Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
133 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Local Time-Stepping for the Shallow Water Equations using CFL Optimized Forward-Backward Runge-Kutta Schemes (2405.10505v1)

Published 17 May 2024 in math.NA, cs.NA, and physics.comp-ph

Abstract: The Courant-Friedrichs-Lewy (CFL) condition is a well known, necessary condition for the stability of explicit time-stepping schemes that effectively places a limit on the size of the largest admittable time-step for a given problem. We formulate and present a new local time-stepping (LTS) scheme optimized, in the CFL sense, for the shallow water equations (SWEs). This new scheme, called FB-LTS, is based on the CFL optimized forward-backward Runge-Kutta schemes from Lilly et al. (2023). We show that FB-LTS maintains exact conservation of mass and absolute vorticity when applied to the TRiSK spatial discretization (Ringler et al., 2010), and provide numerical experiments showing that it retains the temporal order of the scheme on which it is based (second order). In terms of computational performance, we show that when applied to a real-world test case on a highly-variable resolution mesh, the MPAS-Ocean implementation of FB-LTS is up to 10 times faster than the classical four-stage, fourth-order Runge-Kutta method (RK4), and 2.3 times faster than an existing strong stability preserving Runge-Kutta based LTS scheme (LTS3). Despite this significant increase in efficiency, the solutions produced by FB-LTS are qualitatively equivalent to those produced by both RK4 and LTS3.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (18)
  1. Solution of the tidal equations for the M2 and S2 tides in the world oceans from a knowledge of the tidal potential alone. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 290, 235–266. doi:10.1098/rsta.1978.0083.
  2. Computational Design of the Basic Dynamical Processes of the UCLA General Circulation Model, in: Chang, JULIUS. (Ed.), Methods in Computational Physics: Advances in Research and Applications. Elsevier. volume 17 of General Circulation Models of the Atmosphere, pp. 173–265. doi:10.1016/B978-0-12-460817-7.50009-4.
  3. Primer on Global Internal Tide and Internal Gravity Wave Continuum Modeling in HYCOM and MITgcm. New Frontiers in Operational Oceanography , 307–392.
  4. Global barotropic tide modeling using inline self-attraction and loading in MPAS-Ocean. Journal of Advances in Modeling Earth Systems n/a, e2022MS003207. doi:10.1029/2022MS003207.
  5. Local time stepping for the shallow water equations in MPAS. Journal of Computational Physics 449, 110818. doi:10.1016/j.jcp.2021.110818.
  6. Review of Drag Coefficients over Oceans and Continents. Monthly Weather Review 105, 915–929. doi:10.1175/1520-0493(1977)105<0915:RODCOO>2.0.CO;2.
  7. The DOE E3SM Model Version 2: Overview of the Physical Model and Initial Model Evaluation. Journal of Advances in Modeling Earth Systems 14, e2022MS003156. doi:10.1029/2022MS003156.
  8. A two-level time-stepping method for layered ocean circulation models: Further development and testing. Journal of Computational Physics 206, 463–504. doi:10.1016/j.jcp.2004.12.011.
  9. Conservative explicit local time-stepping schemes for the shallow water equations. Journal of Computational Physics 382, 152–176. doi:10.1016/j.jcp.2019.01.006.
  10. Parameterizing tidal dissipation over rough topography. Geophysical Research Letters 28, 811–814. doi:10.1029/2000GL012044.
  11. Voronoi Tessellations and Their Application to Climate and Global Modeling, in: Lauritzen, P., Jablonowski, C., Taylor, M., Nair, R. (Eds.), Numerical Techniques for Global Atmospheric Models. Springer, Berlin, Heidelberg. Lecture Notes in Computational Science and Engineering, pp. 313–342. doi:10.1007/978-3-642-11640-7_10.
  12. Storm Surge Modeling as an Application of Local Time-Stepping in MPAS-Ocean. Journal of Advances in Modeling Earth Systems 15, e2022MS003327. doi:10.1029/2022MS003327.
  13. CFL Optimized Forward–Backward Runge–Kutta Schemes for the Shallow-Water Equations. Monthly Weather Review 151, 3191–3208. doi:10.1175/MWR-D-23-0113.1.
  14. Spatial Tessellations, in: International Encyclopedia of Geography. John Wiley & Sons, Ltd, pp. 1–11. doi:10.1002/9781118786352.wbieg0601.
  15. An Evaluation of the Ocean and Sea Ice Climate of E3SM Using MPAS and Interannual CORE-II Forcing. Journal of Advances in Modeling Earth Systems 11, 1438–1458. doi:10.1029/2018MS001373.
  16. A multi-resolution approach to global ocean modeling. Ocean Modelling 69, 211–232. doi:10.1016/j.ocemod.2013.04.010.
  17. A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids. Journal of Computational Physics 229, 3065–3090. doi:10.1016/j.jcp.2009.12.007.
  18. Time-Splitting Methods for Elastic Models Using Forward Time Schemes. Monthly Weather Review 130, 2088–2097. doi:10.1175/1520-0493(2002)130<2088:TSMFEM>2.0.CO;2.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now