I recently appeared in an interview where a few questions were asked regarding numerical instabilities. I was asked the following, though I tried but couldn't convince. Can someone help in this regard. What is numerical instability. Why and when do they arise and how do you address them in equations.
Looking for expert comments.
Dear Deewakar,
As you know, numerical stability is a desirable property of a numerical method. The precise definition of stability depends on the context, but it is derived from the accuracy of the algorithm.While opposite phenomenon is instability. Typically, numerical method would approach the right solution in the range, if there were no round-off or truncation errors, but depending on the specific computational method, errors can be magnified, instead of damped, causing the error to grow exponentially.
In numerical differential equations effect of these errors (like truncation) would be negligible. However, Discretization method is matter of importance.
As Athul said, various concepts of numerical stability would be used ,depending on our problem. For example ''Von Neumann '' stability analysis is a commonly used procedure for the stability analysis of finite difference schemes as applied to linear partial differential equations.
Regards
Only 'implicit' numerical schemes have unconditional stability. Whereas, explicit and sometimes Crank-Nicolson numerical schemes, can give unstable or unrealistic results.
The stability criterion for an explicit scheme is usually written in terms of a maximum value for the numerical time step, for example for a simple 1-dimensional conduction heat equation the stability criterion is given by:
time step < Density*SpecificHeatCapacity*(GridResolution^2) / 2*Conductivity
Where GridResolution is normally known as 'dx' and time step is normally known at 'dt' or Greek letter Delta-t.
A time step larger that this value will lead to an instability (oscillation) in the computation results. This criterion is linked directly to the sign of the coefficient of the central control volume (or node) term in the discretised version of the equation being solved. If this term can become negative then the solution can become unstable.
I recommend the seminal text of Patankar if you are studying heat transfer or fluid flow!
http://www.ewp.rpi.edu/hartford/~ernesto/F2012/CFD/Readings/Patankar-NHTFF-1980.pdf
Hope that helps.
I agree with Robin.The Courant criterion applies for explicit schemes and Crank Nicolson.A good text Malalasekera CFD
- Badges
- Science topic