A linear thermal stability analysis of discretized fluid equations

Abstract

The effects of discretization on the equations, and their solutions, describing Rayleigh–Bénard convection are studied through linear stability analysis and numerical integration of the discretized equations. Linear stability analyses of the discretized equations were conducted in the usual manner except that the assumed solution contained discretized components (e.g., spatial grid interval in the x direction, ${\Delta x}$). As the resolution became infinitely high (${\Delta x \rightarrow 0}$), the solutions approached those obtained from the continuous equations. The wavenumber of the maximum growth rate increased with increasing ${\Delta x}$ until the wavenumber reached a minimum resolvable resolution, ${\pi \Delta x^{-1}}$. Therefore, the discretization of equations tends to reproduce higher-wavenumber structures than those predicted by the continuous equations. This behavior is counter intuitive and opposed to the expectation of ${\Delta x}$ leading to blurred simulated convection structures. However, when the analysis is conducted for discretized equations that are not combined into a single equation, as is the case for practically solved numerical models, the maximum growing wavenumber rather tends to decrease with increasing ${\Delta x}$ as intuitively expected. The degree of the decrease depends on the discretization accuracy of the first-order differentials. When the accuracy of the discretization scheme is of low order, the wavenumber monotonically decreases with increasing ${\Delta x}$. On the other hand, when higher-order schemes are used for the discretization, the wavenumber does increase with increasing ${\Delta x}$, a similar trend to that in the case of the single-discretized equation for smaller ${\Delta x}$.