Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T04:29:34.818Z Has data issue: false hasContentIssue false

Solving Two-Mode Shallow Water Equations Using Finite Volume Methods

Published online by Cambridge University Press:  03 June 2015

Manuel Jesús Castro Diaz*
Affiliation:
Dpto. de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Malaga, Spain
Yuanzhen Cheng*
Affiliation:
Mathematics Department, Tulane University, New Orleans, LA 70118, USA
Alina Chertock*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
Alexander Kurganov*
Affiliation:
Mathematics Department, Tulane University, New Orleans, LA 70118, USA
*
Get access

Abstract

In this paper, we develop and study numerical methods for the two-mode shallow water equations recently proposed in [S. STECHMANN, A. MAJDA, and B. KHOUIDER, Theor. Comput. Fluid Dynamics, 22 (2008), pp. 407-432]. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches—two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme—and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2014

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]Abgrall, R. and Kami, S., Two-layer shallow water system: a relaxation approach, SIAM J. Sci. Comput. 31 (2009), no. 3,16031627.Google Scholar
[2]Abgrall, R. and Karni, S., A comment on the computation of non-conservative products, J. Comput. Phys. 229 (2010), no. 8, 27592763.CrossRefGoogle Scholar
[3]Alouges, F. and Merlet, B., Approximate shock curves for non-conservative hyperbolic systems in one space dimension, J. Hyperbolic Differ. Equ. 1 (2004), no. 4, 769788.CrossRefGoogle Scholar
[4]Bouchut, F. and Luna, T. Morales de, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment, M2AN Math. Model. Numer. Anal. 42 (2008), 683698.Google Scholar
[5]Bouchut, F. and Zeitlin, V., A robust well-balanced scheme for multi-layer shallow water equations, Discrete Contin. Dyn. Syst. Ser. B 13 (2010), no. 4, 739758.Google Scholar
[6]Castro, M., Macias, J., and Pares, C., A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system, M2AN Math. Model. Numer. Anal. 35 (2001), no. 1,107127.Google Scholar
[7]Castro, M.J., LeFloch, P.G., Munoz-Ruiz, M.L., and Pares, C., Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, J. Comput. Phys. 227 (2008), no. 17, 81078129.Google Scholar
[8]Castro, M.J., Pares, C., Puppo, G., and Russo, G., Central schemes for nonconservative hyperbolic systems, SIAM J. Sci. Comput. 34 (2012), no. 5, B523B558.Google Scholar
[9]Castro Diaz, M. J., Rebollo, T. Chacon, Fernandez-Nieto, E. D., and Pares, C., On well-balanced finite volume methods for nonconservative nonhomogeneous hyperbolic systems, SIAM J. Sci. Comput. 29 (2007), no. 3,10931126.Google Scholar
[10]Castro Diaz, M.J., Fernandez-Nieto, E.D., de Luna, T. Morales, Narbona-Reina, G., and Pares, C., A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport, M2AN Math. Model. Numer. Anal. 47 (2013), no. 01, 132.Google Scholar
[11]Castro Diaz, M.J., Kurganov, A., and de Luna, T. Morales, Path-conservative central-upwind schemes for nonconservative hyperbolic systems, In preparation.Google Scholar
[12]Chalmers, N. and Lorin, E., On the numerical approximation of one-dimensional nonconser-vative hyperbolic systems, J. Comput. Science 4 (2013), 111124.Google Scholar
[13]Maso, G. Dal, Lefloch, P.G., and Murat, F., Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483548.Google Scholar
[14]Einfeld, B., On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal. 25 (1988), 294318.Google Scholar
[15]Gottlieb, S., Shu, C.-W., and Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001), 89112.Google Scholar
[16]Harten, A., Lax, P., and Leer, B. van, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983), 3561.Google Scholar
