On relaxation systems and their relation to discrete velocity Boltzmann models for scalar advection–diffusion equations

Abstract

The connection of relaxation systems and discrete velocity models is essential to the progress of stability as well as convergence results for lattice Boltzmann methods. In the present study we propose a formal perturbation ansatz starting from a scalar one-dimensional target equation, which yields a relaxation system specifically constructed for its equivalence to a discrete velocity Boltzmann model as commonly found in lattice Boltzmann methods. Further, the investigation of stability structures for the discrete velocity Boltzmann equation allows for algebraic characterizations of the equilibrium and collision operator. The methods introduced and summarized here are tailored for scalar, linear advection–diffusion equations, which can be used as a foundation for the constructive design of discrete velocity Boltzmann models and lattice Boltzmann methods to approximate different types of partial differential equations. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

Linear and brute force stability of orthogonal moment multiple-relaxation-time lattice Boltzmann methods applied to homogeneous isotropic turbulence

“…A suitable discrete velocity Boltzmann equation to approximate a given TEQ can be obtained for example by perturbative construction [21] or moment matching [37]. In the case of approximating the incompressible NSE, its weakly compressible counterpart is used as a starting point [45].…”

“…From a mathematically abstracted point of view, LBM belongs to a broader class of methods which is based on a relaxation principle [21,22]. Via the reformulation of the target equations (TEQ), a relaxation system is constructed which exhibits structural advantages compared to the initial partial differential equation system.…”

“…From a historical point of view, the LBM is mostly derived by discretizing a Bhatnagar–Gross–Krook (BGK) simplification of the Boltzmann equation in a discrete velocity space [23–27]. Under specific choices of perturbation terms and generalized Maxwellians [21] the relaxation system coincides with the BGK moment system [28]. A complete discretization then leads to the typical relaxation scheme—or in the BGK setting to the LBM—which under the amalgamation of relaxation and discretization limit converges back to the TEQ [21,22].…”

“…Under specific choices of perturbation terms and generalized Maxwellians [21] the relaxation system coincides with the BGK moment system [28]. A complete discretization then leads to the typical relaxation scheme—or in the BGK setting to the LBM—which under the amalgamation of relaxation and discretization limit converges back to the TEQ [21,22]. The structural advantages gained in both the initial perturbation of the TEQ with relaxation terms as well as the simplistic discretization, offer an increased parallelizability for the implementation on high performance computers [29,30].…”

“…First, though AS constitutes an acceptable compromise between computational effort and desired accuracy from an application-oriented point of view [10], diffusive scaling (DS) is a proven necessity to completely reach the incompressible NSE in the relaxation limit. Thus, to formulate an LBM which is mathematically consistent to a diffusivity-containing TEQ, mesh convergence in DS is required for diminishing all sources of errors in line with the discretization error [21]. With respect to the incompressible NSE as TEQ, this consistency can solely be obtained when particularly including also the incompressibility error.…”