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Re: [ESPResSo-users] LBM, speed of sound, stability

From: Ulf Schiller
Subject: Re: [ESPResSo-users] LBM, speed of sound, stability
Date: Wed, 17 Dec 2014 18:10:08 +0000
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On 17/12/14 12:12, Ivan Cimrak wrote:
> Hi all,
> In one of his emails Ulf Shiller explained that:
> "you need to make sure that h*c_s^2/\nu is small to avoid nonlinear
> instabilities. h is the LB timestep, c_s is the speed of sound, and \nu
> is the kinematic viscosity. In the D3Q19 model, c_s^2=1/3*a^2/h^2, so
> a^2/(3*\nu*h) must be small. It may work with values O(1) but it is not
> guaranteed."
> Ulf, could you please give me the reason why this is necessary? And what
> does it mean "is small"? Are the values 0.1 - 0.99 ok?

Hi Ivan,

the standard lattice Boltzmann algorithm is typically thought to be
second order accurate in time, however, if you look at the
discretisation of the collision operator (usually Crank-Nicolson), the
error is actually of the order O((h/\tau)^3) where \tau is the viscous
relaxation time (or BGK relaxation time). The latter is related to the
viscosity by \nu=c_s^2*\tau where c_s is the speed of sound. Hence the
grid Reynolds number h/\tau=h*c_s^2/\nu needs to be small. Now, in LB
there is a subtle cancellation of errors of the Crank-Nicolson
discretisation and the splitting error, such that the standard LB
algorithm approximates the slow manifold of solutions to the discrete
velocity model even at values of \tau/h beyond unity (an intriguing side
effect of this is that the exact solution of the collision operator does
produce excessive decay of shear waves due to the lack of said
cancellation). Another way to phrase it is that the LBM disconnects from
kinetic theory and can work in the over-relaxation regime (i.e. negative
eigenvalues of the collision operator). Some details of the derivation
are given in and references
therein (in particular Brownlee et al. and Paul Dellar). In practise,
instabilities may arise at the higher moments and couple into the
Navier-Stokes dynamics. I'll mention in passing that coupling particles
to the LB fluid involves singular forces that may also affect stability.
If this actually occurs will depend on the characteristics of the flow
under consideration; for laminar flow and non-stiff coupling there is
probably no problem.

Best wishes,

Dr Ulf D Schiller
Centre for Computational Science
University College London
20 Gordon Street
London WC1H 0AJ
United Kingdom

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