High-order regularization in lattice-Boltzmann equations

Abstract

A lattice-Boltzmann equation (LBE) is the discrete counterpart of a continuous kinetic model. It can be derived using a Hermite polynomial expansion for the velocity distribution function. Since LBEs are characterized by discrete, finite representations of the microscopic velocity space, the expansion must be truncated and the appropriate order of truncation depends on the hydrodynamic problem under investigation. Here we consider a particular truncation where the non-equilibrium distribution is expanded on a par with the equilibrium distribution, except that the diffusive parts of high-order non-equilibrium moments are filtered, i.e. only the corresponding advective parts are retained after a given rank. The decomposition of moments into diffusive and advective parts is based directly on analytical relations between Hermite polynomial tensors. The resulting, refined regularization procedure leads to recurrence relations where high-order non-equilibrium moments are expressed in terms of low-order ones. The procedure is appealing in the sense that stability can be enhanced without local variation of transport parameters, like viscosity, or without tuning the simulation parameters based on embedded optimization steps. The improved stability properties are here demonstrated using the perturbed double periodic shear layer flow and the Sod shock tube problem as benchmark cases.

Central-moment-based Galilean-invariant multiple-relaxation-time collision model

“…Similar improvements was indeed achieved by the "regularized" models which trim the under-resolved moments [10][11][12]25]. More recently, the regularization approach was extended to high-order LB [13,14,26], leading to further enhanced numerical stabilities.…”

“…Aiming at eliminating, or at least alleviating, these deficiencies, a number of efforts have been made to improve the collision model, including the multiple-relaxation-time (MRT) model [7,8] and its central-moment (CM) version [9], the "regularized" models [10][11][12][13][14], and the Hermite expansion based high-order MRT model [15,16]. These models, suggested with their own purposes and assumptions, all enjoyed success of various degrees and shared the commonality that the moments of the distribution are individually manipulated.…”

Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations

“…Apart from that, there is no fundamental reason for the use of central moments instead of raw ones. Furthermore, formulas for post collision populations are more complex in the CM framework (73b) than in the RM one (71). In the 2D case, it is then not clear if the use of central moments should be preferred or not.…”

A block triple-relaxation-time lattice Boltzmann model for nonlinear anisotropic convection–diffusion equations

“…Recently, Xiang et al [56] introduced a tunable parameter β to keep the dimensionless relaxation time τ away from 0.5 such that the stability of LBGK model can be improved.However, it is not convenient to choose a proper β, and the improvement of stability is not significant. Different from aforementioned models that introduce some additional parameters, a series of regularized lattice Boltzmann models (RLBMs) for fluid dynamics have also been proposed [57,58,59,60]. The main idea of RLBM is to regularize the pre-collision distribution functions so as to achieve better accuracy and stability.…”

“…">Regularized lattice Boltzmann modelIn the standard LBM, the equilibrium distribution function is projected onto a truncated Hilbert subspace H q which is spanned by a series of Hermite polynomials, while the non-equilibrium part is not. The regularization procedure work this out by projecting the relevant non-equilibrium moments onto the same subspace H q while filtering out the nonhydrodynamic moments [60]. With the aid of regularization procedure, both the equilibrium and non-equilibrium effects are limited to the subspace so that the stability of model is improved.…”