Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-25T06:21:02.883Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical approximation of the inviscid 3D primitive equations in a limited domain

Published online by Cambridge University Press:  11 January 2012

Qingshan Chen ,
Ming-Cheng Shiue ,
Roger Temam  and
Joseph Tribbia
Qingshan Chen
Affiliation:
Institute for Scientific Computing and Applied Mathematics, Indiana University, Rawles Hall, Bloomington, IN 47405, USA. mingcheng.shiue@gmail.com Department of Scientific Computing, Florida State University, 400 Dirac Science Library, Tallahassee, FL 32306, USA
Ming-Cheng Shiue
Affiliation:
Institute for Scientific Computing and Applied Mathematics, Indiana University, Rawles Hall, Bloomington, IN 47405, USA. mingcheng.shiue@gmail.com
Roger Temam
Affiliation:
Institute for Scientific Computing and Applied Mathematics, Indiana University, Rawles Hall, Bloomington, IN 47405, USA. mingcheng.shiue@gmail.com
Joseph Tribbia
Affiliation:
National Center for Atmospheric Research, CGD-NCAR, 1850 Table Mesa Drive, Boulder, CO 80305, USA
Get access

Abstract

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.

Keywords

Nonviscous primitive equationslimited domainsboundary conditionstransparent boundary conditionsfinite difference methods
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 3: Special volume in honor of Professor David Gottlieb , May 2012 , pp. 619 - 646
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Cao, C. and Titi, E.S., Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann. Math. 166 (2007) 245–267. Google Scholar
Chen, Q., Laminie, J., Rousseau, A., Temam, R. and Tribbia, J., A 2.5D model for the equations of the ocean and the atmosphere. Anal. Appl. (Singap.) 5 (2007) 199–229. Google Scholar
Chen, Q., Temam, R. and Tribbia, J.J., Simulations of the 2.5D inviscid primitive equations in a limited domain. J. Comput. Phys. 227 (2008) 9865–9884. Google Scholar
Chen, Q., Shiue, M.-C. and Temam, R., The barotropic mode for the primitive equations. J. Sci. Comput. 45 (2010) 167-199. Google Scholar
Chorin, A.J., Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745–762. Google Scholar
Engquist, B. and Majda, A., Absorbing boundary conditions for the numerical simulation of waves. Math. Comp. 31 (1977) 629–651. Google Scholar
Givoli, D. and Neta, B., 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). Google Scholar
Guermond, J.L., Minev, P. and Shen, J., An overview of projection methods for incompressible flow. Comput. Methods Appl. Mech. Engrg. 195 (2006) 6011–6045. Google Scholar
Higdon, R.L., Absorbing boundary conditions for difference approximations to the multidimensional wave equation. Math. Comput. 47 (1986) 437–459. Google Scholar
Kobelkov, G., 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. Google Scholar
Kobelkov, G.M., Existence of a solution “in the large” for ocean dynamics equations. J. Math. Fluid Mech. 9 (2007) 588–610. Google Scholar
Lions, J.L., Temam, R. and Wang, S.H., New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5 (1992) 237–288. Google Scholar
Lions, J.L., Temam, R. and Wang, S.H., On the equations of the large-scale ocean. Nonlinearity 5 (1992) 1007–1053. Google Scholar
Marchuk, G., Methods and problems of computational mathematics, Actes du Congres International des Mathematiciens (Nice, 1970) 1 (1971) 151–161. Google Scholar
M. Marion and R. Temam, Navier-Stokes equations : theory and approximation, Handb. Numer. Anal. VI. North-Holland, Amsterdam (1998) 503–688.
McDonald, A., Transparent boundary conditions for the shallow water equations : testing in a nested environment. Mon. Wea. Rev. 131 (2003) 698–705. Google Scholar
Navon, I.M., Neta, B. and Hussaini, M.Y., A perfectly matched layer approach to the linearized shallow water equations models. Mon. Wea. Rev. 132 (2004) 1369–1378. Google Scholar
Oliger, J. and Sundström, A., Theoretical and practical aspects of some initial boundary value problems in fluid dynamics. SIAM J. Appl. Math. 35 (1978) 419–446. Google Scholar
J. Pedlosky, Geophysical fluid dynamics, 2nd edition. Springer (1987).
M. Petcu, R. Temam and M. Ziane, 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).
Rousseau, A., Temam, R. and Tribbia, J., Boundary conditions for the 2D linearized PEs of the ocean in the absence of viscosity. Discrete Contin. Dyn. Syst. 13 (2005) 1257–1276. Google Scholar
A. Rousseau, R. Temam and J. Tribbia, 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).
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. Pure Appl. 89 (2008) 297–319. Google Scholar
A. Rousseau, R. Temam and J. Tribbia, 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).
Temam, R., 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. Google Scholar
Temam, R. and Tribbia, J., Open boundary conditions for the primitive and Boussinesq equations. J. Atmos. Sci. 60 (2003) 2647–2660. Google Scholar
R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, in Handbook of mathematical fluid dynamics, edited by S. Friedlander and D. Serre. North-Holland (2004).
van Kan, J., A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Stat. Comput. 7 (1986) 870–891. Google Scholar
Warner, T.T., Peterson, R.A. and Treadon, R.E., 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. Google Scholar
W. Washington and C. Parkinson, An introduction to three-dimensional climate modelling, 2nd edition. Univ. Sci. Books, Sausalito, CA (2005).
N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables. Springer-Verlag (1971) English translation.