766. | Zbl 0923.76223
[34] and, A Kinetic Scheme for the Saint-Venant System with a Source Term. École Normale Supérieure, Report DMA-01-13. Calcolo 38 (2001) 201-301. | Zbl 1008.65066
[35] , Central Schemes for Balance Laws, Proceedings of HYP2000. Magdeburg (to appear).
[36] , Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit. C. R. Acad. Sci. Paris 73 (1871) 147-154. | JFM 03.0482.04
[37] , Total-Variation-Diminishing Time Discretizations. SIAM J. Sci. Comput. 6 (1988) 1073-1084. | Zbl 0662.65081
[38] and, Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes. J. Comput. Phys. 77 (1988) 439-471. | Zbl 0653.65072
[39] , High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws. SIAM J. Numer. Anal. 21 (1984) 995-1011. | Zbl 0565.65048
[40] , Convenient Total Variation Diminishing Conditions for Nonlinear Difference Schemes. SIAM J. Numer. Anal. 25 (1988) 1002-1014. | Zbl 0662.65082