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.
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Adamy, K., Bousquet, A., Faure, S., et al., A multilevel method for finite-volume discretization of the two-dimensional nonlinear shallow-water equations, Ocean Modelling, 33, 2010, 235–256. DOI: 10.1016/j.ocemod.2010.02.006
Adamy, K. and Pham, D., A finite-volume implicit Euler scheme for the linearized shallow water equations: stability and convergence, Numerical Functional Analysis and Optimization, 27(7–8), 2006, 757–783.
Bellanger, M., Traitement du Signal, Dunod, Paris, 2006.
Bousquet, A., Marion, M. and Temam, R., Finite volume multilevel approximation of the shallow water equations II, in preparation.
Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zhang, T. A., Spectral Methods, Evolution to Complex Geometries and Applications to Fluid Dynamics, Scientific Computation, Springer-Verlag, Berlin, 2007.
Chen, Q., Shiue, M. C. and Temam, R., The barotropic mode for the primitive equations, Special issue in memory of David Gottlieb, Journal of Scientific Computing, 45, 2010, 167–199. DOI: 10.1007/s10915-009-9343-8
Chen, Q., Shiue, M. C., Temam, R. and Tribbia, J., Numerical approximation of the inviscid 3D Primitive equations in a limited domain, Math. Mod. and Num. Anal. (M2AN), 45, 2012, 619–646. DOI: 10.105/m2an/2011058
Dubois, T., Jauberteau, F. and Temam, R., Dynamic, Multilevel Methods and the Numerical Simulation of Turbulence, Cambridge University Press, Cambridge, 1999.
Dautray, R. and Lions, J. L., Mathematical analysis and numerical methods for science and technology, Springer-Verlag, Berlin, 1990–1992.
Eymard, R., Gallouet, T. and Herbin, R., Finite volume methods, Handbook of Numerical Analysis, P. G. Ciarlet, J. L. Lions (eds.), Vol. VII, North-Holland, Amsterdam, 2002, 713–1020.
Gie, G. M. and Temam, R., Cell centered finite-volume methods using Taylor series expansion scheme without fictitious domains, International Journal of Numerical Analysis and Modeling, 7(1), 2010, 1–29.
Huang, A. and Temam, R., The linearized 2D inviscid shallow water equations in a rectangle: boundary conditions and well-posedness, to appear.
Leveque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
Lions, J. L., Temam, R. and Wang, S., Models of the coupled atmosphere and ocean (CAO I), Computational Mechanics Advances, 1, 1993, 5–54.
Lions, J. L., Temam, R. and Wang, S., Numerical analysis of the coupled models of atmosphere and ocean (CAO II), Computational Mechanics Advances, 1, 1993, 55–119.
Lions, J. L., Temam, R. and Wang, S., Splitting up methods and numerical analysis of some multiscale problems, Computational Fluid Dynamics Journal, special issue dedicated to A. Jameson, 5(2), 1996, 157–202.
Marchuk, G. I., Methods of numerical mathematics, 2nd edition, Translated from the Russian by Arthur A. Brown, Applications of Mathematics, 2, Springer-Verlag, New York, Berlin, 1982.
Marion, M. and Temam, R., Nonlinear Galerkin Methods, SIAM J. Num. Anal., 26, 1989, 1139–1157.
Marion, M. and Temam, R., Navier-Stokes equations, Theory and Approximation, Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions (eds.), North-Holland, Amsterdam, VI, 1998, 503–689.
Rousseau, A., Temam, R. and 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), 2008, 297–319. DOI: 10.1016/j.matpur.2007.12.001
Rousseau, A., Temam, R. and Tribbia, J., Boundary value problems for the inviscid primitive equations in limited domains, Computational Methods for the Atmosphere and the Oceans, Handbook of Numerical Analysis, Special Volume, Vol. XIV, R. M. Temam, J. J. Tribbia (Guest Editors), P. G. Ciarlet (Editor), Elsevier, Amsterdam, 2008.
Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations, 2nd edition, SIAM, philadelphia, PA, 2004.
Temam, R., Inertial manifolds and multigrid methods, SIAM J. Math. Anal., 21, 1990, 154–178.
Temam, R. and Tribbia, J., Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sciences, 60, 2003, 2647–2660.
Yanenko, N. N., The method of fractional steps, The solution of problems of mathematical physics in several variables, Translated from the Russian by T. Cheron, English translation edited by M. Holt, Springer-Verlag, New York, Heidelberg, 1971.
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In honor of the scientific heritage of Jacques-Louis Lions
Project supported by the National Science Foundation (Nos. DMS 0906440, DMS 1206438) and the Research Fund of Indiana University.
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Bousquet, A., Marion, M. & Temam, R. Finite Volume Multilevel Approximation of the Shallow Water Equations. Chin. Ann. Math. Ser. B 34, 1–28 (2013). https://doi.org/10.1007/s11401-012-0760-x
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DOI: https://doi.org/10.1007/s11401-012-0760-x