[17]Harten, A. and Lax, P.D., A random choice finite difference scheme for hyperbolic conservation laws, SIAM J. Numer. Anal. 18 (1981), no. 2, 289315.Google Scholar
[18]Kurganov, A. and Levy, D., Central-upwind schemes for the saint-venant system, M2AN Math. Model. Numer. Anal. 36 (2002), 397425.Google Scholar
[19]Kurganov, A. and Lin, C.-T., On the reduction of numerical dissipation in central-upwind schemes, Commun. Comput. Phys. 2 (2007), 141163.Google Scholar
[20]Kurganov, A., Noelle, S., and Petrova, G., Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput. 23 (2001), 707740.Google Scholar
[21]Kurganov, A. and Petrova, G., A second-order well-balanced positivity preserving central-upwind scheme for the saint-venant system, Commun. Math. Sci. 5 (2007), 133160.Google Scholar
[22]Kurganov, A. and Petrova, G., A central-upwind scheme for nonlinear water waves generated by submarine landslides, Hyperbolic problems: theory, numerics, applications (Lyon 2006) (Benzoni-Gavage, S. and Serre, D., eds.), Springer, 2008, pp. 635642.CrossRefGoogle Scholar
[23]Kurganov, A. and Petrova, G., Central-upwind schemes for two-layer shallow equations, SIAM J. Sci. Comput. 31 (2009), 17421773.Google Scholar
[24]Kurganov, A. and Tadmor, E., New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys. 160 (2000), 241282.Google Scholar
[25]Lie, K.-A. and Noelle, S., On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws, SIAM J. Sci. Comput. 24 (2003), no. 4,11571174.Google Scholar
[26]Macias, J., Pares, C., and Castro, M.J., Improvement and generalization of a finite element shallow-water solver to multi-layer systems, Internat. J. Numer. Methods Fluids 31 (1999), no. 7,10371059.Google Scholar
[27]Mignotte, M. and Stefanescu, D., On an estimation of polynomial roots by lagrange, Tech. Report 025/2002, pp. 117, IRMA Strasbourg, http://hal.archives-ouvertes.fr/hal-00129675/en/, 2002.Google Scholar
[28]Munoz-Ruiz, M.L. and Pares, C., Godunov method for nonconservative hyperbolic systems, M2AN Math. Model. Numer. Anal. 41 (2007), no. 1,169185.Google Scholar
[29]Munoz-Ruiz, M.L. and Pares, C., On the convergence and well-balanced property of path-conservative numerical schemes for systems of balance laws, J. Sci. Comput. 48 (2011), no. 13, 274295.Google Scholar
[30]Nessyahu, H. and Tadmor, E., Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys. 87 (1990), no. 2,408463.Google Scholar
[31]Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. Numer. Anal. 44 (2006), no. 1,300321.Google Scholar
[32]Parés, C., Path-conservative numerical methods for nonconservative hyperbolic systems, Numerical methods for balance laws, Quad. Mat., vol. 24, Dept. Math., Seconda Univ. Napoli, Caserta, 2009, pp. 67121.Google Scholar
[33]Parés, C. and Muñoz-Ruiz, M.L., On some difficulties of the numerical approximation of nonconservative hyperbolic systems, Bol. Soc. Esp. Mat. Apl. SMA (2009), no. 47, 2352.Google Scholar
[34]Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. Sci. Comput. 6 (1988), 10731084.Google Scholar
[35]Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), 439471.Google Scholar
[36]Stechmann, S., Majda, A., and Khouider, B., Nonlinear dynamics of hydrostatic internal gravity waves, Theor. Comput. Fluid Dyn. 22 (2008), 407432.Google Scholar
[37]Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), 506517.CrossRefGoogle Scholar
[38]Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21 (1984), no. 5,9951011.Google Scholar
[39]Vallis, G.K., Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation, Cambridge University Press, 2006.Google Scholar
[40]Leer, B. van, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys. 32 (1979), no. 1,101136.Google Scholar