Positivity preservation of implicit discretizations of the advection equation

Abstract

We analyze, from the viewpoint of positivity preservation, certain discretizations of a fundamental partial differential equation, the one-dimensional advection equation with periodic boundary condition. The full discretization is obtained by coupling a finite difference spatial semi-discretization (the second- and some higher-order centered difference schemes, or the Fourier spectral collocation method) with an arbitrary $\theta$-method in time (including the forward and backward Euler methods, and a second-order method by choosing $\theta\in [0,1]$ suitably). The full discretization generates a two-parameter family of circulant matrices $M\in\mathbb{R}^{m\times m}$, where each matrix entry is a rational function in $\theta$ and $\nu$. Here, $\nu$ denotes the CFL number, being proportional to the ratio between the temporal and spatial discretization step sizes. The entrywise non-negativity of the matrix $M$ -- which is equivalent to the positivity preservation of the fully discrete scheme -- is investigated via discrete Fourier analysis and also by solving some low-order parametric linear recursions. We find that positivity preservation of the fully discrete system is impossible if the number of spatial grid points $m$ is even. However, it turns out that positivity preservation of the fully discrete system is recovered for \emph{odd} values of $m$ provided that $\theta\ge 1/2$ and $\nu$ are chosen suitably. These results are interesting since the systems of ordinary differential equations obtained via the spatial semi-discretizations studied are \emph{not} positivity preserving.