Summary
We present a systematic development of energy-stable approximations of the two-dimensional shallow water (SW) equations, which are based on the general framework of entropy conservative schemes introduced in [Tad03, TZ06]. No artificial numerical viscosity is involved: stability is dictated solely by eddy viscosity. In particular, in the absence of any dissipative mechanism, the resulting numerical schemes precisely preserve the total energy, which serves as an entropy function for the SW equations. We demonstrate the dispersive nature of such entropy conservative schemes with a series of scalar examples, interesting for their own sake. We then turn to the SW equations. Numerical experiments of the partial-dam-break problem with energy-preserving and energy stable schemes, successfully simulate the propagation of circular shock and the vortices formed on the both sides of the breach.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Akio Arakawa and Vivian R. Lamb. Computational design of the basic dynamical process of the ucla general circulation model. Meth. Comput. Phys., 17:173–265, 1977.
Akio Arakawa and Vivian R. Lamb. A potential enstrophy and energy conserving scheme for the shallow water equations. Mont. Weat. Rev., 109:18–36, 1981.
Akio Arakawa. Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. I [J. Comput. Phys. 1 (1966), no. 1, 119–143]. J. Comput. Phys., 135(2):101–114, 1997. With an introduction by Douglas K. Lilly, Commemoration of the 30th anniversary {of J. Comput. Phys.}.
Alina Chertock and Alexander Kurganov. On a hybrid finite-volume-particle method. M2AN Math. Model. Numer. Anal., 38(6):1071–1091, 2004.
Constantine M. Dafermos. Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2000.
Bjorn Engquist and Stanley Osher. Stable and entropy satisfying approximations for transonic flow calculations. Math. Comp., 34(149):45–75, 1980.
Robert J. Fennema and M. Hanif Chaudhry. Explicit methods for 2d transient free-surface flows. J. Hydraul. Eng. ASGE, 116(8):1013–1034, 1990.
Jonathan Goodman and Peter D. Lax. On dispersive difference schemes. I. Comm. Pure Appl. Math., 41(5):591–613, 1988.
Paul Glaister. Difference schemes for the shallow water equations. Numerical Analysis Report 9/87, University of Reading, Department of Mathematics, 1987.
Sergei K. Godunov. An interesting class of quasi-linear systems. Dokl. Akad. Nauk SSSR, 139:521–523, 1961.
Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor. Strong stabilitypreserving high-order time discretization methods. SIAM Rev., 43(1):89–112 (electronic), 2001.
Thomas Y. Hou and Peter D. Lax. Dispersive approximations in fluid dynamics. Comm. Pure Appl. Math., 44(1):1–40, 1991.
Stanislav N. Kružkov. First order quasilinear equations with several independent variables. Mat. Sb. (N.S.), 81 (123):228–255, 1970.
Alexander Kurganov and Eitan Tadmor. New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys., 160(1):241–282, 2000.
Peter D. Lax. Shock waves and entropy. Acaemic Press, New York, 1971. Contributions to Nonlinear Functional Analysis, (E.A.Zarantonello, ed.).
Peter D. Lax. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11.
C. David Levermore and Jian-Guo Liu. Large oscillations arising in a dispersive numerical scheme. Phys. D, 99(2–3):191–216, 1996. 94
Philippe G. Lefloch, J. M. Mercier, and C. Rohde. Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal., 40(5):1968–1992 (electronic), 2002.
Michael S. Mock. Systems of conservation laws of mixed type. J. Differential Equations, 37(1):70–88, 1980.
Phil L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Corn-put. Phys., 43(2):357–372, 1981.
Denis Serre. Systems of Conservation Laws, 1: Hyperbolicity, Entropies, Shock Waves. Cambridge University Press, 1999.
Eitan Tadmor. The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comp., 49(179):91–103, 1987.
Eitan Tadmor. Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer., 12:451–512, 2003.
Eitan Tadmor and Weigang Zhong. Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity. J. Hyperbolic Differ. Equ., 3(3):529–559, 2006.
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tadmor, E., Zhong, W. (2008). Energy-Preserving and Stable Approximations for the Two-Dimensional Shallow Water Equations. In: Munthe-Kaas, H., Owren, B. (eds) Mathematics and Computation, a Contemporary View. Abel Symposia, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68850-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-68850-1_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-68848-8
Online ISBN: 978-3-540-68850-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)