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A Local Timestepping Runge–Kutta Discontinuous Galerkin Method for Hurricane Storm Surge Modeling

  • Clint DawsonEmail author
Chapter
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 157)

Abstract

In this paper, we describe a local timestepping (LTS) approach within the Runge–Kutta discontinuous Galerkin (RKDG) method, and the application of this method to the modeling of hurricane storm surge. Modeling storm surge requires the numerical solution of the shallow water equations with wind and atmospheric pressure forcing, over complex domains which include wet and dry regions. The RKDG method is well suited for these applications; however, well-resolved simulations of storm surge can require highly graded meshes, which can lead to severe global CFL constraints. The LTS approach allows for elements to use timesteps which approximately satisfy only local CFL conditions. We describe a fully parallel implementation of the LTS method within an RKDG shallow water simulator, developed by the author and collaborators over a period of several years. We demonstrate that, for a specific hurricane, namely Hurricane Ike, the LTS method can reduce parallel run-times by nearly a factor of two with no degradation in accuracy.

Keywords

Local timestepping Multirate methods Shallow water equations Runge–Kutta discontinuous Galerkin methods Hurricane storm surge 

Notes

Acknowledgements

The author acknowledges the support of National Science Foundation grant DMS-0915118.

References

  1. [1]
    J. B. Bell, C. N. Dawson, and G. R. Shubin, An unsplit, higher-order Godunov method for scalar conservation laws, Journal of Computational Physics, 74 (1988), pp. 1–24.CrossRefzbMATHGoogle Scholar
  2. [2]
    M. Berger, D. L. George, R. J. LeVeque, and K. T. Mandli, The GeoClaw software for depth-averaged flows with adaptive refinement, Advances in Water Resources, (2011).Google Scholar
  3. [3]
    M. Berger and R. J. LeVeque, Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems, SIAM Journal on Numerical Analysis, 35 (1998), pp. 2298–2316.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    S. Bunya, E. Kubatko, J. Westerink, and C. Dawson, A wetting and drying treatment for the runge-kutta discontinuous galerkin solution to the shallow water equations., Computer Methods in Applied Mechanics and Engineering, 198 (2009), pp. 1548 – 1562.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    E. M. Constantinescu and A. Sandu, Multirate timestepping methods for hyperbolic conservation laws, Journal of Scientific Computing, 33 (2007), pp. 239–278.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    C. Dawson, High resolution upwind-mixed finite element methods for advection-diffusion equations with variable time-stepping., Numer. Methods Partial Differential Equations, 11 (1995), pp. 525–538.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    C. Dawson and R. Kirby, High resolution schemes for conservation laws with locally varying time steps, SIAM J. Sci. Comput., 22 (2000), pp. 2256–2281.MathSciNetCrossRefGoogle Scholar
  8. [8]
    C. Dawson, E. Kubatko, J. Westerink, C. Trahan, C. Mirabito, C. Michoski, and N. Panda, Discontinuous Galerkin methods for modeling hurricane storm surge, Advances in Water Resources, doi:10.1016/j.advwatres.2010.11.004 (2010).Google Scholar
  9. [9]
    C. Dawson, C. Trahan, E. Kubatko, and J. J. Westerink, A parallel local timestepping Runge-Kutta Discontinuous Galerkin method with appliations to coastal ocean modeling, submitted, (2012).Google Scholar
  10. [10]
    J. Garratt, Review of drag coefficients over oceans and continents, Monthly Weather Review, 105 (1977), pp. 915–929.CrossRefGoogle Scholar
  11. [11]
    G. Karypis and V. Kumar, METIS: A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices, University of Minnesota, Department of Computer Science/Army HPC Research Center, Minneapolis, MN, (1998).Google Scholar
  12. [12]
    E. M. Constantinescu and, A fast and high quality scheme for partitioning irregular graphs, SIAM Journal on Scientific Computing, 20 (1999), pp. 359–392.Google Scholar
  13. [13]
    A. Kennedy, U. Gravois, B. Zachry, J. Westerink, M. Hope, J. Dietrich, M. Powell, A. Cox, J. R.A. Luettich, and R. Dean, Origin of the hurricane ike forerunner surge, Geophysical Research Letters, 38 (2011).Google Scholar
  14. [14]
    R. Kirby, On the convergence of high resolution methods with multiple time scales for hyperbolic conservation laws., Mathematics of Computation, 72 (2002), pp. 1239–1250.MathSciNetCrossRefGoogle Scholar
  15. [15]
    E. Kubatko, S. Bunya, C. Dawson, and J. Westerink, Dynamic p-adaptive Runge-Kutta discontinuous Galerkin methods for the shallow water equations, Computer Methods in Applied Mechanics and Engineering, 198 (2009), pp. 1766–1774.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    E. J. Kubatko, S. Bunya, C. Dawson, and J. J. Westerink, A performance comparison of continuous and discontinuous finite element shallow water models, Journal of Scientific Computing, 40 (2009), pp. 315–339.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    E. J. Kubatko, J. J. Westerink, and C. Dawson, hp discontinuous Galerkin methods for advection dominated problems in shallow water flow, Comput. Methods Appl. Mech. Engrg., 196 (2006), pp. 437–451.CrossRefzbMATHGoogle Scholar
  18. [18]
    C. Michoski, C. Mirabito, C. Dawson, D. Wirasaet, E. J. Kubatko, and J. J. Westerink, Adaptive hierarchi transformations over dynamic p-enriched schemes applied to generalized DG systems, Journal of Computational Physics, 230 (2011), pp. 8028–8056.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    S. Osher and R. Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids., Mathematics of Computation, 41 (1983), pp. 321 – 336.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    B. F. Sanders, Integration of a shallow water model with a local time-step., Journal of Hydraulic Research, 46 (2008), pp. 466 – 475.CrossRefGoogle Scholar
  21. [21]
    C. J. Trahan and C. Dawson, Local time-stepping in Runge-Kutta discontinuous Galerkin finite element methods applied to the shallow water equations, Comput. Methods Appl. Mech. Engrg., 217–220 (2012), pp. 139–152.MathSciNetCrossRefGoogle Scholar
  22. [22]
    J. Westerink, R. Luettich, J. Feyen, J. Atkinson, C. Dawson, H. Roberts, M. Powell, J. Dunion, E. Kubatko, and H. Pourtaheri, A basin to channel scale unstructured grid hurricane storm surge model applied to sourthern louisiana., American Meteorological Society, 136 (2008), pp. 833 – 864.Google Scholar
  23. [23]
    D. Wirasaet, S. Tanaka, E. J. Kubatko, J. J. Westerink, and C. Dawson, A performance comparison of nodal discontinuous Galerkin methods on triangles and quadrilaterals, Internal Journal of Numerical Methods in Fluids, 64 (2010), pp. 1336–1362.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.The Institute for Computational Science and EngineeringThe University of Texas at AustinAustinUSA

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