we all know how to compute the equilibrium populations f_i^eq and - after that - the non-equilibrium populations f_i^neq = f_i - f_i^eq, which is really sufficient in most cases. But is it also possible to compute the first order corrections f_i^1 without the higher order terms, i. e. to separate f_i^1 from f_i^(>1)?

Indeed, there is a possibility to compute the f_i^1 from the f_i^0:
f_i^1 = - \tau (\partial t + c_{i \alpha} \partial x_\alpha) f_i^0
But for this I need derivatives and a finite difference scheme, which I do not want to use. Is there a local way to estimate the f_i^1?

I know this approximation, but I need another definition, since I want to calculate Pi from f_i^1 and thus cannot use Pi for computing f_i^1. And, moreover, I do not want to use a finite difference approach, but only a purely local scheme. And I think that it is not possible to do that.