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

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.