Constructing relaxation systems for lattice Boltzmann methods

We present the first top-down ansatz for constructing lattice Boltzmann methods (LBM) in d dimensions. In particular, we construct a relaxation system (RS) given scalar, linear, d-dimensional advection–diffusion equation. Subsequently, RS is linked to discrete velocity model (DVBM) on zeroth and energy shell. Algebraic characterizations of equilibrium, moment space, collision operator are carried out. Further, closed equation form expresses added terms as prefactored higher order derivatives conserved quantity. Here, generalized (2d+1)×(2d+1) DdQ(2d+1) DVBM which, upon complete discretization, yields an LBM with second accuracy space time. A rigorous convergence result arbitrary scaling RS, conclusively also final proven. The constructed numerically tested multiple GPUs smooth non-smooth initial data d=3 dimensions several grid-normalized non-dimensional numbers.