Skip to main content
SpringerLink
Log in

Emerging Numerical Methods for Atmospheric Modeling

  • Chapter
  • First Online:
Numerical Techniques for Global Atmospheric Models

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 80))

Abstract

This chapter discusses the development of discontinuous Galerkin (DG) schemes for the hyperbolic conservation laws relevant to atmospheric modeling. Two variants of the DG spatial discretization, the modal and nodal form, are considered for the one- and two-dimensional cases. The time integration relies on a second- or third-order explicit strong stability-preserving Runge–Kutta method. Several computational examples are provided, including a solid-body rotation test, a deformational flow problem and solving the barotropic vorticity equation for an idealized cyclone. A detailed description of various limiters available for the DG method is given, and a new limiter with positivity-preservation as a constraint is proposed for two-dimensional transport. The DG method is extended to the cubed-sphere geometry and the transport and shallow water models are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arakawa A, Lamb VR (1977) Computational design of the basic dynamical process of the UCLA general circulation model. In: Chang J (ed) Methods in Computational Physics, Academic Press, pp 173–265

    Google Scholar 

  2. Atkins HL, Shu CW (1996) Quadrature-free implementation of the discontinuous Galerkin method for hyperbolic equations. In: 2nd AIAA/CEAS Aeroacoustic Conference, Paper 96-1683.

    Google Scholar 

  3. Balsara DS, Altman C, Munz CD, Dumbser M (2007) Sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG + HWENO schemes. J Comput Phys 226(1):586–620

    Article  MATH  MathSciNet  Google Scholar 

  4. Bassi F, Rebay S (1997) A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J Comput Phys 131:267–279

    Article  MATH  MathSciNet  Google Scholar 

  5. Bassi F, Ghidoni A, Rebay S, Tesini P (2009) High-order accurate p-multigrid discontinuous Galerkin solution of the Euler equations. Int J Numer Meth Fluids 60:847–865, doi:10.1002/fld.1917

    Article  MATH  MathSciNet  Google Scholar 

  6. Biswas R, Devine K, Flaherty J (1994) Parallel adaptive finite-element methods for conservation laws. Appl Num Math 14:255–283

    Article  MATH  MathSciNet  Google Scholar 

  7. Boris JP, Book DL (1973) Flux-Corrected Transport. I. SASHSTA, a fluid transport algorithm that works. J Comput Phys 11(1):38–69

    Google Scholar 

  8. Butcher JC (2008) Numerical Methods for Ordinary Differential equations, Second edn. Wiley, ISBN 978-0-470-72335-7 463 pp.

    Google Scholar 

  9. Canuto C, Hussaini MY, Quarteroni A, Zang TA (2007) Spectral Methods: Evolution of Complex Geometries and Application to Fluid Dynamics. Springer, ISBN 978-3-540-30727-3, 596 pp.

    Google Scholar 

  10. Chen C, Xiao F (2008) Shallow water model on cubed-sphere by multi-moment finite volume method. J Comput Phys 227(10):5019–5044

    Article  MATH  MathSciNet  Google Scholar 

  11. Cheruvu V, Nair RD, Tufo HM (2007) A spectral finite volume transport scheme on the cubed-sphere. Appl Num Math 57:1021–1032

    Article  MATH  MathSciNet  Google Scholar 

  12. Cockburn B, Shu CW (1989) TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservative laws II. Math Comp 52:411–435

    MATH  MathSciNet  Google Scholar 

  13. Cockburn B, Shu CW (1998) The local discontinuous Galerkin method for time-dependent convection-diffusion schemes. SIAM J Numer Anal 35:2440–2463

    Article  MATH  MathSciNet  Google Scholar 

  14. Cockburn B, Shu CW (2001) The Runge-Kutta discontinuous Galerkin method for convection-dominated problems. J Sci Computing 16:173–261

    Article  MATH  MathSciNet  Google Scholar 

  15. Cockburn B, Hou S, Shu CW (1990) TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV. Math Comp 54:545–581

    MATH  MathSciNet  Google Scholar 

  16. Cockburn B, Johnson C, Shu CW, Tadmor E (1997) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Springer, LNM 1697

    Google Scholar 

  17. Cockburn B, Karniadakis GE, Shu CW (2000) The development of discontinuous Galerkin methods. In: Cockburn B, Karniadakis GE, Shu CW (eds) Discontinuous Galerkin Methods: Theory, Computation, and Applications. Lecture Notes in Computational Science and Engineering, vol 11, Springer, 470 pp.

