When does the moment basis matter in lattice Boltzmann methods?

Abstract

Whether orthogonalising the moment basis changes a central-moment lattice Boltzmann scheme is decided by the geometry of the velocity set. The basis enters the collision only through the conjugate $\mathit{T}^{-1} Λ\mathit{T}$, so two bases related by a constant matrix $\mathit{A}$ define exactly the same scheme if and only if $[Λ,\mathit{A}]_{ij}=\mathit{A}_{ij}(λ_i-λ_j)$ vanishes, and on physical states only its columns for the non-conserved moments matter. We prove that on lattices with velocities in $\{0,\pm1\}^d$ and symmetric weights orthogonalisation reaches the shear moments only through the face and body diagonals: whatever the orthogonalisation, the shear rate is free on D2Q9 and D3Q15, whereas on D3Q19 and D3Q27 the scheme changes whenever the fourth-order moments do not relax at the shear rate. On rectangular lattices, where the symmetry is broken, an independent bulk rate yields an orthogonal counterpart that does not share the transport coefficients of the published scheme, which is itself orthogonal in the inner product of the rest equilibrium up to the one relaxation entry that consistency requires. Where the formulations differ but share their hydrodynamics, a basis orthogonal in the inner product of the rest equilibrium, which guarantees linear stability at rest, was never less robust in the least stable flow direction by more than a few per cent, often much more robust, and the more accurate against a Taylor-Green DNS. Computed through the non-orthogonal transforms with a conjugated relaxation matrix, it costs at most $2\%$ more on D3Q27.