Multistep Methods
Linear multistep methods have the form
Applied to the test equation, they become
which can be simplified to
where z = hk. This is a linear recurrence relation. The method is A-stable if all solutions {yn} of the recurrence relation converge to zero when Re z < 0. The characteristic polynomial is
All solutions converge to zero for a given value of z if all solutions w of Φ(z,w) = 0 lie in the unit circle..
The region of absolute stability for a multistep method of the above form is then the set of all for which all w such that Φ(z,w) = 0 satisfy |w| < 1. Again, if this set contains the left-half plane, the multi-step method is said to be A-stable.
Read more about this topic: Stiff Equation
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—Henry David Thoreau (1817–1862)