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Mixed finite elements for global tide models.

Colin J Cotter1, Robert C Kirby2

  • 1Imperial College London, South Kensington Campus, London, SW7 2AZ UK.

Numerische Mathematik
|June 16, 2017
PubMed
Summary

This study proves long-time stability for rotating shallow water equations using mixed finite element methods. Numerical results confirm energy damping and accurate convergence rates for these geophysical flow models.

Keywords:
35Q8665M1265M60

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Area of Science:

  • Computational fluid dynamics
  • Geophysical fluid dynamics
  • Numerical analysis

Background:

  • The linearized rotating shallow water equations are fundamental in geophysical fluid dynamics.
  • Understanding long-time stability and error estimates is crucial for accurate numerical simulations.
  • Previous methods may face challenges with energy accumulation or convergence.

Purpose of the Study:

  • To analyze the long-time stability of mixed finite element methods for linearized rotating shallow water equations.
  • To derive a priori error estimates for key variables.
  • To validate theoretical findings through numerical simulations.

Main Methods:

  • Application of mixed finite element methods.
  • Development of a strong energy estimate for a second-order formulation of linearized momentum.
  • Derivation of a priori error estimates in L² norm and for time derivatives and divergence.

Main Results:

  • Proof of long-time stability without energy accumulation (geostryptic state).
  • Established a priori error estimates for linearized momentum and free surface elevation.
  • Demonstrated convergence rates for numerical solutions.

Conclusions:

  • The proposed mixed finite element methods ensure long-time stability for the studied equations.
  • Theoretical error estimates are confirmed by numerical experiments.
  • The methods are effective for simulating geophysical flows with drag and forcing.