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Simulation of the Navier–Stokes equations in three dimensions with a spectral collocation method
Author(s) -
Subich Christopher J.,
Lamb Kevin G.,
Stastna Marek
Publication year - 2013
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.3788
Subject(s) - spectral method , preconditioner , collocation (remote sensing) , multigrid method , coordinate system , chebyshev polynomials , mathematics , collocation method , nonlinear system , navier–stokes equations , conformal map , boundary (topology) , mathematical analysis , chebyshev filter , compressibility , partial differential equation , geometry , computer science , differential equation , linear system , physics , ordinary differential equation , quantum mechanics , machine learning , thermodynamics
SUMMARY This paper describes a nonlinear, three‐dimensional spectral collocation method for the simulation of the incompressible Navier–Stokes equations under the Boussinesq approximation, motivated by geophysical and environmental flows. These flows are driven by the interaction of stratified fluid with topography, which this model accurately accounts for by using a mapped coordinate system. The spectral collocation method is implemented with both a Fourier trigonometric expansion and the Chebyshev polynomials, as appropriate for the domain boundary conditions. The coordinate mapping prohibits the use of existing, fast solution methods that rely on the separation of variables, so a preconditioner based on the approximate solution of a corresponding finite‐difference problem with geometric multigrid is used. The model is parallelized with the Message Passing Interface library, and it runs effectively on shared and distributed‐memory systems. Copyright © 2013 John Wiley & Sons, Ltd.