Could anyone explain:
We can conclude the LBM equation from an special first order finite difference scheme of the velocity-discrete Boltzmann equation as, for instance, depicted by Wolf-Gladrow in his book. On the other hand we can extract the velocity-discrete Boltzmann equation from the Taylor series expansion of the LBM equation, as it can be seen that the first order (in Ma) of Taylor series is exactly the velocity-discrete Boltzmann equation.
The question is: which equation is the more general one? The LBM equation or the velocity-discrete Boltzmann equation?
Could anyone explain:
The most fundamental equation is the Boltzmann equation, with continuous velocities. To get a numerical scheme which can be implemented on the computer, the velocities are first discretised to get the velocity-discrete Boltzmann equation. Then, space and time is discretised as you say to get the LB equation. This means that the velocity-discrete Boltzmann equation is more general than the LB equation, but only the LB equation can be directly implemented as a simulation method on the computer.
Hope that helps!
Hi Erlend M,
This is correct, but as you know, in the Chapman-Enskog expansion, we use the Taylor series expansion of the LB equation and use the first and second order terms to derive the N-s equation in low-Ma limit. But as I said, the first order term in the Taylor series is the discrete-velocity Boltzmann equation, and as you said is the more general equation. So it appears that the first order term is the most accurate term and there is no need to further account for the higher order terms!
could you please explain this.
Ah yes, I see what you mean now. It’s a good question!
When you discretise any equation, you’re going to get a certain discretisation error. For a nth order discretisation, the error is going to show up at order n+1 in the Taylor expansion of the discrete scheme. As you’ve already seen, the discretisation of the discrete-velocity Boltzmann equation (DVBE) is first order, and the Taylor expansion of the lattice Boltzmann equation (LBE) recovers the DVBE at first order. It would then be natural to assume that the second order Taylor expansion term is an error term.
However, this is only partly true. If you include this second order term in the Chapman-Enskog analysis, you will find the same equations of motion as for the DVBE. There is one significant difference; for the DVBE the viscosity is proportional with \tau, whereas for the LBE the viscosity is proportional with \tau-1/2. This is the effect of the second order discretisation error term in the Taylor expansion.
Does that make it clearer?
Another way of looking at this is the realise that the LBE can be obtained by using the forward Euler scheme to discretise the discrete velocity Boltzmann equation, which is a first order method. You could instead use Crank-Nicolson, which is second order.
Your answer is quite obvious.
Thank you very much for giving attention.
Good point, pleb01; the second order discretisation gives the same equations of motion as the DVBE, with a viscosity correctly proportional to \tau (instead of \tau-1/2), and a more accurate momentum flux tensor. If you’re interested, hmusavi, it’s described in e.g. section IV of this article.
I’ve never tried to do the Taylor expansion on the second order scheme myself, but I guess you would get the DVBE at first order, zero at second order, and error terms at third order?
Another questions may arises here. If we take the higher order terms in the Taylor series as the error terms, will it be correct to consider higher order terms to recover higher Ma (or Kn) regimes as for example what’s called Burnett or super Burnett equations?
No, I don’t think that would be correct. The higher order terms in the Taylor series are purely errors from the discretisation of the DVBE. The Burnett and super-Burnett terms are found from another expansion; the Chapman-Enskog expansion of the Boltzmann equation.
What do you mean from "I’ve never tried to do the Taylor expansion on the second order scheme myself, but I guess you would get the DVBE at first order, zero at second order, and error terms at third order? "
If we do a Taylor expansion of the first order discretisation of the DVBE, we find the DVBE at first order in the expansion (the single derivatives), and we get non-DVBE terms at second order (the double derivatives). These particular error terms are not so harmful; we don’t usually see their effects other than that they make the viscosity proportional to \tau-1/2 instead of \tau. Increasing the order of the discretisation by one would give us these error terms at one order higher, so that they would in this case appear first at third order. The second order would be error free.
If you are familiar with error analysis of finite difference schemes, it’s essentially like that.