A unified MRT-LB framework for Navier-Stokes and nonlinear convection-diffusion equations and beyond: moment equations, auxiliary moments, multispeed lattices, and Hermite matrices

TL;DR

This work presents a unified RMRT-LB framework for simulating Navier–Stokes and nonlinear convection–diffusion equations on rectangular multispeed lattices, grounded in discrete Hermite matrices to enforce weighted orthogonality. Macroscopic NSE/NCDE moment equations are derived via direct Taylor expansion, and recovery of target equations relies on carefully chosen fundamental moments plus auxiliary moments (notably M2G for the source term and M30 for NSEs, as well as M1G and M20 for NCDEs). The framework yields a family of multispeed lattices (e.g., rD2Q25, rD3Q53, and variants) with generalized third-order EDFs and rectified Hermite matrices to ensure isotropy on rectangular grids. By unifying equilibrium, auxiliary, and source distributions through Hermite representations, the approach extends MRT-LB to compressible, incompressible, single- and multiphase flows, and NCDEs, offering a principled path to higher-order accuracy and broader applicability in fluid and transport simulations.

Abstract

We develop a unified multi-relaxation-time lattice Boltzmann (MRT-LB) framework based on discrete Hermite polynomials (Hermite matrices) for the Navier-Stokes equations (NSEs) and nonlinear convection-diffusion equations (NCDEs), using multispeed rectangular lattice (rD$d$Q$b$) models. For NSEs, the proposed MRT-LB model simulates incompressible and compressible isothermal flows in both single-phase and multiphase systems. Macroscopic moment equations are derived from the MRT-LB model via the direct Taylor expansion method. By selecting appropriate fundamental moments, the target NSEs and NCDE are recovered from these moment equations. Critically, the elimination of spurious terms and/or the recovery of the desired terms relies on specific auxiliary moments: the second-order auxiliary moment ($\mathbf{M}_{2G}$) of the source term distribution function (SDF) and the third-order auxiliary moment ($\mathbf{M}_{30}$) of the equilibrium distribution function (EDF) for NSEs, as well as the first-order auxiliary moment ($\mathbf{M}_{1G}$) of the SDF and the second-order auxiliary moment ($\mathbf{M}_{20}$) of the EDF for NCDE. Furthermore, using the weighted orthogonality of Hermite matrices, we establish essential relations for weight coefficients and construct several multispeed rectangular lattice models, including rD2Q25 and rD3Q53, with subgroup models rD2Q21, rD2Q17, rD2Q13, rD3Q45, and rD3Q33. A generalized third-order equilibrium distribution function is derived. We emphasize that for rectangular lattices, specific elements of the Hermite matrix corresponding to third-order discrete Hermite polynomials require correction to satisfy weighted orthogonality.

Figures (1)

• Figure 1: Modeling and elements of MRT-DVBE/LBE, where $f$, $f^{eq}$, $\mathbf M$, $m$, $\Omega$, $\mathbf F$ and $\Lambda$ are the distribution function (DF), equilibrium, transform matrix, moments, collision term, source DF and collision matrix, respectively.