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A higher order Finite Volume resolution method for a system related to the inviscid primitive equations in a complex domain

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Abstract

We construct the cell-centered Finite Volume discretization of the two-dimensional inviscid primitive equations in a domain with topography. To compute the numerical fluxes, the so-called Upwind Scheme (US) and the Central-Upwind Scheme (CUS) are introduced. For the time discretization, we use the classical fourth order Runge–Kutta method. We verify, with our numerical simulations, that the US (or CUS) is a robust first (or second) order scheme, regardless of the shape or size of the topography and without any mesh refinement near the topography.

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References

  1. Gill, A.E.: Atmosphere-Ocean Dynamics. Academic Press, New York (1982)

    Google Scholar 

  2. Haltiner, G.J.: Numerical Weather Prediction. Wiley, New York (1971)

    Google Scholar 

  3. Rogers, R.R., Yau, M.K.: A Short Course in Cloud Physics, 3rd edn. Pergamon Press Oxford, New York (1989)

    Google Scholar 

  4. Oliger, Joseph, Sundström, Arne: Theoretical and practical aspects of some initial boundary value problems in fluid dynamics. SIAM J. Appl. Math. 35(3), 419–446 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Temam, R., Tribbia, J.: Open boundary conditions for the primitive and Boussinesq equations. J. Atmos. Sci. 60(21), 2647–2660 (2003)

    Article  MathSciNet  Google Scholar 

  6. Chen, Q.S., Laminie, J., Rousseau, A., Temam, R., Tribbia, J.: A 2.5D model for the equations of the ocean and the atmosphere. Anal. Appl. (Singap.) 5(3), 199–229 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, Q., Shiue, M.-C., Temam, R.: The barotropic mode for the primitive equations. J. Sci. Comput. 45(1–3), 167–199 (2010)

    Article  MathSciNet  Google Scholar 

  8. Chen, Q., Temam, R., Tribbia, J.J.: Simulations of the 2.5D inviscid primitive equations in a limited domain. J. Comput. Phys. 227(23), 9865–9884 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Rousseau, A., Temam, R., Tribbia, J.: Boundary conditions for the 2D linearized PEs of the ocean in the absence of viscosity. Discrete Contin. Dyn. Syst. 13(5), 1257–1276 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rousseau, A., Temam, R., Tribbia, J.: Numerical simulations of the inviscid primitive equations in a limited domain. In: Calgaro, C., Coulombel, J.-F., Goudon, T. (eds.) Analysis and Simulation of Fluid Dynamics, Advances in Mathematical Fluid Mechanics, pp. 163–181. Birkhäuser, Basel (2007)

  11. Rousseau, A., Temam, R., Tribbia, J.: The 3D primitive equations in the absence of viscosity: boundary conditions and well-posedness in the linearized case. J. Math. Pures Appl. (9) 89(3), 297–319 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rousseau, A., Temam, R.M., Tribbia, J.J.: Boundary value problems for the inviscid primitive equations in limited domains. In: Handbook of Numerical Analysis, vol. XIV. Special Volume: Computational Methods for the Atmosphere and the Oceans, vol. 14 of Handbook of Numerical Analysis, pp. 481–575. Elsevier, North-Holland, Amsterdam (2009)

  13. Chen, Q., Shiue, M.-C., Temam, R., Tribbia, J.: Numerical approximation of the inviscid 3D primitive equations in a limited domain. ESAIM Math. Model. Numer. Anal. 46(3), 619–646 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hong, Y., Temam, R., Bousquet, A., Tribbia, J.: Numerical weather prediction with primitive equations with humidity in two dimension using finite volume method (preprint)

