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Linear Lattice Boltzmann Schemes for Acoustic: parameters choices and isotropy properties

In this paper, we investigate the numerous parameters choices for linear lattice Boltzmann schemes according to the definition of the isotropic order given in \cite{ADG11}. This property---written in a general framework including all of the \ddqq schemes---can be read through a group operation. It implies some relations on the parameters of the scheme (equilibrium states and relaxation times) that give rigorous methodology to select them according to the desired order of isotropy. For acoustic applications in two spaces dimensions (namely \ddqn and \ddqt schemes) this methodology is used to propose a full description of the sets of parameters that involve isotropy of order $m$ ($m\in\{1,2,3,5\}$ for \ddqn and $m\in\{1,2\}$ for \ddqt). We then propose numerical illustrations for the \ddqn scheme.