  In this paper, we first present the general propagation
multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the
corresponding macroscopic finite-difference (GPMFD) scheme on conservative
moments. Then based on the Maxwell iteration method, we conduct the analysis on
the truncation errors and modified equations (MEs) of the GPMRT-LB model and
GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear
anisotropic convection-diffusion equation (NACDE) and Navier-Stokes equations
(NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model
and GPMFD scheme. In particular, for the one-dimensional convection-diffusion
equation (CDE) with the constant velocity and diffusion coefficient, we can
develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding
fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, two
benchmark problems, Gauss hill problem and Poiseuille flow in two-dimensional
space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found
that the numerical results are not only in good agreement with corresponding
analytical solutions, but also have a second-order convergence rate in space.
Additionally, a numerical study on one-dimensional CDE also demonstrates that
the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in
space, which is consistent with our theoretical analysis.
