Note on the von Neumann stability of explicit one-dimensional advection schemes

Abstract

There is a wide-spread belief that most explicit one-dimensional advection schemes need to satisfy the condition that the Courant number, c = uΔt/ Δx, must be less than or equal to 1, for stability in the von Neumann sense. This puts severe limitations on the time-step in high-speed, fine-grid calculations and is an impetus for the development of implicit schemes, which often require less restrictive time-step conditions for stability, but are more expensive per time-step. However, it turns out that, if explicit schemes are formulated in a consistent flux-based conservative finite-volume form, von Neumann stability analysis does not place any restriction on the allowable Courant number. Any explicit scheme that is stable for c < 1, with a complex amplitude ratio, G (c) , can be easily extended to arbitrarily large Courant number. The complex amplitude ratio is then given by exp (—ιNθ) G (Δc) , where N is the integer part of c, and Δc = c - N (<1); this is clearly stable. The unity-Courant-number limitation is, in fact, not a stability condition at all, but, rather, a ‘range restriction’ on the ‘pieces’ in a piecewise polynomial interpolation. When a global view is taken of the interpolation, the need for a Courant-number restriction evaporates. A number of well-known explicit advection schemes are considered and thus extended to large Δt. The analysis also includes a simple interpretation of (large Δt) TVD constraints.