Derivations of Exact Lattice Boltzmann Evolution Equation

A comparative analysis on the schemes for exact lattice Boltzmann (LB) evolution equation is presented in this paper. It includes two classical exact LB schemes, i.e., Bosch-Karlin (BK) scheme and He-Luo (HL) scheme, and the present Taylor-expansion (TE) scheme. TE scheme originates from the extension of BK scheme. The mathematical mechanism and the equilibrium distribution evolution behind these exact schemes have been detailedly addressed. After that, an analysis is carried out to discuss the cause of the LB equation difference among the schemes, which offers an insight of the exactness in these schemes and brings up their continuity precondition. At last, the schemes are systematically addressed for their pros and cons in the further development of LB equations.