Linear lattice Boltzmann schemes for acoustic: Parameter choices and isotropy properties

In this paper we investigate the numerous possible parameter choices for linear lattice Boltzmann schemes according to the definition of the order of isotropy given in Augier et al. (in press) [4]. This property–written in a general framework that includes all of the DdQq schemes–can be understood through a group operation. The resulting relationships between the parameters of the scheme (defining the equilibrium states and relaxation times) yield a rigorous methodology that should be followed if one is to ensure isotropy at a given order. For acoustic applications in two spacial dimensions (namely D2Q9 and D2Q13 schemes) this methodology is used to propose a full description of the sets of parameters that are isotropic at order mm (m∈{1,2,3,4,5}m∈{1,2,3,4,5} for D2Q9 and m∈{1,2}m∈{1,2} for D2Q13). We include numerical illustrations for the D2Q9 scheme obtained with the code LBMpy developed in the laboratory of Mathematics of the University Paris-Sud.