Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T23:02:07.888Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary Conditions for Limited Area Models Based on the Shallow Water Equations

Published online by Cambridge University Press:  03 June 2015

Arthur Bousquet ,
Madalina Petcu ,
Ming-Cheng Shiue ,
Roger Temam  and
Joseph Tribbia
Arthur Bousquet*
Affiliation:
The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, USA
Madalina Petcu*
Affiliation:
Laboratoire de Mathematiques et Applications, UMR 6086, Universite de Poitiers, France The Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Ming-Cheng Shiue*
Affiliation:
The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, USA Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan
Roger Temam*
Affiliation:
The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, USA
Joseph Tribbia*
Affiliation:
National Center for Atmospheric Research, Boulder, Colorado, USA
*
Email:madalina.petcu@math.univ-poitiers.fr
Email:mingcheng.shiue@gmail.com
Corresponding author.Email:temam@indiana.edu
Email:arthbous@indiana.edu
Email:tribbia@ucar.edu
Get access

Abstract

A new set of boundary conditions has been derived by rigorous methods for the shallow water equations in a limited domain. The aim of this article is to present these boundary conditions and to report on numerical simulations which have been performed using these boundary conditions. The new boundary conditions which are mildly dissipative let the waves move freely inside and outside the domain. The problems considered include a one-dimensional shallow water system with two layers of fluids and a two-dimensional inviscid shallow water system in a rectangle.

Keywords

35L0435Q3565M0865M55Boundary conditionsfinite volumesshallow water
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 3 , September 2013 , pp. 664 - 702
Copyright
Copyright © Global Science Press Limited 2013

