Stability analysis in the unsteady problem

How does an unstable solution look like?

If the error obtained at one time step of the calculation causes the errors to be magnified as the computations are continued, then the solution is said to be unstable.

If the error eventually die out without any further magnification, then the solution is said to be stable.

If the errors remain constant as the computations are carried out, then the solution is said to be neutrally stable.

1. The factor by which the stability of the explicit and implicit form varies is totally different.

2. The stability of the explicit form of the 2D heat conduction equation is dependent on the CFL number, r = alpha*dt/(dx^2), which should be less than or equal to 0.25. The figure below is an example for the unstable condition of the explicit form at which the CFL number is equal to 0.40500.

3. Implicit form is stable even for large values of dt, and the value of \'r\' wont affect the stability of the solution. This is beacuse implicit schemes use the entire domain to calculate each timestep. The drawback of implicit form is, eventhough it is stable for large values of dt, the time accuracy decreases as the value of time step increases.