A Structured Approach to the Construction of Stable Linear Lattice-Boltzmann Collision Operators

Abstract: We introduce a structured approach to the construction of linear BGK-type collision operators ensuring that the resulting Lattice-Boltzmann methods are stable with respect to a weighted $L^2$-norm. The results hold for particular boundary conditions including periodic, bounce-back, and bounce-back with flipping of sign. This construction uses the equivalent moment-space definition of BGK-type collision operators and the notion of stability structures as guiding principle for the choice of the equilibrium moments for those moments influencing the error term only but not the order of consistency. The presented structured approach is then applied to the 3D isothermal linearized Euler equations with non-vanishing background velocity. Finally, convergence results in the strong discrete $L^\infty$-norm highlight the suitability of the structured approach introduced in this manuscript.