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Central-Upwind Schemes for the Saint-Venant System

Published online by Cambridge University Press:  15 August 2002

Alexander Kurganov  and
Doron Levy
Alexander Kurganov
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109 and Mathematics Department, Tulane University, New Orleans, LA 70118, USA. [email protected].
Doron Levy
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA. [email protected].
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Abstract

We present one- and two-dimensional central-upwind schemesfor approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutionsin which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preservethis delicate balance with numerical schemes.Small perturbations of these states are also very difficultto compute. Our approach is based on extending semi-discrete central schemes forsystems of hyperbolic conservation laws to balance laws.Special attention is paid to the discretization of the sourceterm such as to preserve stationary steady-statesolutions. 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 problemsthat include dry areas.

Keywords

Saint-Venant systemshallow water equationshigh-order central-upwind schemesbalance lawsconservation lawssource terms.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 3 , May 2002 , pp. 397 - 425
Copyright
© EDP Sciences, SMAI, 2002

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References

Abdulle, A., Fourth Order Chebyshev Methods with Recurrence Relation. SIAM J. Sci. Comput. 23 (2002) 2041-2054. CrossRef
Abdulle, A. and Medovikov, A., Second Order Chebyshev Methods Based on Orthogonal Polynomials. Numer. Math. 90 (2001) 1-18. CrossRef
P. Arminjon and M.-C. Viallon, 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.
Arminjon, P., Viallon, M.-C. and Madrane, A., Finite Volume Extension, A of the Lax-Friedrichs and Nessyahu-Tadmor Schemes for Conservation Laws on Unstructured Grids. Int. J. Comput. Fluid Dyn. 9 (1997) 1-22. CrossRef
E. Audusse, M.O. Bristeau and B. Perthame, Kinetic Schemes for Saint-Venant Equations With Source Terms on Unstructured Grids. INRIA Report RR-3989 (2000).
Bermudez, A. and Vasquez, M.E., Upwind Methods for Hyperbolic Conservation Laws With Source Terms. Comput. & Fluids 23 (1994) 1049-1071. CrossRef
Bianco, F., Puppo, G. and Russo, G., High Order Central Schemes for Hyperbolic Systems of Conservation Laws. SIAM J. Sci. Comput. 21 (1999) 294-322. CrossRef
Buffard, T., Gallouët, T. and Hérard, J.-M., Sequel, A to a Rough Godunov Scheme. Application to Real Gas Flows. Comput. & Fluids 29-7 (2000) 813-847. CrossRef
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
Friedrichs, K.O. and Lax, P.D., Systems of Conservation Equations with a Convex Extension. Proc. Nat. Acad. Sci. USA 68 (1971) 1686-1688. CrossRef
T. Gallouët, J.-M. Hérard and N. Seguin, Some Approximate Godunov Schemes to Compute Shallow-Water Equations with Topography. Computers and Fluids (to appear).
J.F. Gerbeau and B. Perthame, Derivation of Viscous Saint-Venant System for Laminar Shallow Water; Numerical Validation. Discrete Contin. Dynam. Systems Ser. B 1 (2001) 89-102.
Gosse, L., Well-Balanced Scheme Using Non-Conservative Products De, Asigned for Hyperbolic Systems of Conservation Laws With Source Terms. Math. Models Methods Appl. Sci. 11 (2001) 339-365. CrossRef
Harten, A., Engquist, B., Osher, S. and Chakravarthy, S.R., Uniformly High Order Accurate Essentially Non-Oscillatory Schemes III. J. Comput. Phys. 71 (1987) 231-303. CrossRef
Jiang, G.-S. and Tadmor, E., Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws. SIAM J. Sci. Comput. 19 (1998) 1892-1917. CrossRef
Jin, S., Steady-state Capturing Method, A for Hyperbolic System with Geometrical Source Terms. ESAIM: M2AN 35 (2001) 631-645. CrossRef
Kurganov, A. and Levy, D., Third-Order Semi-Discrete Scheme, A for Conservation Laws and Convection-Diffusion Equations. SIAM J. Sci. Comput. 22 (2000) 1461-1488. 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
Kurganov, A. and Petrova, G., Third-Order Semi-Discrete Genuinely Multidimensional Central Scheme, A for Hyperbolic Conservation Laws and Related Problems. Numer. Math. 88 (2001) 683-729. CrossRef
Kurganov, A. and Petrova, G., Central Schemes and Contact Discontinuities. ESAIM: M2AN 34 (2000) 1259-1275. 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.
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
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.J. LeVeque and D.S. Bale, 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.
Levy, D., Puppo, G. and Russo, G., Central WENO Schemes for Hyperbolic Systems of Conservation Laws. ESAIM: M2AN 33 (1999) 547-571. CrossRef
Levy, D., Puppo, G. and Russo, G., Compact Central WENO Schemes for Multidimensional Conservation Laws. SIAM J. Sci. Comput. 22 (2000) 656-672. CrossRef
S.F. Liotta, V. Romano and G. Russo, 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.
Liu, X.-D. and Osher, S., Nonoscillatory High Order Accurate Self Similar Maximum Principle Satisfying Shock Capturing Schemes. I. SIAM J. Numer. Anal. 33 (1996) 760-779. CrossRef
Liu, X.-D., Osher, S. and Chan, T., Weighted Essentially Non-Oscillatory Schemes. J. Comput. Phys. 115 (1994) 200-212. CrossRef
Liu, X.-D. and Tadmor, E., Third Order Nonoscillatory Central Scheme for Hyperbolic Conservation Laws. Numer. Math. 79 (1998) 397-425. CrossRef
Medovikov, A., High Order Explicit Methods for Parabolic Equations. BIT 38 (1998) 372-390. CrossRef
Nessyahu, H. and Tadmor, E., Non-Oscillatory Central Differencing for Hyperbolic Conservation Laws. J. Comput. Phys. 87 (1990) 408-463. CrossRef
S. Noelle, 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.
Perthame, B. and Simeoni, C., A Kinetic Scheme for the Saint-Venant System with a Source Term. École Normale Supérieure, Report DMA-01-13. Calcolo 38 (2001) 201-301. CrossRef
G. Russo, Central Schemes for Balance Laws, Proceedings of HYP2000. Magdeburg (to appear).
de Saint-Venant, A.J.C., 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.
Shu, C.-W., Total-Variation-Diminishing Time Discretizations. SIAM J. Sci. Comput. 6 (1988) 1073-1084. CrossRef
Shu, C.-W. and Osher, S., Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes. J. Comput. Phys. 77 (1988) 439-471. CrossRef
Sweby, P.K., High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws. SIAM J. Numer. Anal. 21 (1984) 995-1011. CrossRef
Tadmor, E., Convenient Total Variation Diminishing Conditions for Nonlinear Difference Schemes. SIAM J. Numer. Anal. 25 (1988) 1002-1014. CrossRef