In this work, we first develop a unified lattice Boltzmann  (LB) model with the single-relaxation-time collision operator for the $d$($\geq1$)-dimensional linear hyperbolic equation (LHE), where the natural moments and the D$d$Q$(2d^2+1)$ [($2d^2+1$) discrete velocities in $d$-dimensional space] lattice structure are considered. Then, at the acoustic scaling, we perform an accuracy analysis on the developed LB model by the direct Taylor expansion method, and present the second- and third-order moments of the equilibrium distribution functions to ensure that the LB model can be fourth-order consistent with the LHE. In addition, when considering the  Dirichlet boundary condition, the fourth-order full-way and half-way boundary schemes are proposed to determine the unknown distribution functions such that the LB model can achieve a fourth-order accuracy. Subsequently, based on the kinetic entropy theory, we derive the conditions that the fourth-order moments of the equilibrium distribution functions should satisfy to ensure the microscopic entropy stability of the LB model. On the other hand, with the aid of the von Neumann stability analysis, we also discuss the $L^2$ stability of the LB model and numerically plot the stability regions. In particular, from a numerical perspective, we find that the region of the microscopic entropy stability is identical to that of the $L^2$ stability. We carry out some numerical experiments to test the accuracy and stability of the LB model, and the numerical results are in agreement with our theoretical analysis. Finally, we also conduct a comparison of the full-way and half-way boundary schemes for the Dirichlet boundary condition, and find that the full-way boundary scheme is more stable.    

Lattice Boltzmann