Abstract
We are interested in the numerical simulation of large scale phenomena in geophysical flows. In these cases, Coriolis forces play an important role and the circulations are often perturbations of the so-called geostrophic equilibrium. Hence, it is essential to design a numerical strategy that preserves a discrete version of this equilibrium. In this article we work on the shallow water equations in a finite volume framework and we propose a first step in this direction by introducing an auxiliary pressure that is in geostrophic equilibrium with the velocity field and that is computed thanks to the solution of an elliptic problem. Then the complete solution is obtained by working on the deviating part of the pressure. Some numerical examples illustrate the improvement through comparisons with classical discretizations.
MSC2010: 65M08, 76U05, 86A05
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. Audusse, R.Klein, A. Owinoh, Conservative discretization of Coriolis force in a finite volume framework, Journal of Computational Physics, 228, 2934-2950 (2009).
E. Audusse, F. Bouchut, M.O. Bristeau, R. Klein, B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25, 2050–2065 (2004).
N. Botta, R. Klein, S. Langenberg, S. Lützenkirchen, Well-balanced finite volume methods for nearly hydrostatic flows, JCP, 196, 539–565 (2004).
F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Birkaüser (2004).
F. Bouchut, J. Le Sommer, V. Zeitlin, Frontal geostrophic adjustment and nonlinear wave phenomena in one dimensional rotating shallow water. Part 2: high-resolution numerical simulations, J. Fluid Mech., 513, 35–63 (2004).
M.J. Castro, J.A. Lopez, C. Pares, Finite Volume Simulation of the Geostrophic Adjustment in a Rotating Shallow-Water System SIAM J. on Scientific Computing, 31, 444–477 (2008).
A. Harten, P. Lax, B. Van Leer, On upstream differencing and Godunov type schemes for hyperbolic conservation laws, SIAM Review, 25, 235–261 (1983).
A. Kuo, L. Polvani, Time-dependent fully nonlinear geostrophic adjustment, J. Phys. Oceanogr. 27, 1614–1634 (1997).
R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Camb. Univ. Press (2002).
N. Panktratz, J.R. Natvig, B. Gjevik, S. Noelle, High-order well-balanced finite-volume schemes for barotropic flows. Development and numerical comparisons, Ocean Modelling, 18, 53–79 (2007).
J. Pedlosky, Geophysical Fluid Dynamics, Springer, 2nd edition (1990).
G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, Cambridge University Press (2006).
S. Vater, R. Klein, Stability of a Cartesian Grid Projection Method for Zero Froude Number Shallow Water Flows, Numerische Mathematik, 113, 123–161 (2009).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Audusse, E., Klein, R., Nguyen, D.D., Vater, S. (2011). Preservation of the Discrete Geostrophic Equilibrium in Shallow Water Flows. In: Fořt, J., Fürst, J., Halama, J., Herbin, R., Hubert, F. (eds) Finite Volumes for Complex Applications VI Problems & Perspectives. Springer Proceedings in Mathematics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20671-9_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-20671-9_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20670-2
Online ISBN: 978-3-642-20671-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)