    Google Scholar 

  18. Colella P, Woodward PR (1984) The Piecewise Parabolic Method (PPM) for gas-dynamical simulations. J Comput Phys 54:174–201

    Article  MATH  MathSciNet  Google Scholar 

  19. Crowell S, Williams D, Marviplis C, Wicker L (2009) Comparison of traditional and novel discretization methods for advection models in numerical weather prediction. In: Lecture Notes in Computer Science, vol 5545, Springer-Verlag, pp 263–272, ICCS 2009, Part II.

    Google Scholar 

  20. DeMaria M (1985) Tropical cyclone motion in a nondivergent barotropic model. Mon Wea Rev 113:119–1210

    Article  Google Scholar 

  21. Dennis J, Fournier A, Spotz WF, St-Cyr A, Taylor MA, Thomas SJ, Tufo H (2005) High-resolution mesh convergence properties and parallel efficiency of a spectral element atmospheric dynamical core. Int J High Perf Computing Appl 19(3):225–235

    Article  Google Scholar 

  22. Deville MO, Fisher PF, Mund EM (2002) High-Order Methods for Incompressible Fluid Flow. Cambridge University Press, ISBN 0-521-45309-7, 499 pp.

    Google Scholar 

  23. Diosady LT, Darmofal DL (2009) Preconditioning methods for discontinuous Galerkin solutions of the compressible Navier-Stokes equations. J Comput Phys 228:3917–3835

    Article  MATH  MathSciNet  Google Scholar 

  24. Galewsky J, Polvani LM, Scott RK (2004) An initial-value problem to test numerical models of the shallow water equations. Tellus 56A:429–440

    Google Scholar 

  25. Ghostine R, Kessewani G, Mosé R, Vazquez J, Ghenaim A (2009) An improvement of classical slope limiters for high-order discontinuous Galerkin method. Int J Numer Meth Fluids 59: 423–442

    Article  MATH  Google Scholar 

  26. Giraldo FX, Restelli M (2008) A study of spectral element and discontinuous Galerkin methods for the Navier-Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases. J Comput Phys 227:3849–3877

    MATH  MathSciNet  Google Scholar 

  27. Giraldo FX, Hesthaven JS, Wartburton T (2002) Nodal high-order discontinous Galerkin methods for the shallow water equations. J Comput Phys 181:499–525

    Article  MATH  MathSciNet  Google Scholar 

  28. Godunov SK (1959) A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat Sb 47:271–306

    MathSciNet  Google Scholar 

  29. Gottlieb S, Shu CW, Tadmor E (2001) Strong stability preserving high-order time discretization methods. SIAM Rev 43:89–112

    Article  MATH  MathSciNet  Google Scholar 

  30. Harten A, Engquist B, Osher S, Chakravarthy S (1987) Uniformly high order essentially non-oscillatory schemes, III. J Comput Phys 71:231–303

    Article  MATH  MathSciNet  Google Scholar 

  31. Hesthaven JS, Warburton T (2008) Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, ISBN 978-0-387-72065-4, 500 pp.

    Google Scholar 

  32. Iskandarani M, Levin J, Choi BJ, Haidgovel D (2005) Comparison of advection schemes for high-order h-p finite element and finite volume methods. Ocean Modeling 10:233–252

    Article  Google Scholar 

  33. Kanevsky A, Carpenter MH, Gottlieb D, Hasthevan JS (2007) Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes. J Comput Phys 225: 1753–1781

    Article  MATH  MathSciNet  Google Scholar 

  34. Karniadakis GE, Sherwin S (2005) Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press, ISBN 0-19-852869-8, 657 pp.

    Google Scholar 

  35. Käser M, Dumbser M, Puente J, Igel H (2007) An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes-III. Geophys J Int 168:224–242

    Article  Google Scholar 

  36. Kopriva DA (2009) Implementing Spectral Methods for Partial Differential Equations. Springer, ISBN 978-90-481-2260-8, 394 pp.

    Google Scholar 

  37. Kopriva DA, Gassner G (2010) On the quadrature and weak form choices in collocation type discontinuous Galerkin spectral element methods. J Sci Comput 44:136–155

    Article  MATH  MathSciNet  Google Scholar 

  38. Krivodonova L (2007) Limiters for high-order dicontinuous Galerkin methods. J Comput Phys 226:879–896

    Article  MATH  MathSciNet  Google Scholar 

  39. Kubatko EJ, Bunya S, Dawson C, Westerink JJ (2009) Dynamic p-adaptive Runge–Kutta discontinuous Galerkin methods for the shallow water equations. Comput Methods Appl Mech Engrg 198:1766–1774

    Article  MathSciNet  Google Scholar 

  40. Lauritzen PH, Nair RD, Ullrich PA (2010) A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid. J Comput Phys 229:1401–1424

    Article  MATH  MathSciNet  Google Scholar 

  41. Läuter M, Giraldo FX, Handorf D, Dethloff K (2008) A discontinuous Galerkin method for shallow water equations in spherical traingular coordinates. J Comput Phys 227:10, 226–10, 242

