Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T08:24:58.769Z Has data issue: false hasContentIssue false
  • English
  • Français

IMEX Large Time Step Finite Volume Methods for Low Froude Number Shallow Water Flows

Published online by Cambridge University Press:  03 June 2015

Georgij Bispen ,
K. R. Arun ,
Mária Lukáčová-Medvid’ová  and
Sebastian Noelle
Georgij Bispen*
Affiliation:
Institute of Mathematics, University of Mainz, Germany
K. R. Arun*
Affiliation:
School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, India
Mária Lukáčová-Medvid’ová*
Affiliation:
Institute of Mathematics, University of Mainz, Germany
Sebastian Noelle*
Affiliation:
IGPM, RWTH Aachen, Germany
*
Email:bispeng@uni-mainz.de
Email:arun@iisertvm.ac.in
Corresponding author.Email:lukacova@uni-mainz.de
Email:noelle@igpm.rwth-aachen.de
Get access

Abstract

We present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate fast linear waves implicitly in time and in space by means of a genuinely multidimensional evolution operator. On the other hand, we approximate nonlinear advection part explicitly in time and in space by means of the method of characteristics or some standard numerical flux function. Time integration is realized by the implicit-explicit (IMEX) method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit. Numerical experiments demonstrate stability, accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.

Keywords

35L6576B1565M0865M0635L4535L65Low Froude number flowsasymptotic preserving schemesshallow water equationslarge time stepsemi-implicit approximationevolution Galerkin schemes.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 2 , August 2014 , pp. 307 - 347
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]Arun, K.R., Kraft, M., Lukáčová-Medvid’ová, M., and Prasad, Ph.Finite volume evolution Galerkin method for hyperbolic conservation laws with spatially varying flux functions. J. Comput. Phys., 228: 565–590, 2009.Google Scholar
[2]Arun, K.R. and Noelle, S.An asymptotic preserving scheme for low Froude number shallow flows. Accepted to the Proceedings of the 14th International Conference on Hyperbolic Problems. Theory, Numerics and Applications, American Institute of Mathematical Sciences, 2013.Google Scholar
[3]Arun, K.R., Noelle, S., Lukáčová-Medvid’ová, M., and Munz, C.-D.Asymptotic preserving all Mach number scheme for the Euler equations of gas dynamics. submitted to SIAM J. Sci. Comput. 2012.Google Scholar
[4]Audusse, E., Bouchut, F., Bristeau, M., Klein, R., and Perthame, B.A fast and stable well- balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Com- put., 25(6): 2050–2065, 2004.Google Scholar
[5]Bermudez, A. and Vazquez, M.E.Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids, 23(8): 1049–1071, 1994.Google Scholar
[6]Bijl, H. and Wesseling, P.A unified method for computing incompressible and compressible flows in boundary-fitted coordinates. J. Comput. Phys., 141(2): 153–173, 1998.Google Scholar
[7]Botta, N., Klein, R., Langenberg, S., and Lutzenkirchen, S.Well balanced finite volume methods for nearly hydrostatic flows. J. Comput. Phys., 196(2): 539–565, 2004.Google Scholar
[8]Bouchut, F., Le Sommer, J., and Zeitlin, V.Frontal geostrophic adjustment and nonlinear wave phenomena in one-dimensional rotating shallow water. II: High-resolution numerical simulations. J. Fluid Mech., 514: 35–63, 2004.Google Scholar
[9]Bresch, D., Klein, R., and Lucas, C.Multiscale analyses for the shallow water equations. Computational Science and High Performance Computing IV, Notes on Numerical Fluid Mechanics and Multidisciplinary Design 115, 149–164, 2011.Google Scholar
[10]Canestrelli, A., Dumbser, M., Siviglia, A., and Toro, E.F.Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed. Advanced in Water Resources, 33: 291–303, 2010.Google Scholar
[11]Castro Diaz, Manuel J., Gonzalez-Vida, J. M., Macias, Jorge, and Pares, Carlos. Realistic application of a tidal 2D two-layer shallow water model to the Strait of Gibraltar. AIP Conf. Proceedings, 168, 1429–1432, 2009.Google Scholar
[12]Degond, P. and Tang, M.All speed scheme for the low Mach number limit of the isentropic Euler equations. Commun. Comput. Phys., 10: 1–31, 2011.Google Scholar
[13]Feistauer, M., Felcman, J., and Straskraba, I.Mathematical and Computational Methods for Compressible Flow. Oxford University Press, 2003.Google Scholar
[14]Giraldo, F.X. and Restelli, M.High-order semi-implicit time-integrators for a triangular discontinuous Galerkin oceanic shallow water model. Int. J. Numer. Methods Fluids, 63(9): 1077–1102, 2010.Google Scholar
[15]Giraldo, F.X., Restelli, M., and Lauter, M.Semi-implicit formulations of the Navier-Stokes equations: application to nonhydrostatic atmospheric modeling. J. Sci. Comput., 32(6): 3394–3425, 2010.Google Scholar
[16]Greenberg, J.M. and Le Roux, A.-Y.A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal., 33(1): 1–16, 1996.Google Scholar
[17]Haack, J., Jin, S., and Liu, J.An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equations. Commun. Comput. Phys., 12: 955–980, 2012.Google Scholar
