Equilibrium-distribution-function based mesoscopic finite-difference methods for partial differential equations: Modeling and Analysis

In this paper, based on the idea of direct discrete modeling (DDM) with equilibrium distribution functions (EDFs), we develop a general framework of the mesoscopic numerical method (MesoNM) 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 MesoNM is an EDF-based mesoscopic finite-difference (MesoFD) method, and by taking the moments of the MesoFD scheme, its macroscopic version, called MMFD method, can be derived directly. Both MesoFD scheme and MMFD schemes are multi-level FD methods, MesoFD scheme being mesoscopic, and MMFD scheme being its macroscopic form which has the form of the central FD scheme. They are unified FD schemes for PDEs and can be in implicit or explicit forms as needed. The macroscopic moment equations (MEs) can be derived from the MesoFD or MMFD scheme through the Taylor expansion method, and the common PDEs can be recovered from the MEs by using the direct Taylor expansion method. Moreover, the stability of the MMFD scheme is analyzed for linear CDE and liner wave equation with anisotropic diffusion, and the stability conditions of a two-level explicit MMFD scheme, a two-level $θ$-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 (LB) based macroscopic FD models for the NSEs and NCDE are the special cases of present MMFD, which can be considered as a unified framework of FD schemes for PDEs, from this point of view.