    Google Scholar 

  42. van Leer B (1974) Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J Comput Phys 14:361–370

    Google Scholar 

  43. van Leer B (1977) Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection. J Comput Phys 23:276–299

    Google Scholar 

  44. Lesaint P, Raviart P (1974) Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, New York, chap On a finite element method for solving neutron transport eqaution, pp 89–123

    Google Scholar 

  45. LeVeque RJ (2002) Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, ISBN 0-19-00924-3, 558 pp.

    Google Scholar 

  46. Levy MN (2009) A high-order element-based Galerkin method for the global shallow water equations. PhD thesis, University of Colorado at Boulder, Department of Applied Mathematics, 108 pp.

    Google Scholar 

  47. Levy MN, Nair RD, Tufo HM (2007) High-order Galerkin methods for scalable global atmospheric models. Computers & Geosciences 33(8):1022–1035

    Article  Google Scholar 

  48. Levy MN, Nair RD, Tufo HM (2009) A high-order element-based Galerkin method for the barotropic vorticity equation. Int J Numer Meth Fluids 59(12):1369–1387

    Article  MATH  MathSciNet  Google Scholar 

  49. Liu XD, Osher S, Chan T (1994) Weighted essentially non-oscillatory schemes. J Comput Phys 115:200–212

    Article  MATH  MathSciNet  Google Scholar 

  50. Lomtev I, Kirby RM, Karniadakis GE (2000) A discontinuous Galerkin method in moving domains. In: Cockburn B, Karniadakis GE, Shu CW (eds) Discontinuous Galerkin Methods: Theory, Computation, and Applications. Lecture Notes in Computational Science and Engineering, vol 11, Springer, 470 pp.

    Google Scholar 

  51. Luo H, Baum JD, Löhner R (2007) A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J Comput Phys 225(1):686–713

    Article  MATH  MathSciNet  Google Scholar 

  52. Lynch P (2008) The origins of computer weather prediction and climate modeling. J Comput Phys 227:3431–3444

    Article  MATH  MathSciNet  Google Scholar 

  53. Nair RD (2009) Diffusion experiments with a global discontinuous Galerkin shallow-water model. Mon Wea Rev 137:3339–3350

    Article  Google Scholar 

  54. Nair RD, Lauritzen PH (2010) A class of deformational flow test cases for linear transport problems on the sphere. J Comput Phys 229:8868–8887

    Article  MATH  MathSciNet  Google Scholar 

  55. Nair RD, Thomas SJ, Loft RD (2005a) A discontinuous Galerkin global shallow water model. Mon Wea Rev 133:876–888

    Article  Google Scholar 

  56. Nair RD, Thomas SJ, Loft RD (2005b) A discontinuous Galerkin transport scheme on the cubed-sphere. Mon Wea Rev 133:814–828

    Article  Google Scholar 

  57. Nair RD, Choi HW, Tufo HM (2009) Computational aspects of a scalable high-order discontinuous Galerkin atmospheric dynamical core. Computers and Fluids 38:309–319

    Article  MathSciNet  Google Scholar 

  58. Prather MJ (1986) Numerical advection by conservation of second-order moments. J Geophys Res 91:6671–6681

    Article  Google Scholar 

  59. Qiu J, Khoo BC, Shu CW (2006) A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes. J Comput Phys 212(2):540–565

    Article  MATH  MathSciNet  Google Scholar 

  60. Qui J, Shu CW (2005a) Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin methods II: Two dimesional case. Computers & Fluids 34:642–663

    Article  MathSciNet  Google Scholar 

  61. Qui J, Shu CW (2005b) Runge-Kutta discontinuous Galerkin methods using WENO limiters. SIAM J Sci Computing 26:907–927

    Article  Google Scholar 

  62. Rančić M, Purser R, Mesinger F (1996) A global shallow water model using an expanded spherical cube. Q J R Meteorol Soc 122:959–982

    Google Scholar 

  63. Reed WH, Hill TR (1973) Triangular mesh method for neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory

    Google Scholar 

  64. Remacle JF, Flaherty JE, Sheppard MS (2003) An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems. SIAM Review 45:53–72

    Article  MATH  MathSciNet  Google Scholar 

  65. Restelli M, Giraldo FX (2009) A conservative discontinuous Galerkin semi-implicit formulation for the Navier-Stokes equations in nonhydrostatic mesoscale modeling. SIAM J Sci Comput 31:2231–2257

    Article  MATH  MathSciNet  Google Scholar 

  66. Rivière B (2008) Discontinuous Galerkin Method for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, ISBN 978-0-898716-56-6, 187 pp.

