Stability and consistency of kinetic upwinding for advection–diffusion equations

Abstract

Numerical methods based on kinetic models of fluid flows, like the so-called BGK scheme, are becoming increasingly popular for the solution of convection-dominated viscous fluid equations in a finite-volume approach due to their accuracy and robustness. Based on kinetic-gas theory, the BGK scheme approximately solves the BGK kinetic model of the Boltzmann equation at each cell interface and obtains a numerical flux from integration of the distribution function. This paper provides the first analytical investigations of the BGK-scheme and its stability and consistency applied to a linear advection–diffusion equation. The structure of the method and its limiting cases are discussed. The stability results concern explicit time marching and demonstrate the upwinding ability of the kinetic method. Furthermore, its stability domain is larger than that of common finite-volume methods in the under-resolved case, i.e. where the grid Reynolds number is large. In this regime, the BGK scheme is shown to allow the time step to be controlled from the advection alone. We show the existence of a third-order ‘super-convergence’ on coarse grids independent of the initial condition. We also prove a limiting order for the local consistency error and show the error of the BGK scheme to be asymptotically first order on very fine grids. However, in advection-dominated regimes super-convergence is responsible for the high accuracy of the method.