I strongly object to the above statement that bounce back boundary conditions are not needed anymore. The strongest statement we can make at present is in some special flows alternatives to bounce back work better, or just as well, as bounce back. There are many alternative methods, the first that I’m aware of being Noble et al 1994, but there may be even earlier contributions. The most popular alternative is Zou and He’s non-equilibrium bounce-back. The paper referenced above seems to combine Zou and He with a different corner implementation, and the corner treatment appears to be identical to the earlier work of Bennet (Bennett 2010, Bennett et al 2012, Reis and Dellar 2012). The “moment-method” of Bennett appears in his 2010 thesis (freely available on the internet) and extra details can be found in material given at https://www.maths.ox.ac.uk/groups/occam/events/occam-lattice-boltzmann-workshop and https://www.maths.ox.ac.uk/groups/occam/events/lattice-boltzmann-workshop (it may also be worth noting that Bennett showed what conditions are needed for corners when Zou and He method is used at boundaries)

Besides the point of giving credit where it is due, it can not be said that these alternatives, in general, are better than bounce back. They can all eliminate the slip error of bounce back, but so can an MRT/TRT method (when used with bounce back). Both are equally simple, but bounce back is more stable. If there is to be an advantage of the method of Bennett, then it is its generality. Still, its weaker stability for 2D flows remains. Note also that all these on-node methods that I have mentioned, including the one referenced above, are currently limited to very simple geometries which have boundaries sitting on grid points. Bounce-back and interpolation methods, however, have a simple extension.

My apologies for the negative tone of the previous long paragraphs, but I feel compelled to summarise: it is not fair nor justified to make a sweeping statement about bounce back. At the very least, a comparison of the methods using a simple 2D flow (say lid driven cavity or flow over an obstacle) with identical parameters is needed. And even then, appropriate conditions are likely to be problem-dependent. No method is perfect, despite what the marketing and advertising might try to suggest, and you have to be aware of the pros and cons of your model.

Back to the original post. Erlend’s reply is completely correct: f^1 contains gradients of the conserved variables. With reference again to “alternative” boundary conditions, this time for inflow/outflow, Zou and He schemes , Bennett’s moment method, and the referenced, rely on us knowing what either the velocity profile or the pressure (density) is at the entrance and exit. This is often not know. However, Inlet/outlet based on f can interpolate/extrapolate to impose zero flux conditions (while being consistent with the Chapman-Enskog solution). I think the moment method could also be used to impose a no flow exit condition, but as far as know its success has not yet be assessed. Moreover, none of the “alternatives” (ie all those other then inflow/outflow based on f and the Chapman-Enskog solution of f) take higher order contributions (from Chapman-Enskog/asymptotic analysis) and moments into consideration (they cannot - you have 3 unknowns but 9 moments) and are thus often unstable and/or inaccurate for all but this simplest flows).

People like Junk have done some very nice work on consistent inflow/outflow conditions which you may find useful.

Good luck!