Question

Which of these iterative processes is unstable? (Note: \(\epsilon_n\) is the error in the n^th iteration).

(a) \(\frac{\epsilon_{n+1}}{\epsilon_n} = 0.5\)

(b) \(\frac{\epsilon_{n+1}}{\epsilon_n} = 0.75\)

(c) \(\frac{\epsilon_{n+1}}{\epsilon_n} = 1.25\)

(d) \(\frac{\epsilon_{n+1}}{\epsilon_n} = 1\)

I got this question in semester exam.

I would like to ask this question from Discretization Aspects in chapter Basic Aspects of Discretization, Grid Generation with Appropriate Transformation of Computational Fluid Dynamics

Answer 1

Correct option is (c) \(\frac{\epsilon_{n+1}}{\epsilon_n} = 1.25\)

Explanation: For a system to be stable, the error should be decreasing or at least maintained.

Representing this mathematically,

εn+1 ≤ εn

\(\frac{\epsilon_{n+1}}{\epsilon_n} ≤ 1\)

For an unstable system,

\(\frac{\epsilon_{n+1}}{\epsilon_n} > 1\)

This happens in \(\frac{\epsilon_{n+1}}{\epsilon_n} = 1.25\). So, this system is unstable.