Skip to main content
×
Home
    • Aa
    • Aa

Runge-Kutta Discontinuous Local Evolution Galerkin Methods for the Shallow Water Equations on the Cubed-Sphere Grid

  • Yangyu Kuang (a1), Kailiang Wu (a2) and Huazhong Tang (a1)
Abstract
Abstract

The paper develops high order accurate Runge-Kutta discontinuous local evolution Galerkin (RKDLEG) methods on the cubed-sphere grid for the shallow water equations (SWEs). Instead of using the dimensional splitting method or solving one-dimensional Riemann problem in the direction normal to the cell interface, the RKDLEG methods are built on genuinely multi-dimensional approximate local evolution operator of the locally linearized SWEs on a sphere by considering all bicharacteristic directions. Several numerical experiments are conducted to demonstrate the accuracy and performance of our RKDLEG methods, in comparison to the Runge-Kutta discontinuous Galerkin method with Godunov's flux etc.

Copyright
Corresponding author
*Corresponding author. Email addresses: kyy@pku.edu.cn (Y. Y. Kuang), wukl@pku.edu.cn (K. L. Wu), hztang@math.pku.edu.cn (H. Z. Tang)
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] K. R. Arun , M. Kraft , M. Lukáčová-Medvid’ová , and P. Prasad , Finite volume evolution Galerkin method for hyperbolic conservation laws with spatially varying flux functions, J. Comput. Phys., 228 (2009), pp. 565590.

[2] J. R. Bates , F. H. M. Semazzi , and R. W. Higgins , Integration of the shallow water equations on the sphere using a vector semi-Lagrangian scheme with a multigrid solver, Mon. Wea. Rev., 118 (1990), pp. 16151627.

[3] B. J. Block , M. Lukáčová-Medvid’ová , P. Virnau , and L. Yelash , Accelerated GPU simulation of compressible flow by the discontinuous evolution Galerkin method, Eur. Phys. J. Spec. Top., 210 (2012), pp. 119132.

[4] A. Bollermann , S. Noelle , and M. Lukáčová-Medvid’ová , Finite volume evolution Galerkin methods for the shallow water equations with dry beds, Commun. Comput. Phys., 10 (2011), pp. 371404.

[5] D. S. Butler , The numerical solution of hyperbolic systems of partial differential equations in three independent variables, Proc. R. Soc. Lond. A., 255 (1960), pp. 232252.

[6] C. G. Chen , X. L. Li , X. S. Shen , and F. Xiao , Global shallow water models based on multi-moment constrained finite volume method and three quasi-uniform spherical grids, J. Comput. Phys., 271 (2014), pp. 191223.

[7] C. G. Chen and F. Xiao , Shallow water model on cubed-sphere by multi-moment finite volume method, J. Comput. Phys., 227 (2008), pp. 50195044.

[8] B. Cockburn and C.-W. Shu , The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199224.

[9] M. Dudzinski and M. Lukáčová-Medvid’ová , Well-balanced bicharacteristic-based scheme for multilayer shallow water flows including wet/dry fronts, J. Comput. Phys. 235 (2013), pp. 82113.

[10] J. Galewsky , R. K. Scott , and L. M. Polvani , An initial-value problem for testing numerical models of the global shallow-water equations, Tellus A, 56 (2004), pp. 429440.

[11] F. X. Giraldo , J. S. Hesthaven , and T. Wartburton , Nodal high-order discontinuous Galerkin methods for the shallow water equations, J. Comput. Phys., 181 (2002), pp. 499525.

[12] F. X. Giraldo and T. Warburton , A nodal triangle-based spectral element method for the shallow water equations on the sphere, J. Comput. Phys., 207 (2005), pp. 129150.

[13] F. X. Giraldo and T. Warburton , A high-order triangular discontinuous Galerkin oceanic shallow water model, Int. J. Numer. Meth. Fluids, 56 (2008), pp. 899925.

