On the Convective and Absolute Nature of Instabilities in Finite Difference Numerical Simulations of Open Flows

A linear focusing mechanism for dispersive and non-dispersive wave problems

“…For the case of central spatial discretization schemes the highest value of jGj always appear where group velocity is zero, which is not the case for upwind spatial discretization schemes. Hence the focusing phenomena is not related to absolute instability as suggested in [21]. Fig.…”

“…chosen numerical method of discretization and the error wave-packet is at times focused corresponding to a wavenumber at which numerical group velocity is zero. This has led to authors in [21] to declare such instability as absolute instability. We emphasize that in the cases displayed here, this is not a necessary and sufficient condition of all spectacular error growth.…”

Weakly nonlinear interaction of mixed convection patterns in porous media heated from below

“…As it is well known, the CrankNicholson scheme is always numerically stable when the solution is physically stable. However, Cossu and Loiseleux [29] showed that when the solution becomes physically unstable, a Crank-Nicholson scheme with a spatial grid size of x and a time resolution of t may experience a numerical transition from absolute to convective instability and vice versa. In the framework of the Landau-Ginzburg equation, these authors derived conditions on the discretization parameters x and t in order to avoid such numerical transition.…”

“…In the framework of the Landau-Ginzburg equation, these authors derived conditions on the discretization parameters x and t in order to avoid such numerical transition. Therefore, our numerical experiments are performed with x sufficiently small in order to respect the criterium derived in [29]. The length of the porous medium is imposed to be 50.…”

A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow

“…A similar concept was used by Tam & Webb [38] in creating Dispersion Relation Preserving (DRP) finite-difference schemes, where the finite-difference stencils were designed so that the numerical dispersion relation was a good approximation to the continuous dispersion relation of the continuous system being solved. More recently, Cossu & Loiseleux [13] considered the dispersion relation of three finitedifference schemes for solving the linearised Ginzburg-Landau equation, and interpreted their results in terms of convective and absolute instability by considering the numerical group velocity. However, each of these studies performed a dispersion analysis of only the spatial and temporal finite-difference schemes.…”