    Google Scholar 

  67. Sadourny R (1972) Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids. Mon Wea Rev 100:136–144

    Article  Google Scholar 

  68. Shu CW (1997) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Springer, chap Essentially non-oscillatory and weighted essentially non-oscillatory schemes for conservation laws, pp 324–432. LNM 1697

    Google Scholar 

  69. Simmons AJ, Burridge DM (1981) An energy and angular-momentum conserving vertical finite-difference scheme and hybrid vertical coordinates. Mon Wea Rev 109:758–766

    Article  Google Scholar 

  70. Smolarkiewicz PK (1982) The multi-dimensional Crowley advection scheme. Mon Wea Rev 110:1968–1983

    Article  Google Scholar 

  71. Smolarkiewicz PK (1984) A fully muliti-dimensional positive definite advection transport algorithm with small implicit diffusion. J Comput Phys 54:325–362

    Article  Google Scholar 

  72. St-Cyr A, Neckels D (2009) A fully implicit Jacobian-free high-order discontinuous Galerkin mesoscale flow solver. In: Lecture Notes in Computer Science, vol 5545, Springer-Verlag, pp 243–252, ICCS 2009, Part II.

    Google Scholar 

  73. St-Cyr A, Jablonowski C, Dennis JM, Tufo HM, Thomas SJ (2008) A comparison of two shallow water models with nonconforming advaptive grids. Mon Wea Rev 136:1898–1922

    Article  Google Scholar 

  74. Staniforth A, Coté J, Pudickiewicz J (1987) Comments on “Smolarkiewicz’s deformational flow”. Mon Wea Rev 115:894–900

    Article  Google Scholar 

  75. Suresh A (2000) Positivity preserving schemes in multidimensions. SIAM J Sci Comput 22:1184–1198

    Article  MATH  MathSciNet  Google Scholar 

  76. Toro EF (1999) Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction (2nd Ed.). Springer-Verlag, New York

    Google Scholar 

  77. Toro EF (2001) Shock-Capturing Methods for Free-Surface Shallow Flows. John Wiley & Sons, England, ISBN 0-471-98766-2, 305 pp.

    Google Scholar 

  78. Vallis GK (2006) Atmopsheric and Ocenaic Fluid Dynamics. Cambridge University Press, ISBN 978-0-521-84969-2, 745 pp.

    Google Scholar 

  79. Williamson DL (2007) The evolution of dynamical cores for global atmospheric models. J Meteo Soc of Japan 85:241–269

    Article  Google Scholar 

  80. Williamson DL, Drake JB, Hack JJ, Jakob R, Swarztrauber PN (1992) A standard test set for numerical approximations to the shallow water equations in spherical geometry. J Comput Phys 102:211–224

    Article  MATH  MathSciNet  Google Scholar 

  81. Zhang X, Shu CW (2010) On positivity-preserving high-order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J Comput Phys 229(23):8918–8934, doi:10.1016/j.jcp.2010.08.016

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are thankful to IMAGe (NCAR) colleagues, particularly Dr. Duane Rosenberg for an internal review of the manuscript. The authors would also like to thank two anonymous reviewers for several helpful suggestions. This project is partially supported by the U.S. Department of Energy under the awards DE-FG02-07ER64464 and DE-SC0001658. The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Author information

Authors and Affiliations

  1. National Center for Atmospheric Research, 1850 Table Mesa Drive, Boulder, CO, 80305, USA

    Ramachandran D. Nair & Peter H. Lauritzen

  2. Sandia National Laboratories, Albuquerque, NM, 87185, USA

    Michael N. Levy

Authors
  1. Ramachandran D. Nair
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael N. Levy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peter H. Lauritzen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ramachandran D. Nair .

Editor information

Editors and Affiliations

  1. Atmospheric Research, Dept. Climate/Global Dyn., National Center for, Boulder, Colorado, USA

    Peter Lauritzen

  2. Dept. Atmospheric, Oceanic &, Space Sciences, University of Michigan, Hayward St. 2455, Ann Arbor, 48109, Michigan, USA

    Christiane Jablonowski

  3. Organization 1433, MS-0370, Sandia National Laboratories, Albuquerque, New Mexico, USA

    Mark Taylor

  4. (NCAR), Inst. for Mathematics Applied to, National Center for Atmospheric Research, Boulder, Colorado, USA

    Ramachandran Nair

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Nair, R.D., Levy, M.N., Lauritzen, P.H. (2011). Emerging Numerical Methods for Atmospheric Modeling. In: Lauritzen, P., Jablonowski, C., Taylor, M., Nair, R. (eds) Numerical Techniques for Global Atmospheric Models. Lecture Notes in Computational Science and Engineering, vol 80. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11640-7_9

Download citation

Publish with us

Policies and ethics

Search

Navigation