Non-hydrodynamic modes and a priori construction of shallow water
lattice Boltzmann equations
P. J. Dellar Non-hydrodynamic
modes and a priori construction of shallow water lattice Boltzmann
equations Phys. Rev. E
65 036309 (12 pages). DOI: 10.1103/PhysRevE.65.036309
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Abstract
Lattice Boltzmann equations for the isothermal Navier-Stokes equations
have been constructed systematically using a truncated moment expansion
of the equilibrium distribution function from continuum kinetic theory.
Applied to the shallow water equations, with its different equation of
state, the same approach yields discrete equilibria that are subject to
a grid scale computational instability. Different and stable equilibria
were previously constructed by Salmon [J. Marine Res. 57 p.503].
The two sets of equilibria differ through a non-hydrodynamic or ``ghost''
mode that has no direct effect on the hydrodynamic behavior derived in
the slowly varying limit. However, Salmon's equilibria eliminate a coupling
between hydrodynamic and ghost modes, one that leads to instability with
a growth rate increasing with wavenumber. Previous work has usually assumed
that truncated moment expansions lead to stable schemes. Such instabilities
have implications for lattice Boltzmann equations that simulate other non-ideal
equations of state, or that simulate fully compressible, non-isothermal
fluids using additional particles.
[Full reference is Salmon, R. (1999) The lattice Boltzmann method
as a basis for ocean circulation modeling J. Marine Research 57
pp 503-535]
BibTeX citation information:
@article{Dellar02Shallow,
author = "P. J. Dellar",
title = "Non-hydrodynamic modes and a priori construction of
shallow water lattice {Boltzmann} equations",
year = "2002",
journal = "Phys. Rev. E",
volume = "65",
pages = "036309 (12 pages)",
URL = "http://link.aps.org/abstract/PRE/v65/e036309",
DOI = "10.1103/PhysRevE.65.036309"}
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