Interdisciplinary Applied Mathematics

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In particular, the    following    form    of    the    H    function    was    introduced    by


(Karlin et al., 1999):


N

h = Y, fi Hfi/Wi),


i=1


where Wi    are    velocity-dependent    weights    with    Wi    =    fie4    at zero flow.    An


example in two dimensions for the nine-velocity lattice (2D9V) is


4    4


H = Hb + ln(3/8)fo + ln(3/2) £ fli + ln(6) £ f2i,


i=1    i=1


where Hb =Y11=^24=1 fki(ln fki — 1), and at equilibrium, 82


4 = з exp [a], Q = — exp [a + Aaciia],


/2? = g exp[a + AaC2ia.


Also, a = ln(p/6) —u2/(2c2), and Aa = ua/c2. Constructing the corresponding local equilibrium from these expressions, (Karlin et al., 1999) recovered the equilibrium distribution used in (Qian et al., 1992), on the 2D9V lattice. Such H functions enforce Galilean invariance up to O(M4), where is the Mach number. Given, however, that the lattice Boltzmann equation is an approximation to the Navier-Stokes equations within O(M2), such H functions lead to very accurate results for the local equilibrium distributions.


A second-order O(M2) H function of different form was employed in (Boghosian et al., 2003), where uniform contributions to the H function from the lattice sites are assumed of the form


N

H = £ h(fi).


i=1


This H function is also minimized subject to the usual hydrodynamic constraints. In this case h is not a Boltzmann entropy, but its form depends on the space dimension. For example, in three dimensions it follows the Tsallis form, which is typically associated with lack of ergodicity (Boghosian et al., 2003). Many other entropy functions are possible depending on the specific kinetic model, i.e., single- or multispeed. For example, in (Boghosian et al.,    2004),    h(fi)    is    obtained    from    a certain    functional    differential    equa

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