One step of Forward Euler method: u_j+1=u_j+hf_j Find the region of absolute statibility for the Forward Euler method?

Tiana Hill 2022-09-30 Answered
One step of Forward Euler method: u j + 1 = u j + h f j
Find the region of absolute statibility for the Forward Euler method?
I'm struggling to solve this problem, could I please get a hint?
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Answers (1)

Salma Baird
Answered 2022-10-01 Author has 8 answers
Let's consider the so-called linear test equation
d u d t = f ( u ) = λ u ,
where λ C is a system parameter which mimics the eigenvalues of linear systems of differential equations.
The Forward Euler discretization will get you
u j + 1 = u j + h f j = u j + h λ u j = ( 1 + h λ ) u j
Note that
u 1 = ( 1 + h λ ) u 0 u 2 = ( 1 + h λ ) u 1 = ( 1 + h λ ) 2 u 0 u 3 = ( 1 + h λ ) u 2 = ( 1 + h λ ) 2 u 1 = ( 1 + h λ ) 3 u 0
and so on.
For absolute stability, we require
| 1 + z | 1
where z = h λ C . It follows that z must lie inside a disk of radius 1 centblack at z = −1 in the complex plane. We call that region the absolute stability of the Forward Euler’s method.

Facit: For stable ODEs with a fast decaying solution (Real( λ) << −1) or highly oscillatory modes (Im( λ) >> 1) the explicit Euler method demands small step sizes. This makes the method inefficient for these so-called stiff systems.
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