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In this paper, we recall the linear version of the lattice Boltzmann schemes in the  framework proposed by d’Humières. According to the equivalent equations we introduce a  definition for a scheme to be isotropic at some order. This definition is chosen such that  the equivalent equations are preserved by orthogonal transformations of the frame. The  property of isotropy can be read through a group operation and then implies a sequence of  relations on relaxation times and equilibrium states that characterizes a lattice  Boltzmann scheme. We propose a method to select the parameters of the scheme according to  the desired order of isotropy. Applying it to the D2Q9 scheme yields the classical  constraints for the first and second orders and some non classical for the third and  fourth orders

Isotropy conditions for lattice Boltzmann schemes. Application to D2Q9