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Well-Balanced Central Schemes on Overlapping Cells with Constant Subtraction Techniques for the Saint-Venant Shallow Water System

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Abstract

We develop well-balanced finite-volume central schemes on overlapping cells for the Saint-Venant shallow water system and its variants. The main challenge in deriving the schemes is related to the fact that the Saint-Venant system contains a geometric source term due to nonflat bottom topography and therefore a delicate balance between the flux gradients and source terms has to be preserved. We propose a constant subtraction technique, which helps one to ensure a well-balanced property of the schemes, while maintaining arbitrary high-order of accuracy. Hierarchical reconstruction limiting procedure is applied to eliminate spurious oscillations without using characteristic decomposition. Extensive one- and two-dimensional numerical simulations are conducted to verify the well-balanced property, high-order of accuracy, and non-oscillatory high-resolution for both smooth and nonsmooth solutions.

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References

  1. Atkinson, K.E.: An Introduction to Numerical Analysis, 2nd edn. Wiley, New York (1989)

    MATH  Google Scholar 

  2. Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050–2065 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bermudez, A., Vazquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cea, L., Garrido, M., Puertas, J.: Experimental validation of two-dimensional depth-averaged models for forecasting rainfall-runoff from precipitation data in urban areas. J. Hydrol. 382, 88–102 (2010)

    Article  Google Scholar 

  5. Cea, L., Vázquez-Cendón, M.E.: Unstructured finite volume discretisation of bed friction and convective flux in solute transport models linked to the shallow water equations. J. Comput. Phys. 231, 3317–3339 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chertock, A., Cui, S., Kurganov, A., Wu, T.: Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms. Int. J. Numer. Meth. Fluids (submitted)

  7. Chevaugeon, N., Xin, J., Hu, P., Li, X., Cler, D., Flaherty, J.E., Shephard, M.S.: Discontinuous Galerkin methods applied to shock and blast problems. J. Sci. Comput. 22(23), 227–243 (2005)

    Article  MathSciNet  Google Scholar 

  8. de Saint-Venant, A.: Thèorie du mouvement non-permanent des eaux, avec application aux crues des rivière at à l’introduction des marèes dans leur lit. C.R. Acad. Sci. Paris 73, 147–154 (1871)

    MATH  Google Scholar 

  9. Flamant, A.: Mécanique appliquée: Hydraulique. Baudry éditeur, Paris (France) (1891)

    Google Scholar 

  10. Gallouët, T., Hérard, J.-M., Seguin, N.: Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32, 479–513 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gottlieb, S., Ketcheson, D., Shu, C.-W.: Strong stability preserving Runge–Kutta and multistep time discretizations. World Scientific Publishing Co Pte. Ltd., Hackensack (2011)

    Book  MATH  Google Scholar 

  12. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high-order accurate essentially nonoscillatory schemes. III. J. Comput. Phys. 71, 231–303 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  14. Jin, S.: A steady-state capturing method for hyperbolic systems with geometrical source terms. M2AN Math. Model. Numer. Anal. 35, 631–645 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kurganov, A., Levy, D.: Central-upwind schemes for the Saint-Venant system. M2AN Math. Model. Numer. Anal. 36, 397–425 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kurganov, A., Petrova, G.: A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5, 133–160 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. LeVeque, R.J.: Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146, 346–365 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Liu, Y.: Central schemes on overlapping cells. J. Comput. Phys. 209, 82–104 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Liu, Y., Shu, C.-W., Tadmor, E., Zhang, M.: Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction. SIAM J. Numer. Anal. 45, 2442–2467 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Liu, Y., Shu, C.-W., Tadmor, E., Zhang, M.: Non-oscillatory hierarchical reconstruction for central and finite volume schemes. Commun. Comput. Phys. 2, 933–963 (2007)

    MATH  MathSciNet  Google Scholar 

  21. Lukáčová-Medviďová, M., Noelle, S., Kraft, M.: Well-balanced finite volume evolution Galerkin methods for the shallow water equations. J. Comput. Phys. 221, 122–147 (2007)

    Article  MathSciNet  Google Scholar 

  22. Nessyahu, H., Tadmor, E.: Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  23. Noelle, S., Pankratz, N., Puppo, G., Natvig, J.R.: Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213, 474–499 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Noelle, S., Xing, Y., Shu, C.-W.: High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226, 29–58 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Russo, G.: Central schemes for conservation laws with application to shallow water equations. In: Rionero, S., Romano, G. (eds.) Trends and Applications of Mathematics to Mechanics, pp. 225–246. Springer, Milan (2005)

  26. Shu, C.-W.: Numerical experiments on the accuracy of ENO and modified ENO schemes. J. Sci. Comput. 5, 127–149 (1990)

    Article  MATH  Google Scholar 

  27. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Cetraro, 1997), Lecture Notes in Mathematics, pp. 325–432. Springer, Berlin 1697 (1998)

  28. Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  30. Shu, C.-W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. J. Comput. Phys. 83, 32–78 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  31. van Leer, B.: Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys. 14, 361–370 (1974)

    Article  MATH  Google Scholar 

  32. van Leer, B.: Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection. J. Comput. Phys. 23, 276–299 (1977)

    Article  MATH  Google Scholar 

  33. van Leer, B.: Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)

    Article  Google Scholar 

  34. Vázquez-Cendón, M.E.: Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 148, 497–526 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  35. Vukovic, S., Sopta, L.: ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J. Comput. Phys. 179, 593–621 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  36. Xing, Y., Shu, C.-W.: High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208, 206–227 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  37. Xing, Y., Shu, C.-W.: High-order well-balanced finite difference WENO schemes for a class of hyperbolic systems with source terms. J. Sci. Comput. 27, 477–494 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  38. Xing, Y., Shu, C.-W.: High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys. 214, 567–598 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  39. Xu, Z., Liu, Y., Du, H., Lin, G., Shu, C.-W.: Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws. J. Comput. Phys. 230, 6843–6865 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Suo Yang & Yingjie Liu

  2. Mathematics Department, Tulane University, New Orleans, LA, 70118, USA

    Alexander Kurganov

Authors
  1. Suo Yang
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  2. Alexander Kurganov
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  3. Yingjie Liu
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Correspondence to Yingjie Liu.

Additional information

Suo Yang and Yingjie Liu: Research supported in part by NSF Grant DMS-1115671.

Alexander Kurganov: Research supported in part by NSF Grant DMS-1216957 and ONR Grant N00014-12-1-0833.

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Yang, S., Kurganov, A. & Liu, Y. Well-Balanced Central Schemes on Overlapping Cells with Constant Subtraction Techniques for the Saint-Venant Shallow Water System. J Sci Comput 63, 678–698 (2015). https://doi.org/10.1007/s10915-014-9908-z

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