Lattice Boltzmann method with multiple relaxation times require a transformation of the particle distribution function into an equivalent set of moments such that each non-conserved moment can be assigned its own relaxation rate. The choice of the moment set is not unique in serval aspects. For example, moments can be taken in the laboratory frame of reference or in the frame co-moving with the fluid. They can be grouped in different ways, e.g. based on physical association (e.g. heat flux) or according to their symmetry and transformation properties.

Most importantly, moments have to be independent of each other if they evolve at different time scales. Unfortunately, this independence is often only vaguely defined in the LBM literature. Pioneering lattice Boltzmann methods with multiple relaxation rates [Phil. Trans. R. Soc. A 360, 1792 pp. 437-451] use unweighted orthogonalization of the moment base in an attempt to enforce mutual independence. In [JCP 190, 2 pp. 351-370] Dellar presented a weighted orthogonal moment basis but did not address the mathematical consequences of this decision. More recently also non-orthogonal moments, Hermite moments and cumulants have been proposed as candidates.

The choice of the moment basis and in particular the type of orthogonality (if any) has profound and often surprising consequences on stability and accuracy. However, the topic is underrepresented in the scientific literature and educational material on the lattice Boltzmann method. In this course I will demonstrate that unweighted orthogonality is incompatible with certain lattices, in particular in three dimensions. Hermite moments and cumulants lead to favorable types of orthogonality.