I discover this phrase in a journal and I would like to know what is this nonlinear instability is about?

I define it as the occurrence of numerical oscillation in the solution of a set of partial differential equations due to the fact that these partial differential equations are nonlinear, am I right?

i do not believe that nonlinear stability has anything to do with NUMERICAL oscillations.

in principle there is no such thing as a well defined nonlinear stability (or instability for that matter). we can only speak of stability in the linear sense. however, this term does float around and one is essentially referring to second (or third) order perturbation analysis. many systems are amenable to this analysis and show remarkably different solutions at even second order.

THe instabilities (linear or non-linear) refer to physical instabilities and not numerical.

-1) A linear instability is when the flow is unstable to an infinitesimally small perturbation, which means that if you make a simulation of this flow, the flow will always be unstable, no matter what is the initial condition, and how far is the initial condition from the steady solution. Even if you put in the steady solution, because of numerical small errors, it will be unstable (it is like trying to put a sharp pen vertically up side down on a table, it will always fall).

-2) A non linear instability is when the flow is unstable only when the perturbation is larger than a given value (this value usually might be an increasing function of the viscosity - for a very viscous flow you will need a strong perturbation to make the flow unstable, while for a small perturbation you will need only a small perturbation). It is like puting a pen (with a flat end) vertically on a table - if you push enough, it will fall down.

-3) Usually, these can be inferred from first order linear perturbation theory. Analytically you can predict the instability in its linear regime (where the perturbation grows exponentially as a function of time), but you need numerical simulations to follow the evolution of the instability in the non linear regime (when the perturbation starts to saturate and different modes interact to transfer energy between different scales for example).

(3) Linear and non linear regimes of an instablity should not be confused with the intrinsic property of an instability which can be itself linear (1) of non-linear (2) !!

Linear can refer to the (first) order pertubation analysis (e.g. linear analysis of the stability of the flow - an analytical analysis); it can refer to the instability itself or to something else even.

Example: a flow can be linearly stable, which means that if left alone as it is, it will stay that way for ever. The same linearly stable flow can however be non-linearly unstable to a given perturbation of a given size (finite amplitude perturbation, non-linear instability). ONce the flow is perturbed with the finite-amplitude perturbation it will be unstable and the perturbation will growth exponentially. During the early phase of the growth, the perturbation grows exponentially, and one can use linear (first order) analysis to find an analytical expression for the growth rate, etc.... Later, as the flow grows more unstable, the linear analysis is not valid any more and one needs to follow the instablity in its non-linear regime (when the grow rates do not follow anymore any exponential growth or decay, but vary with time in different manners) using numerical simulations.

I hope I made myself clear,

Cheers,

Patrick Godon

What should be responsible for the NUMERICAL oscillations?

How does mesh refinement affect NUMERICAL oscillations? Does Time increment or relaxation factor has anything to do with NUMERICAL oscillations?

Thanks in advance.