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Well-Balanced Numerical Schemes for Shallow Water Equations with Horizontal Temperature Gradient

  • Mai Duc ThanhEmail author
  • Nguyen Xuan Thanh
Article
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Abstract

A class of well-balanced numerical schemes for the one-dimensional shallow water equations with temperature gradient is constructed. The construction of the schemes is based on two steps: the first step to absorb the nonconservative term and the second step to deal with the evolution of the system. Algorithms for computing contact waves which absorb the nonconservative term are developed. Furthermore, to improve the accuracy, the underlying numerical fluxes can be formed as convex combinations of a pair of numerical fluxes of a low and stable scheme and a higher and fast scheme. The schemes are well balanced and can retain the positivity of the water height and the water temperature. Numerical tests show that the schemes are stable and have a good accuracy.

Keywords

Shallow water equations Ripa system Conservation law Well-balanced scheme Convergence Accuracy 

Mathematics Subject Classification

35L65 65M08 76B15 

Notes

Acknowledgements

The authors would like to thank the reviewers for their very valuable comments and suggestions. This research is funded by the Vietnam National University HoChiMinh City (VNU-HCM) under Grant No. B2018-28-01.

References

  1. 1.
    Ambroso, A., Chalons, C., Coquel, F., Galié, T.: Relaxation and numerical approximation of a two-fluid two-pressure diphasic model. Math. Mod. Numer. Anal. 43, 1063–1097 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambroso, A., Chalons, C., Raviart, P.-A.: A Godunov-type method for the seven-equation model of compressible two-phase flow. Comput. Fluids 54, 67–91 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baudin, M., Coquel, F., Tran, Q.-H.: A semi-implicit relaxation scheme for modeling two-phase flow in a pipeline. SIAM J. Sci. Comput. 27, 914–936 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Botchorishvili, R., Perthame, B., Vasseur, A.: Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comput. 72, 131–157 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Botchorishvili, R., Pironneau, O.: Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws. J. Comput. Phys. 187, 391–427 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chertock, A., Kurganov, A., Liu, Y.: Central-upwind schemes for the system of shallow water equations with horizontal temperature gradients. Numer. Math. 127, 595–639 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coquel, F., Godlewski, E., Perthame, B., In, Rascle, P.: Some new Godunov and relaxation methods for two-phase flow problems. In: Toro, E. F. (ed.) Godunov Methods (Oxford, 1999), pp. 179–188. Kluwer/Plenum, New York (2001)Google Scholar
  9. 9.
    Cuong, D.H., Thanh, M.D.: A Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section. Appl. Math. Comput. 256, 602–629 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Coquel, F., Hérard, J.-M., Saleh, K., Seguin, N.: Two properties of two-velocity two-pressure models for two-phase flows. Commun. Math. Sci. 12, 593–600 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dubroca, B.: Positively conservative Roe’s matrixfor Euler equations. In: 16th International Conference on Numerical Methods in Fluid Dynamics (Arcachon, 1998), Lecture Notes in Phys., vol. 515, pp. 272–277. Springer, Berlin.  https://doi.org/10.1007/BFb0106594
  12. 12.
    Dal Maso, G., LeFloch, P.G., Murat, F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483–548 (1995)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gallardo, J.M., Parés, C., Castro, M.: On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J. Comput. Phys. 227, 574–601 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gallouet, T., Herard, J.-M., Seguin, N.: Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32, 479–513 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Greenberg, J.M., Leroux, A.Y.: A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33, 1–16 (1996)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Han, X., Li, G.: Well-balanced finite difference WENO schemes for the Ripa model. Comput. Fluids 134–135, 1–10 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Isaacson, E., Temple, B.: Convergence of the \(2\times 2\) Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55, 625–640 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kröner, D., Thanh, M.D.: Numerical solutions to compressible flows in a nozzle with variable cross-section. SIAM J. Numer. Anal. 43, 796–824 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kröner, D., LeFloch, P.G., Thanh, M.D.: The minimum entropy principle for fluid flows in a nozzle with discntinuous crosssection. Math. Mod. Numer. Anal. 42, 425–442 (2008)CrossRefzbMATHGoogle Scholar
  20. 20.
    LeFloch, P.G., Thanh, M.D.: The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Commun. Math. Sci. 1, 763–797 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    LeFloch, P.G., Thanh, M.D.: A Godunov-type method for the shallow water equations with variable topography in the resonant regime. J. Comput. Phys. 230, 7631–7660 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Li, G., Caleffi, V., Qi, Z.K.: A well-balanced finite difference WENO scheme for shallow water flow model. Appl. Math. Comput. 265, 1–16 (2015)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Li, G., Song, L.N., Gao, J.M.: High order well-balanced discontinuous Galerkin methods based on hydrostatic reconstruction for shallow water equations. J. Comput. Appl. Math. 340, 546–560 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Qian, S.G., Shao, F.J., Li, G.: High order well-balanced discontinuous Galerkin methods for shallow water flow under temperature fields. Comput. Appl. Math. (2018).  https://doi.org/10.1007/s40314-018-0662-y
  25. 25.
    Ripa, P.: Conservation laws for primitive equations models with inhomogeneous layers. Geophys. Astrophys. Fluid Dyn. 70, 85–111 (1993)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ripa, P.: On improving a one-layer ocean model with thermodynamics. J. Fluid Mech. 303, 169–201 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rosatti, G., Begnudelli, L.: The Riemann Problem for the one-dimensional, free-surface shallow water equations with a bed step: theoretical analysis and numerical simulations. J. Comput. Phys. 229, 760–787 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sanchez-Linares, C., Morales de Luna, T., Castro Diaz, M.J.: A HLLC scheme for Ripa model. Appl. Math. Comput. 72, 369–384 (2016)MathSciNetGoogle Scholar
  29. 29.
    Saurel, R., Abgrall, R.: A multi-phase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425–467 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tian, B., Toro, E.F., Castro, C.E.: A path-conservative method for a five-equation model of two-phase flow with an HLLC-type Riemann solver. Comput. Fluids 46, 122–132 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Thanh, M.D.: A phase decomposition approach and the Riemann problem for a model of two-phase flows. J. Math. Anal. Appl. 418, 569–594 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Thanh, M.D.: The Riemann problem for a non-isentropic fluid in a nozzle with discontinuous cross-sectional area. SIAM J. Appl. Math. 69, 1501–1519 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Thanh, M.D.: The Riemann problem for the shallow water equations with horizontal temperature gradients. Appl. Math. Comput. 325, 159–178 (2018)MathSciNetGoogle Scholar
  34. 34.
    Thanh, M.D.: Building fast well-balanced two-stage numerical schemes for a model of two-phase flows. Commun. Nonlinear Sci. Num. Simul. 19, 1836–1858 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Thanh, M.D., Kröner, D., Chalons, C.: A robust numerical method for approximating solutions of a model of two-phase flows and its properties. Appl. Math. Comput. 219, 320–344 (2012)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Touma, R., Klingenberg, C.: Well-balanced central finite volume methods for the Ripa system. Appl. Numer. Math. 97, 42–68 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  1. 1.Department of Mathematics, International UniversityVietnam National University - Ho Chi Minh CityHo Chi Minh CityVietnam
  2. 2.Department of Mathematics and Computer Science, University of ScienceVietnam National University - Ho Chi Minh CityHo Chi Minh CityVietnam

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