# 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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Salma Baird
Let's consider the so-called linear test equation
$\frac{du}{dt}=f\left(u\right)=\lambda u,$
where $\lambda \in \mathbb{C}$ is a system parameter which mimics the eigenvalues of linear systems of differential equations.
The Forward Euler discretization will get you
${u}_{j+1}={u}_{j}+h{f}_{j}={u}_{j}+h\lambda {u}_{j}=\left(1+h\lambda \right){u}_{j}$
Note that
$\begin{array}{r}{u}_{1}=\left(1+h\lambda \right){u}_{0}\\ {u}_{2}=\left(1+h\lambda \right){u}_{1}=\left(1+h\lambda {\right)}^{2}{u}_{0}\\ {u}_{3}=\left(1+h\lambda \right){u}_{2}=\left(1+h\lambda {\right)}^{2}{u}_{1}=\left(1+h\lambda {\right)}^{3}{u}_{0}\end{array}$
and so on.
For absolute stability, we require
$|1+z|\le 1$
where $z=h\lambda \in \mathbb{C}$. 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($\lambda$) << −1) or highly oscillatory modes (Im($\lambda$) >> 1) the explicit Euler method demands small step sizes. This makes the method inefficient for these so-called stiff systems.