In this paper, based on the idea of directly discrete modeling using equilibrium distribution functions (EDFs), we develop a general framework of the mesoscopic numerical method for macroscopic partial differential equations (PDEs), including but not limited to the nonlinear convection-diffusion equation (NCDE) and the Navier-Stokes equations (NSEs). Unlike the mesoscopic lattice Boltzmann method, this kind of mesoscopic numerical method is an EDF-based mesoscopic finite-difference (MesoFD) scheme. The macroscopic version of the MesoFD (MMFD) method can be derived directly through taking the moments of the MesoFD scheme. Both MesoFD and MMFD schemes are multilevel FD methods and have the form of the central FD scheme. The macroscopic moment equations can be derived from either the MesoFD or MMFD scheme through the Taylor expansion method, and the common PDEs can be recovered from these moment equations by using the direct Taylor expansion method. Furthermore, the stability of the MMFD scheme is analyzed for the linear CDE and linear wave equation with anisotropic diffusion. The stability conditions of a two-level explicit MMFD scheme, a two-level $\theta$-MMFD scheme (hybrid explicit and implicit MMFD scheme) and a three-level MMFD scheme are obtained, respectively. Finally, we note that some existing lattice-Boltzmann-based macroscopic FD schemes for the NSEs and NCDE are the special cases of the present MMFD method, which can be considered as a unified framework of FD schemes for PDEs from this point of view.

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• Lattice-Boltzmann methods