[14] L. C. Huang , Conservative bicharacteristic upwind schemes for hyperbolic conservation laws II, Comput. Math. Appl., 29 (1995), pp. 91107.

[15] A. Hundertmark-Zauškova , M. Lukáčová-Medvid’ová , and F. Prill , Large time step finite volume evolution Galerkin methods, J. Sci. Comput. 48 (2011), pp. 227240.

[16] R. Jakob-Chien , J. J. Hack , and D. L. Williamson , Spectral transform solutions to the shallow water test set, J. Comput. Phys., 119 (1995), pp. 164187.

[17] R. L. Johnston and S. K. Pal , The numerical solution of hyperbolic systems using bicharacteristics, Math. Comput., 26 (1972), pp. 377392.

[18] A. Kageyama and T. Sato , The Yin-Yang grid: An overset grid in spherical geometry, Geochem. Geophys. Geosyst., 5 (2004), pp. Q09005.

[19] M. Läuter , F.X. Giraldo , D. Handorf , and K. Dethloff , A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates, J. Comput. Phys., 227 (2008), pp. 1022610242.

[20] M. Läuter , D. Handorf , and K. Dethloff , Unsteady analytical solutions of the spherical shallow water equations, J. Comput. Phys., 210 (2005), pp. 535553.

[22] X. L. Li , D. H. Chen , X. D. Peng , K. Takahashi , and F. Xiao , A multimoment finite volume shallow-water model on the Yin-Yang overset spherical grid, Mon. Wea. Rev., 136 (2008), pp. 30663086.

[23] X. L. Li , X. S. Shen , X. D. Peng , F. Xiao , Z. R. Zhuang , and C. G. Chen , Fourth order transport model on Yin-Yang grid by multi-moment constrained finite volume scheme, Proc. Comput. Sci., 9 (2012), pp. 10041013.

[24] S. J. Lin and R. B. Rood , An explicit flux-form semi-Lagrangian shallow-water model on th sphere, Quart. J. Roy. Meteor. Soc., 123 (1997), pp. 24772498.

[25] M. Lukáčová-Medvid’ová and K.W. Morton , Finite volume evolution Galerkin methods–A survey, Indian J. Pure Appl. Math., 41 (2010), pp. 329361.

[26] M. Lukáčová-Medvid’ová , K.W. Morton , and G. Warnecke , Finite volume evolution Galerkin methods for Euler equations of gas dynamics, Int. J. Numer. Meth. Fluids, 40 (2002), pp. 425434.

[27] M. Lukáčová-Medvid’ová , K.W. Morton , and G. Warnecke , Evolution Galerkin methods for hyperbolic systems in two space dimensions, Math. Comput., 69 (2000), pp. 13551384.

[28] M. Lukáčová-Medvid’ová , K. W. Morton , and G. Warnecke , Finite volume evolution Galerkin (FVEG) methods for hyperbolic problems, SIAM. J. Sci. Comput., 26 (2004), pp. 130.

[29] M. Lukáčová-Medvid’ová , S. Noelle , and M. Kraft , Well-balanced finite volume evolution Galerkin methods for the shallow water problems, J. Comput. Phys., 221 (2007), pp. 122147.

[30] M. Lukáčová-Medvid’ová , J. Saibertová , and G. Warnecke , Finite volume evolution Galerkin methods for nonlinear hyperbolic systems, J. Comput. Phys., 183 (2002), 533562.

[31] A. McDonald and J. R. Bates , Semi-Lagrangian integration of a gridpoint shallow water model on the sphere, Mon. Wea. Rev., 117 (1989), pp. 130137.

[32] K. W. Morton , On the analysis of finite volume methods for evolutionary problems, SIAM J. Numer. Anal., 35 (1998), pp. 21952222.

[33] R. D. Nair and B. Machenhauer , The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere, Mon. Wea. Rev., 130 (2002), pp. 649667.