[18]Hoffmann, L. B.Ein zeitlich selbstadaptives numerisches Verfahren zur berechnung von Strömungen aller Mach-Zahlen basierend auf Mehrskalenasymptotik und diskreter Datenanalys. PhD thesis, Universitat Hamburg, 2000.Google Scholar
[19]Jin, S.Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput., 21(2): 441–454, 1999.CrossRefGoogle Scholar
[20]Klainerman, S. and Majda, A.Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math., 34: 481–524, 1981.Google Scholar
[21]Klein, R., Botta, N., Schneider, T., Munz, C.-D., Roller, S., Meister, A., Hoffmann, L., and Sonar, T.Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J. Eng. Math., 39(1-4): 261–343, 2001.Google Scholar
[22]Klein, R.An applied mathematical view of meteorological modelling. James M., Hill et al. (eds.), Applied mathematics entering the 21st century, Proceedings in Applied Mathematics, 116: 227–269, 2004.Google Scholar
[23]Klemp, J.B., Skamarock, W.C., and Dudhia, J.Conservative split-explicit time integration methods for the compressible nonhydrostatic equations. Monthly Weather Rev., 135:2897–2913,2007.Google Scholar
[24]Kurganov, A. and Levy, D.Central-upwind schemes for the Saint-Venant system. Math. Model. Numer. Anal., 36(3): 397–425, 2002.Google Scholar
[25]LeVeque, R.J.Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm. J. Comput. Phys., 146(1): 346–365, 1998.Google Scholar
[26]Liang, Q., Borthwick, A.G.L.Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography. Comput. Fluids, 38(2): 221–234, 2009.Google Scholar
[27]Liang, Q., Marche, F.Numerical resolution of well-balanced shallow water equations with complex source terms. Adv. Water Resour., 32(6): 873–884, 2009.CrossRefGoogle Scholar
[28]Lukáčová-Medvid’ová, M., Morton, K.W., and Warnecke, G.Finite volume evolution Galerkin methods for hyperbolic systems. J. Sci. Comput., 26(1): 1–30, 2004.Google Scholar
[29]Lukáčová-Medvid’ová, M., Morton, K.W., and Warnecke, Gerald. Evolution Galerkin meth-ods for hyperbolic systems in two space dimensions. Math. Comput., 69(232):1355–1384, 2000.Google Scholar
[30]Lukáčová Medvid’ová, M., Müller, A., Wirth, V., and Yelash, L.Large time step discontinuous evolution Galerkin methods for multiscale atmospheric flow. submitted to J. Comput. Phys., 2013.Google Scholar
[31]Lukáčová-Medvid’ová, M. and Morton, K.W.Finite volume evolution Galerkin methods - A survey. Indian J. Pure Appl. Math., 41(2): 329–361, 2010.Google Scholar
[32]Lukáčová-Medvid’ová, M., Noelle, S., and Kraft, M.Well-balanced finite volume evolution Galerkin methods for the shallow water equations. J. Comput. Phys., 221(1): 122–147, 2007.Google Scholar
[33]Majda, A.Introduction to PDEs and Waves for the Atmosphere and Ocean, Lecture Notes, Courant Institute, 2003.Google Scholar
[34]Müller, A., Behrens, J., Giraldo, F.X., and Wirth, V.Comparision between adaptive and uni-form discontinuous Galerkin simulations in dry 2d bubble experiments. J. Comput. Phys., 235: 371–393, 2013.Google Scholar
[35]Munz, C.-D., Roller, S., Klein, R., and Geratz, K.J.The extension of incompressible flow solvers to the weakly compressible regime. Comput. Fluids, 32(2): 173–196, 2003.Google Scholar
[36]Park, J.H. and Munz, C.-D.Multiple pressure variables methods for fluid flow at all Mach numbers. Int. J. Numer. Methods Fluids, 49(8): 905–931, 2005.Google Scholar
[37]Prasad, P.Ray theories for hyperbolic waves, kinematical conservation laws (KCL) and applications. Indian J. Pure Appl. Math., 38(5): 467–490, 2007.Google Scholar
[38]Restelli, M.Semi-Lagrangian and Semi-Implicit Discontinuous Galerkin Methods for Atmospheric Modeling Applications. PhD thesis, Politecnico di Milano, 2007.Google Scholar
[39]Restelli, M. and Giraldo, F.X.A conservative discontinuous Galerkin semi-implicit formulation for the Navier-Stokes equations in nonhydrostatic mesoscale modeling. SIAM J. Sci. Comput., 31(3): 2231–2257, 2009.Google Scholar
[40]Ricchiuto, M. and Bollermann, A.Stabilized residual distribution for shallow water simulations. J. Comput. Phys., 228(4): 1071–1115, 2009.Google Scholar
[41]Rogers, B., Borthwick, A.G.L., and Taylor, P.H.Mathematical balancing of flux gradient and source terms prior to using Roe’s approximate Riemann solver. J. Comput. Phys., 192(2):422–451, 2003.CrossRefGoogle Scholar
[42]Rogers, B., Fujihara, M., and Borthwick, A.G.L.Adaptive Q-tree Godunov-type scheme for shallow water equations. Int. J. Numer. Methods Fluids, 35(3): 247–280, 2001.Google Scholar
[43]Sun, Y. and Ren, Y.The finite volume local evolution Galerkin method for solving the hyperbolic conservation laws. J. Comput. Phys., 228(13): 4945–4960, 2009.Google Scholar
[44]Stecca, G., Siviglia, A. and Toro, E. F.A finite volume upwind-biased centred scheme for hyperbolic systems of conservation laws. Application to shallow water equations. Commun. Comput. Phys., 12(4): 1183–1214, 2012.Google Scholar
[45]Vallis, G.K.Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, 2006.Google Scholar