Constructing relaxation systems for lattice Boltzmann methods. (March 2023) - Simonis, Stephan; Frank, Martin; Krause, Mathias J.
Abstract: We present the first top-down ansatz for constructing lattice Boltzmann methods (LBM) in d dimensions. In particular, we construct a relaxation system (RS) for a given scalar, linear, d -dimensional advection–diffusion equation. Subsequently, the RS is linked to a d -dimensional discrete velocity Boltzmann model (DVBM) on the zeroth and first energy shell. Algebraic characterizations of the equilibrium, the moment space, and the collision operator are carried out. Further, a closed equation form of the RS expresses the added relaxation terms as prefactored higher order derivatives of the conserved quantity. Here, a generalized ( 2 d + 1 ) × ( 2 d + 1 ) RS is linked to a D d Q ( 2 d + 1 ) DVBM which, upon complete discretization, yields an LBM with second order accuracy in space and time. A rigorous convergence result for arbitrary scaling of the RS, the DVBM and conclusively also for the final LBM is proven. The top-down constructed LBM is numerically tested on multiple GPUs with smooth and non-smooth initial data in d = 3 dimensions for several grid-normalized non-dimensional numbers. Highlights: First top-down ansatz from d -dimensional PDE to lattice Boltzmann method (LBM). D d Q ( 2 d + 1 ) LBM for advection–diffusion equation via ( 2 d + 1 ) 2 relaxation system. Numerical analysis with closed equation, relaxation stability, and convergence. Algebraic characterization of equilibrium, moment space, and collision scheme. Second order convergence approved for up to 5 . 12 × 1 0 8 cells on multiple GPUs.