Constructing relaxation systems for lattice Boltzmann methods

TL;DR

This paper introduces a novel top-down approach to construct lattice Boltzmann methods (LBM) for d-dimensional scalar advection-diffusion equations, providing a rigorous convergence analysis and numerical validation on GPUs.

Contribution

It presents the first top-down ansatz linking relaxation systems to discrete velocity Boltzmann models for constructing LBMs in arbitrary dimensions.

Findings

Achieves second order accuracy in space and time.

Provides a rigorous convergence proof for the constructed LBMs.

Successfully tests the method on GPU implementations with various initial data.

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 (2d+1)x(2d+1) RS is linked to a DdQ(2d+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.