[34] R. D. Nair , S. J. Thomas , and R. D. Loft , A discontinuous Galerkin transport scheme on the cubed sphere, Mon. Wea. Rev., 133 (2005), pp. 814828.

[35] R. D. Nair , S. J. Thomas , and R. D. Loft , A discontinuous Galerkin global shallow water model, Mon. Wea. Rev., 133 (2005), pp. 876888.

[36] J. A. Pudykiewicz , On numerical solution of the shallow water equations with chemical reactions on icosahedral geodesic grid, J. Comput. Phys., 230 (2011), pp. 19561991.

[37] W. M. Putman and S. J. Lin , Finite-volume transport on various cubed-sphere grid, J. Comput. Phys., 227 (2007), pp. 5578.

[38] C. Ronchi , R. Iacono , and P. S. Paolucci , The cubed sphere: A new method for the solution of partial differential equations in spherical geometry, J. Comput. Phys, 124 (1996), pp. 93114.

[39] R. Sadourny , Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids, Mon. Wea. Rev., 100 (1972), pp. 136144.

[41] C.-W. Shu , Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 10731084.

[42] A. St-Cyr , C. Jablonowski , J. M. Dennis , H. M. Tufo , and S. J. Thomas , A comparison of two shallow-water models with non-conforming adaptive grids, Mon. Wea. Rev., 136 (2008), pp. 18981922.

[43] Y. T. Sun and Y. X. Ren , The finite volume local evolution Galerkin method for solving the hyperbolic conservation laws, J. Comput. Phys., 228 (2009), pp. 49454960.

[44] M. Taylor , J. Tribbia , and M. Iskandarani , The spectral element method for the shallow water equations on the sphere, J. Comput. Phys., 130 (1997), pp. 92108.

[45] S. J. Thomas and R. D. Loft , Semi-implicit spectral element model, J. Sci. Comput., 17 (2002), pp. 339350.

[46] S. J. Thomas and R. D. Loft , The NCAR spectral element climate dynamical core: semi-implicit Eulerian formulation, J. Sci. Comput., 25 (2005), pp. 307322.

[47] J. Thuburn , A PV-based shallow-water model on a hexagonal-icosahedral grid, Mon. Wea. Rev., 125 (1997), pp. 23282347.

[48] H. Tomita , M. Tsugawa , M. Satoh , and K. Goto , Shallow-water model on a modified icosahedral geodesic grid by using spring dynamics, J. Comput. Phys., 174 (2001), pp. 579613.

[49] P. A. Ullrich , C. Jablonowski , and B. Van Leer , High-order finite-volume methods for the shallow water equations on the sphere, J. Comput. Phys., 229 (2010), pp. 61046134.

[50] D. L. Williamson , J. B. Drake J. J. Hack , R. Jakob , and P. N. Swarztrauber , A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys., 102 (1992), pp. 211224.

[51] K. L. Wu and H. Z. Tang , Finite volume local evolution Galerkin method for two-dimensional special relativistic hydrodynamics, J. Comput. Phys., 256 (2014), pp. 277307.

[52] C. Yang , J. W. Cao , and X. C. Cai , A fully implicit domain decomposition algorithm for shallow water equations on the cubed-sphere, SIAM J. Sci. Comput., 32 (2010), pp. 418438.

[53] L. Yelash , A. Müller , M. Lukáčová-Medvid’ová , F. X. Giraldo , and S. V. Wirth , Adaptive discontinuous evolution Galerkin method for dry atmospheric flow, J. Comput. Phys., 268 (2014), pp. 106133.

[54] J. Zhao and H. Z. Tang , Runge-Kutta discontinuous Galerkin methods with WENO limiter for the special relativistic hydrodynamics, J. Comput. Phys., 242 (2013), pp. 138168.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 21 *
Loading metrics...

Abstract views

Total abstract views: 99 *
Loading metrics...

* Views captured on Cambridge Core between 9th May 2017 - 20th September 2017. This data will be updated every 24 hours.