A new set of nonlocal boundary conditions is proposed for the higher modes of the 3D inviscid primitive equations. Numerical schemes using the splitting-up method are proposed for these modes. Numerical simulations of the full nonlinear primitive equations are performed on a nested set of domains, and the results are discussed.
Mots-clés : nonviscous primitive equations, limited domains, boundary conditions, transparent boundary conditions, finite difference methods
@article{M2AN_2012__46_3_619_0,
author = {Chen, Qingshan and Shiue, Ming-Cheng and Temam, Roger and Tribbia, Joseph},
title = {Numerical approximation of the inviscid {3D} primitive equations in a limited domain},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {619--646},
publisher = {EDP-Sciences},
volume = {46},
number = {3},
year = {2012},
doi = {10.1051/m2an/2011058},
mrnumber = {2877368},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2011058/}
}
TY - JOUR AU - Chen, Qingshan AU - Shiue, Ming-Cheng AU - Temam, Roger AU - Tribbia, Joseph TI - Numerical approximation of the inviscid 3D primitive equations in a limited domain JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 619 EP - 646 VL - 46 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2011058/ DO - 10.1051/m2an/2011058 LA - en ID - M2AN_2012__46_3_619_0 ER -
%0 Journal Article %A Chen, Qingshan %A Shiue, Ming-Cheng %A Temam, Roger %A Tribbia, Joseph %T Numerical approximation of the inviscid 3D primitive equations in a limited domain %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 619-646 %V 46 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2011058/ %R 10.1051/m2an/2011058 %G en %F M2AN_2012__46_3_619_0
Chen, Qingshan; Shiue, Ming-Cheng; Temam, Roger; Tribbia, Joseph. Numerical approximation of the inviscid 3D primitive equations in a limited domain. ESAIM: Mathematical Modelling and Numerical Analysis , Special volume in honor of Professor David Gottlieb. Numéro spécial, Tome 46 (2012) no. 3, pp. 619-646. doi : 10.1051/m2an/2011058. http://archive.numdam.org/articles/10.1051/m2an/2011058/
[1] and , Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann. Math. 166 (2007) 245-267. | MR | Zbl
[2] , , , and , A 2.5D model for the equations of the ocean and the atmosphere. Anal. Appl. (Singap.) 5 (2007) 199-229. | MR | Zbl
[3] , and , Simulations of the 2.5D inviscid primitive equations in a limited domain. J. Comput. Phys. 227 (2008) 9865-9884. | MR
[4] , and , The barotropic mode for the primitive equations. J. Sci. Comput. 45 (2010) 167-199. | MR | Zbl
[5] , Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745-762. | MR | Zbl
[6] and , Absorbing boundary conditions for the numerical simulation of waves. Math. Comp. 31 (1977) 629-651. | MR | Zbl
[7] and , High-order nonreflecting boundary conditions for the dispersive shallow water equations. J. Comput. Appl. Math. 158 (2003) 49-60; Selected papers from the Conference on Computational and Mathematical Methods for Science and Engineering, Alicante (2002). | MR | Zbl
[8] , and , An overview of projection methods for incompressible flow. Comput. Methods Appl. Mech. Engrg. 195 (2006) 6011-6045. | MR | Zbl
[9] , Absorbing boundary conditions for difference approximations to the multidimensional wave equation. Math. Comput. 47 (1986) 437-459. | MR | Zbl
[10] , Existence of a solution ‘in the large' for the 3D large-scale ocean dynamics equations. C. R. Math. Acad. Sci. Paris 343 (2006) 283-286. | MR | Zbl
[11] , Existence of a solution “in the large” for ocean dynamics equations. J. Math. Fluid Mech. 9 (2007) 588-610. | MR | Zbl
[12] , and , New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5 (1992) 237-288. | MR | Zbl
[13] , and , On the equations of the large-scale ocean. Nonlinearity 5 (1992) 1007-1053. | MR | Zbl
[14] , Methods and problems of computational mathematics, Actes du Congres International des Mathematiciens (Nice, 1970) 1 (1971) 151-161. | MR | Zbl
[15] and , Navier-Stokes equations : theory and approximation, Handb. Numer. Anal. VI. North-Holland, Amsterdam (1998) 503-688. | MR | Zbl
[16] , Transparent boundary conditions for the shallow water equations : testing in a nested environment. Mon. Wea. Rev. 131 (2003) 698-705.
[17] , and , A perfectly matched layer approach to the linearized shallow water equations models. Mon. Wea. Rev. 132 (2004) 1369-1378.
[18] and , Theoretical and practical aspects of some initial boundary value problems in fluid dynamics. SIAM J. Appl. Math. 35 (1978) 419-446. | MR | Zbl
[19] , Geophysical fluid dynamics, 2nd edition. Springer (1987). | Zbl
[20] , and , Mathematical problems for the primitive equations with viscosity. in Handbook of Numerical Analysis. Special Issue on Some Mathematical Problems in Geophysical Fluid Dynamics, Handb. Numer. Anal., edited by R. Temam, P.G. Ciarlet EDs and J.G. Tribbia. Elsevier, New York (2008). | MR
[21] , and , Boundary conditions for the 2D linearized PEs of the ocean in the absence of viscosity. Discrete Contin. Dyn. Syst. 13 (2005) 1257-1276. | MR | Zbl
[22] , and , Numerical simulations of the inviscid primitive equations in a limited domain, in Analysis and Simulation of Fluid Dynamics, Advances in Mathematical Fluid Mechanics. Caterina Calgaro and Jean-François Coulombel and Thierry Goudon (2007). | MR | Zbl
[23] , and , The 3D primitive equations in the absence of viscosity : boundary conditions and well-posedness in the linearized case. J. Math. Pure Appl. 89 (2008) 297-319. | MR | Zbl
[24] , and , Boundary value problems for the inviscid primitive equations in limited domains, in Computational Methods for the Oceans and the Atmosphere, Special Volume of the Handbook of Numerical Analysis, edited by P.G. Ciarlet, R. Temam and J. Tribbia, Guest. Elsevier, Amsterdam (2009). | MR | Zbl
[25] , Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas frationnaires (ii). Arch. Rational Mech. Anal. 33 (1969) 377-385. | MR | Zbl
[26] and , Open boundary conditions for the primitive and Boussinesq equations. J. Atmos. Sci. 60 (2003) 2647-2660. | MR
[27] and , Some mathematical problems in geophysical fluid dynamics, in Handbook of mathematical fluid dynamics, edited by S. Friedlander and D. Serre. North-Holland (2004). | MR | Zbl
[28] , A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Stat. Comput. 7 (1986) 870-891. | MR | Zbl
[29] , and , A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction. Bull. Amer. Meteor. Soc. 78 (1997) 2599-2617.
[30] and , An introduction to three-dimensional climate modelling, 2nd edition. Univ. Sci. Books, Sausalito, CA (2005). | Zbl
[31] , The method of fractional steps. The solution of problems of mathematical physics in several variables. Springer-Verlag (1971) English translation. | MR | Zbl
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