# Latt's Ph.D. Thesis, p.11

Hi, I have read the following statement in Latt’s thesis, p. 11:

In practice a value of c_s^2 = 1/3 is found to be numerically most stable, and this choice is therefore most

As far as I know the stability of LBM has to do with tau, the dimensionless relaxation time.

Is there any good reference to the above statement, which I failed to find after an extensive
search.

sincerely
moon jung

Hi,

When I read the p. 29 of Latt’s thesis, some hint occurred to me about my question.

The relevant paragraph reads:

In the LB model that was introduced in the text and which is systematically used in the
communy, the speed of sound is a lattice constant. This means that it takes
a constant value (independent of delta x and delta t) when it is expressed in lattice units.

For consistency, it should be remarked that it is principally possible to vary the speed of
sound in LB models. This is done by adapting the rest-particle weights t_0. [
It is because the square of the sound speed is represented as the weighted sum of the squares of
lattice velocities c_i.]
Choosing a speed of sound different than 1/root(3) has however been observed to induce
numerical instabilities and is therefore very uncommon.

==>
So, choosing the dimensionless relaxation time tau to ensure stability is at a different level
than choosing the speed of sound. IT seems that the latter comes down to defining a new LB model.

Is my guess on the right track?

Moon Jung

The first thing to understand about the LB equilibrium is that it is really a quadrature approximation to the Maxwell-Boltzmann distribution. Since that distribution is simply normal by another name the appropriate quadrature is Gauss-Hermite.

Now in the athermal system speed of sound is connected to temperature and hence variance of that normal distribution. Since everybody likes to use the lattice distance 1, that variance has to be chosen such that the three point Gauss-Hermite quadrature nodes are naturally at \pm 1 and 0. That is where the sqrt(1/3) comes from.

Of course it is possible to use a different quadrature than Gauss-Hermite if you like, and indeed all multi speed lattices are of that type since for, say, 5 point Gauss-Hermite you cannot scale the quadrature nodes onto the unit lattice any more.

On the question of relaxation time indeed this parameter will definitely affect stability but usually you don’t have any choice in it since you would like to simulate a system with given physics you have no choice over (viscosity etc.). If you attempt a simulation and find it to be unstable then you would resort to using some kind of stabilization technique which would not affect the physics, such as MRT. If you ever need to do this use Paul Dellar’s variant not the Lallemand/Luo one, Dellar respects the proper Hermite structure of the problem.

Hi,
@dp : “variance has to be chosen such that the three point Gauss-Hermite quadrature nodes are naturally at \pm 1 and 0. That is where the sqrt(1/3) comes from” , can you give a detailed reference on this point please? I didn’t understand well.
Thanks