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

[1]Adamy, K., Bousquet, A., Faure, S., Laminie, J. and Temam, R., A multilevel method for finite volume discretization of the two-dimensional nonlinear shallow-water equations, Ocean Modelling, vol. 33 (2010), Issues 3-4, 235–256.CrossRefGoogle Scholar
[2]Abgrall, R. and Karni, S., Two-layer shallow water system: a relaxation approach, SIAM J. Sci. Comput., vol. 31 (2009), no. 3, 1603–1627.Google Scholar
[3]Audusse, E., A multilayer Saint-Venant System: Derivation and Numerical Validation, Discrete Contin. Dyn. Syst. Ser. B, vol.5 (2005), 189–214.Google Scholar
[4]Baines, P.G., 1998, Topographic effectsin stratified flows, Cambridge Univ. Press, Cambridge, U. K., 1998.Google Scholar
[5]Bennett, A.F. and Chua, B. S., Open boundary conditions for Lagrangian geophysical fluid dynamics, J. Comput. Phys., vol. 153 (1999), no. 2, 418–436.Google Scholar
[6]Blayo, E. and ebreu, DL., Revisiting open boundary conditions from the point of view of characteristic variables, Ocean Modelling, vol. 9 (2005), 231–252.Google Scholar
[7]Bennett, A. F. and Kloeden, P.E., Boundary conditions for limited-area forecasts, J. Atmospheric Sci., vol. 35 (1978), no. 6, 990–996.2.0.CO;2>CrossRefGoogle Scholar
[8]Benzoni-Gavage, S. and Serre, D., Multidimensional hyperbolic partial differential equations,First-order systems and applications. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.Google Scholar
[9]Bouchut, F. and Zeitlin, V., A robust well-balanced scheme for multi-layer shallow water equations, DCDS-B, vol. 13 (2010), no. 4, 739–758.Google Scholar
[10]Cullen, M.J.P., Analysis of the semi-geostrophic shallow water equations, Phys. D, vol. 237 (2008), 1461–1465.Google Scholar
[11]Engquist, B. and Halpern, L., Far field boundary conditions for computation over long time, Appl. Numer. Math., vol. 4 (1988), no. 1, 21–45.Google Scholar
[12]Engquist, B. and Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977), no. 139, 629–651.Google Scholar
[13]Grabowski, W. W., Coupling cloud processes with large-scale dynamics using the cloud-resolving convection parameterization (CRCP), J. Atmospheric Sciences, Vol 58 (2001), 978V997Google Scholar
[14]Halpern, L. and Rauch, J., Absorbing boundary conditions for diffusion equations, Numer. Math., vol. 71 (1995), no. 2, 185–224.Google Scholar
[15]Huang, A., Petcu, M. and Temam, R., The nonlinear 2d supercritical inviscid shallow water equations in a rectangle, in preparation (under completion).Google Scholar
[16]Huang, A. and Temam, R., The linearized 2d inviscid shallow water equations in a rectangle: boundary conditions and well-posedness, submitted, arXiv:1209.3194.Google Scholar
[17]Huang, A. and Temam, R., The nonlinear 2d subcritical inviscid shallow water equations with periodicity in one direction, in preparation (under completion).Google Scholar
[18]Kurganov, A., Noelle, S. and Petrova, G., Semi-discrete central upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., vol. 23(2001), no. 3, 707–740.Google Scholar
[19]Kurganov, A. and Petrova, G., Central-upwind schemes for the Saint-Venant system, Math. Modell. Numer. Anal., vol. 34(2000), 1259–1275.Google Scholar
[20]Kreiss, H.-O., Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., vol. 23 (1970), 277–298.Google Scholar
[21]Lopatinsky, Ya. B., The mixed Cauchy-Dirichlet type problem for equations of hyperbolic type, Dopovfdf Akad. Nauk Ukrain. RSR Ser. A, vol. 668 (1970), 592–594.Google Scholar
[22]Lavelle, J.W. and Thacker, W.C., A pretty good sponge: Dealing with open boundaries in limited-area ocean models, Ocean Modelling, vol. 20 (2008), no. 3, 270–292.Google Scholar
[23]Li, T.-T. and Yu, W.-C., Boundary value problems for quasilinear hyperbolic systems,Duke University Mathematics Series, V. Duke University, Mathematics Department, Durham, NC, 1985Google Scholar
[24]McDonald, A., Transparent boundary conditions for the shallow water equations: testing in a nested environment, Mon. Wea. Rev., vol. 131 (2003), 698–705.Google Scholar
[25]Nycander, J., McC.Hogg, A. and Frankcombe, L. M., Open boundary conditions for nonlinear channel flow, Ocean Modelling, vol. 24 (2008), 108–121.Google Scholar
[26]Navon, I. M., Neta, B. and Hussaini, M. Y., A perfectly matched layer approach to the linearized shallow water equations models, Monthly Weather Review, vol. 132 (2004), 1369–1378.Google Scholar
[27]Oliger, J. and Sundstrom, A., Theoretical and practical aspects of some initial boundary value problems in fluid dynamics, SIAM J. Appl. Math., vol. 35 (1978), 419–446.CrossRefGoogle Scholar
[28]Palma, E. and Matano, R., On the implementation of passive open boundary conditions for a general circulation model: The barotropic mode, J. of Geophysical Research, vol 103 (1998), 1319–1341.Google Scholar
[29]Petcu, M. and Temam, R., The one-dimensional shallow-water equations with transparent boundary conditions, Mathematical Methods in the Applied Sciences, (MMAS)(2011); DOI: 10; 1002/mma, 1482.Google Scholar
[30]Petcu, M. and Temam, R., An interface problem: the two-layer shallow water equations, DCDS-A, 2011, to appear.Google Scholar
[31]Randall, D., Khairoutdinov, M., Arakawa, A. and Grabowski, W., Breaking the cloud parameterization deadlock, Bulletin of the American Meteor. Society (BAMS) (2003), 1547–1564, DOI: 10.1175/BAMS-84-11-1547Google Scholar
[32]Rousseau, A., Temam, R. and Tribbia, J., 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, Ciarlet, P. G., Ed, Temam, R. and Tribbia, J., Guest Eds, Elesevier, Amsterdam (2009), 377–385.Google Scholar
[33]Salmon, R., Numerical solution of the two-layer shallow water equation with bottom topography, Journal of Marine Research, vol. 60 (2002), 605–638.Google Scholar
[34]Shiue, M.-C., Laminie, J., Temam, R. and Tribbia, J., Boundary value problems for the shallow water equations with topography, J. Geophy. Res., vol. 116 (2011), C02015.Google Scholar
[35]Temam, R. and Tribbia, J., Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci., vol. 60 (2003), 2647–2660.Google Scholar
[36]Treguier, A.M., Barnier, B., de Miranda, A.P., Molines, J.M., Grima, N., Imbard, M., Madec, G., Messager, C., Reynald, T. and Michel, S., An eddy-permitting model of the Atlantic circulation: evaluating open boundary conditions. J. Geophys, Res. 106 (2001), 22115V22129.Google Scholar
[37]Whitham, B., Linear and nonlinear waves, Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999.Google Scholar
[38]Warner, T., Peterson, R. and Treadon, R., A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction, Bull. Amer. Meteor. Soc., vol. 78(1997), 2599–2617.Google Scholar