  15. Zelati, M.C., Temam, R.: The atmospheric equation of water vapor with saturation. Boll. Unione Mat. Ital. (9) 5(2), 309–336 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Lions, J.-L., Temam, R., Wang, S.H.: New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5(2), 237–288 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Petcu, M., Temam, R.M., Ziane, M.: Some mathematical problems in geophysical fluid dynamics. In: Handbook of Numerical Analysis, vol. XIV. Special Volume: Computational Methods for the Atmosphere and the Oceans, vol. 14 of Handbook of Numerical Analysis, pp. 577–750. Elsevier, North-Holland, Amsterdam (2009)

  18. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  19. Versteeg, H.K., Malalasekera, W.: An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson Education Limited, London (2007)

    Google Scholar 

  20. Adamy, K., Bousquet, A., Faure, S., Laminie, J., Temam, R.: A multilevel method for finite volume discretization of the two-dimensional nonlinear shallow-water equations. Ocean Model. 33(3–4), 235–256 (2010)

    Article  Google Scholar 

  21. Kurganov, A., Levy, D.: Central-upwind schemes for the Saint-Venant system. M2AN Math. Model. Numer. Anal. 36(3), 397–425 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kurganov, A., Noelle, S., Petrova, G.: Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations. SIAM J. Sci. Comput. 23(3), 707–740 (2001). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kurganov, A., Petrova, G.: Central-upwind schemes for two-layer shallow water equations. SIAM J. Sci. Comput. 31(3), 1742–1773 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kurganov, A., Petrova, G., Popov, B.: Adaptive semidiscrete central-upwind schemes for nonconvex hyperbolic conservation laws. SIAM J. Sci. Comput. 29(6), 2381–2401 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160(1), 241–282 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nessyahu, H., Tadmor, E.: Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87(2), 408–463 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  27. Calhoun, D., LeVeque, R.J.: A Cartesian grid finite-volume method for the advection-diffusion equation in irregular geometries. J. Comput. Phys. 157(1), 143–180 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. LeVeque, R.J.: Cartesian grids and rotated difference methods for multi-dimensional flow. In: Proceedings of International Conference on Scientific Computation (Hangzhou, 1991), vol. 1 of Series Applied Mathematics, pp. 76–85. World Scientific Publishing, River Edge, NJ (1992)

  29. LeVeque, R.J., Calhoun, D.: Cartesian grid methods for fluid flow in complex geometries. In: Fauci, L.J., Gueron, S. (eds.) Computational modeling in biological fluid dynamics (Minneapolis, MN, 1999), vol. 124, pp. 117–143. IMA Volumes in Mathematics and its Applications, Springer, New York (2001)

  30. Huang, W., Kappen, A.M.: A study of cell-center finite volume methods for diffusion equations. Electronic (1998)

  31. Gie, G.-M., Temam, R.: Cell centered finite volume methods using Taylor series expansion scheme without fictitious domains. Int. J. Numer. Anal. Model. 7(1), 1–29 (2010)

    MathSciNet  Google Scholar 

  32. Gie, G.-M., Temam, R.: Convergence of the cell-centered Finite Volume discretization method (submitted)

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Acknowledgments

This work was supported in part by NSF Grants DMS 1206438 and DMS 1212141, and by the Research Fund of Indiana University. The authors would like to thank Professor Roger Temam and Dr. Joseph Tribbia for their suggestion and advice.

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Authors and Affiliations

  1. The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, IN, 47405, USA

    Arthur Bousquet, Gung-Min Gie & Youngjoon Hong

  2. Université des Antilles et de la Guyane, Fouillole, BP 250, 97157 , Pointe-à-Pitre, France

    Jacques Laminie

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  1. Arthur Bousquet
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  2. Gung-Min Gie
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  3. Youngjoon Hong
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  4. Jacques Laminie
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Correspondence to Gung-Min Gie.

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Bousquet, A., Gie, GM., Hong, Y. et al. A higher order Finite Volume resolution method for a system related to the inviscid primitive equations in a complex domain. Numer. Math. 128, 431–461 (2014). https://doi.org/10.1007/s00211-014-0622-4

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  • DOI: https://doi.org/10.1007/s00211-014-0622-4

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