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Partial Differential Equations: Theory, Control and Approximation pp 67–98Cite as

Finite Volume Multilevel Approximation of the Shallow Water Equations

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Abstract

The authors consider a simple transport equation in one-dimensional space and the linearized shallow water equations in two-dimensional space, and describe and implement a multilevel finite-volume discretization in the context of the utilization of the incremental unknowns. The numerical stability of the method is proved in both cases.

Project supported by the National Science Foundation (Nos. DMS 0906440, DMS 1206438) and the Research Fund of Indiana University.

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Notes

  1. 1.

    Including, strictly speaking, the separation points.

  2. 2.

    Including, strictly speaking, the separation edges.

References

  1. Adamy, K., Bousquet, A., Faure, S., et al.: A multilevel method for finite-volume discretization of the two-dimensional nonlinear shallow-water equations. Ocean Model. 33, 235–256 (2010). doi:10.1016/j.ocemod.2010.02.006

    Article  Google Scholar 

  2. Adamy, K., Pham, D.: A finite-volume implicit Euler scheme for the linearized shallow water equations: stability and convergence. Numer. Funct. Anal. Optim. 27(7–8), 757–783 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bellanger, M.: Traitement du Signal. Dunod, Paris (2006)

    Google Scholar 

  4. Bousquet, A., Marion, M., Temam, R.: Finite volume multilevel approximation of the shallow water equations II. (2013, in preparation)

    Google Scholar 

  5. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zhang, T.A.: Spectral Methods, Evolution to Complex Geometries and Applications to Fluid Dynamics, Scientific Computation. Springer, Berlin (2007)

    Google Scholar 

  6. Chen, Q., Shiue, M.C., Temam, R.: The barotropic mode for the primitive equations. J. Sci. Comput. 45, 167–199 (2010). doi:10.1007/s10915-009-9343-8. Special issue in memory of David Gottlieb

    Article  MathSciNet  Google Scholar 

  7. Chen, Q., Shiue, M.C., Temam, R., Tribbia, J.: Numerical approximation of the inviscid 3D Primitive equations in a limited domain. Modél. Math. Anal. Numér. 45, 619–646 (2012). doi:10.105/m2an/2011058

    Article  MathSciNet  Google Scholar 

  8. Dubois, T., Jauberteau, F., Temam, R.: Dynamic, Multilevel Methods and the Numerical Simulation of Turbulence. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  9. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Springer, Berlin (1990–1992)

    Book  Google Scholar 

  10. Eymard, R., Gallouet, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. VII, pp. 713–1020. North-Holland, Amsterdam (2002)

    Google Scholar 

  11. 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 

  12. Huang, A., Temam, R.: The linearized 2D inviscid shallow water equations in a rectangle: boundary conditions and well-posedness. (2013, to appear)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  14. Lions, J.L., Temam, R., Wang, S.: Models of the coupled atmosphere and ocean (CAO I). Comput. Mech. Adv. 1, 5–54 (1993)

    MathSciNet  MATH  Google Scholar 

  15. Lions, J.L., Temam, R., Wang, S.: Numerical analysis of the coupled models of atmosphere and ocean (CAO II). Comput. Mech. Adv. 1, 55–119 (1993)

    MathSciNet  MATH  Google Scholar 

  16. Lions, J.L., Temam, R., Wang, S.: Splitting up methods and numerical analysis of some multiscale problems. Comput. Fluid Dyn. J. 5(2), 157–202 (1996). Special issue dedicated to A. Jameson

    MathSciNet  Google Scholar 

  17. Marchuk, G.I.: Methods of numerical mathematics, 2nd edn. Applications of Mathematics, vol. 2. Springer, New York (1982). Translated from the Russian by Arthur A. Brown

    Book  MATH  Google Scholar 

  18. Marion, M., Temam, R.: Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26, 1139–1157 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Marion, M., Temam, R.: Navier-Stokes equations, theory and approximation. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. VI, pp. 503–689. North-Holland, Amsterdam (1998)

    Google Scholar 

  20. 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. 89(3), 297–319 (2008). doi:10.1016/j.matpur.2007.12.001

    Article  MathSciNet  MATH  Google Scholar 

  21. Rousseau, A., Temam, R., Tribbia, J.: Boundary value problems for the inviscid primitive equations in limited domains. In: Temam, R.M., Tribbia, J.J., Ciarlet, P.G. (eds.) Computational Methods for the Atmosphere and the Oceans, Handbook of Numerical Analysis, vol. XIV. Elsevier, Amsterdam (2008)

    Google Scholar 

  22. Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations, 2nd edn. SIAM, Philadelphia (2004)

    Book  MATH  Google Scholar 

  23. Temam, R.: Inertial manifolds and multigrid methods. SIAM J. Math. Anal. 21, 154–178 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Yanenko, N.N.: The Method of Fractional Steps, The Solution of Problems of Mathematical Physics in Several Variables. Springer, New York (1971). Translated from the Russian by T. Cheron. English translation edited by M. Holt

    MATH  Google Scholar 

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Author information

Authors and Affiliations

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

    Arthur Bousquet, Martine Marion & Roger Temam

  2. Département Mathématique Informatique, Université de Lyon, Ecole Centrale de Lyon, CNRS UMR 5208, 36 avenue Guy de Collongue, 69134, Ecully Cedex, France

    Martine Marion

Authors
  1. Arthur Bousquet
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  2. Martine Marion
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  3. Roger Temam
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Corresponding author

Correspondence to Arthur Bousquet .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, City University of Hong Kong, Hong Kong, Hong Kong SAR

    Philippe G. Ciarlet

  2. School of Mathematical Sciences, Fudan University, Shanghai, People's Republic of China

    Tatsien Li

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie(Paris VI), Paris, France

    Yvon Maday

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Bousquet, A., Marion, M., Temam, R. (2014). Finite Volume Multilevel Approximation of the Shallow Water Equations. In: Ciarlet, P., Li, T., Maday, Y. (eds) Partial Differential Equations: Theory, Control and Approximation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41401-5_3

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