# Central-upwind schemes for the Saint-Venant system

Alexander Kurganov; Doron Levy

- Volume: 36, Issue: 3, page 397-425
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topKurganov, Alexander, and Levy, Doron. "Central-upwind schemes for the Saint-Venant system." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.3 (2002): 397-425. <http://eudml.org/doc/245738>.

@article{Kurganov2002,

abstract = {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.},

author = {Kurganov, Alexander, Levy, Doron},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {Saint-Venant system; shallow water equations; high-order central-upwind schemes; balance laws; conservation laws; source terms},

language = {eng},

number = {3},

pages = {397-425},

publisher = {EDP-Sciences},

title = {Central-upwind schemes for the Saint-Venant system},

url = {http://eudml.org/doc/245738},

volume = {36},

year = {2002},

}

TY - JOUR

AU - Kurganov, Alexander

AU - Levy, Doron

TI - Central-upwind schemes for the Saint-Venant system

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2002

PB - EDP-Sciences

VL - 36

IS - 3

SP - 397

EP - 425

AB - 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.

LA - eng

KW - Saint-Venant system; shallow water equations; high-order central-upwind schemes; balance laws; conservation laws; source terms

UR - http://eudml.org/doc/245738

ER -

## References

top- [1] A. Abdulle, Fourth Order Chebyshev Methods with Recurrence Relation. SIAM J. Sci. Comput. 23 (2002) 2041–2054. Zbl1009.65048
- [2] A. Abdulle and A. Medovikov, Second Order Chebyshev Methods Based on Orthogonal Polynomials. Numer. Math. 90 (2001) 1–18. Zbl0997.65094
- [3] 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. Zbl0831.65091
- [4] 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. Zbl0913.76063
- [5] 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).
- [6] A. Bermudez and M.E. Vasquez, Upwind Methods for Hyperbolic Conservation Laws With Source Terms. Comput. & Fluids 23 (1994) 1049–1071. Zbl0816.76052
- [7] F. Bianco, G. Puppo and G. Russo, High Order Central Schemes for Hyperbolic Systems of Conservation Laws. SIAM J. Sci. Comput. 21 (1999) 294–322. Zbl0940.65093
- [8] T. Buffard, T. Gallouët and J.-M. Hérard, A Sequel to a Rough Godunov Scheme. Application to Real Gas Flows. Comput. & Fluids 29-7 (2000) 813–847. Zbl0961.76048
- [9] S. Gottlieb, C.-W. Shu and E. Tadmor, High Order Time Discretization Methods with the Strong Stability Property. SIAM Rev. 43 (2001) 89–112. Zbl0967.65098
- [10] K.O. Friedrichs and P.D. Lax, Systems of Conservation Equations with a Convex Extension. Proc. Nat. Acad. Sci. USA 68 (1971) 1686–1688. Zbl0229.35061
- [11] T. Gallouët, J.-M. Hérard and N. Seguin, Some Approximate Godunov Schemes to Compute Shallow-Water Equations with Topography. Computers and Fluids (to appear). Zbl1084.76540MR1966639
- [12] 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. Zbl0997.76023
- [13] L. Gosse, 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. Zbl1018.65108
- [14] A. Harten, B. Engquist, S. Osher and S.R. Chakravarthy, Uniformly High Order Accurate Essentially Non-Oscillatory Schemes III. J. Comput. Phys. 71 (1987) 231–303. Zbl0652.65067
- [15] G.-S. Jiang and E. Tadmor, Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws. SIAM J. Sci. Comput. 19 (1998) 1892–1917. Zbl0914.65095
- [16] S. Jin, A Steady-state Capturing Method for Hyperbolic System with Geometrical Source Terms. ESAIM: M2AN 35 (2001) 631–645. Zbl1001.35083
- [17] A. Kurganov and D. Levy, A Third-Order Semi-Discrete Scheme for Conservation Laws and Convection-Diffusion Equations. SIAM J. Sci. Comput. 22 (2000) 1461–1488. Zbl0979.65077
- [18] 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. Zbl0998.65091
- [19] A. Kurganov and G. Petrova, A Third-Order Semi-Discrete Genuinely Multidimensional Central Scheme for Hyperbolic Conservation Laws and Related Problems. Numer. Math. 88 (2001) 683–729. Zbl0987.65090
- [20] A. Kurganov and G. Petrova, Central Schemes and Contact Discontinuities. ESAIM: M2AN 34 (2000) 1259–1275. Zbl0972.65055
- [21] 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. Zbl0987.65085
- [22] 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. Zbl0939.76063
- [23] 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. Zbl0931.76059
- [24] 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. Zbl0927.35062
- [25] D. Levy, G. Puppo and G. Russo, Central WENO Schemes for Hyperbolic Systems of Conservation Laws. ESAIM: M2AN 33 (1999) 547–571. Zbl0938.65110
- [26] D. Levy, G. Puppo and G. Russo, Compact Central WENO Schemes for Multidimensional Conservation Laws. SIAM J. Sci. Comput. 22 (2000) 656–672. Zbl0967.65089
- [27] 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. Zbl0926.35081
- [28] X.-D. Liu and S. Osher, Nonoscillatory High Order Accurate Self Similar Maximum Principle Satisfying Shock Capturing Schemes. I. SIAM J. Numer. Anal. 33 (1996) 760–779. Zbl0859.65091
- [29] X.-D. Liu, S. Osher and T. Chan, Weighted Essentially Non-Oscillatory Schemes. J. Comput. Phys. 115 (1994) 200–212. Zbl0811.65076
- [30] X.-D. Liu and E. Tadmor, Third Order Nonoscillatory Central Scheme for Hyperbolic Conservation Laws. Numer. Math. 79 (1998) 397–425. Zbl0906.65093
- [31] A. Medovikov, High Order Explicit Methods for Parabolic Equations. BIT 38 (1998) 372–390. Zbl0909.65060
- [32] H. Nessyahu and E. Tadmor, Non-Oscillatory Central Differencing for Hyperbolic Conservation Laws. J. Comput. Phys. 87 (1990) 408–463. Zbl0697.65068
- [33] 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. Zbl0923.76223
- [34] B. Perthame and C. Simeoni, A Kinetic Scheme for the Saint-Venant System with a Source Term. École Normale Supérieure, Report DMA–01–13. Calcolo 38 (2001) 201–301. Zbl1008.65066
- [35] G. Russo, Central Schemes for Balance Laws, Proceedings of HYP2000. Magdeburg (to appear). Zbl0926.35081
- [36] A.J.C. de Saint-Venant, 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. Zbl03.0482.04JFM03.0482.04
- [37] C.-W. Shu, Total-Variation-Diminishing Time Discretizations. SIAM J. Sci. Comput. 6 (1988) 1073–1084. Zbl0662.65081
- [38] C.-W. Shu and S. Osher, Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes. J. Comput. Phys. 77 (1988) 439–471. Zbl0653.65072
- [39] P.K. Sweby, High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws. SIAM J. Numer. Anal. 21 (1984) 995–1011. Zbl0565.65048
- [40] E. Tadmor, Convenient Total Variation Diminishing Conditions for Nonlinear Difference Schemes. SIAM J. Numer. Anal. 25 (1988) 1002–1014. Zbl0662.65082

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.