LBM weighting factors and lattice speed

I need to be know where the weights of the force term in LBM come from. i.e. why for the D1Q3 scheme we have 2/3, 1/6 and 1/6; for the D2Q9 scheme we have 4/9,1/9,1/9,1/9,1/9,1/36,1/36,1/36 and 1/36 and so on.
Also I need to know why for the D1Q3 scheme we have Cs=C/sqrt(3) and for D1Q2 it becomes Cs=C/sqrt(2)? The general form appears to be Cs=C/sqrt(D) but why? Thanks for your help in advance.

You can either consider general rules for isotropy together with a Chapman-Enskog expansion, conservation restriction, and specification of the equilibrium momentum flux (sum[f_i(0)c_ic_i]=Pi^0=pressure+rhouu), or discetise the Boltzmann equation using Gauss quadrature.

Thanks for that but honestly I’m an engineer not a physicist or mathematician so your response didn’t help me much. Is there any paper that expalins the origins of those weights and the value of cs?

Try He and Luo’s paper about deriving the LBE from Boltzmann’s equation (1997). There’s a derivation in Wolf-Gladrow’s book, I think, and I imagine in Rothman and Zaleski’s too. You can look at early papers from the mid 90s and before (people like D’humieres and Lallemand, Qian). I think Wagner has some online notes that are not too mathematical and I reckon a lot of theses/dissertations include such details.

Much appreciated for your help.