Notes on Cambridge Maths

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Linear stability

We are considering the problem .
Suppose we have a ODEs with some step size . We define the linear stability domain:

A-stability

We say the method is A-stable when .

Usually when applying the numerical method to this problem, we find that for some (hopefully) analytic function . Then we just need on .

Using the Maximum principle, we find that it is enough to check
and for real :

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