Find the region of absolute statibility for the Forward Euler method.

Question

One step of Forward Euler method: $u_{j+1} = u_j + hf_j$

Find the region of absolute statibility for the Forward Euler method?

I'm struggling to solve this problem, could I please get a hint?

Answer 1

Let's consider the so-called linear test equation

$$ \frac{du}{dt} = f(u) = \lambda u,$$

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

$$ u_{j+1} = u_j + h f_j = u_j + h\lambda u_j = (1 + h \lambda)u_j $$

Note that

$$ \begin{aligned} u_1 = (1 + h \lambda)u_0 \\ u_2 = (1 + h \lambda)u_1 = (1 + h \lambda)^2u_0 \\ u_3 = (1 + h \lambda)u_2 = (1 + h \lambda)^2u_1 = (1 + h \lambda)^3u_0 \end{aligned} $$ and so on.

For absolute stability, we require

$$ \vert{1 + z\vert \leq 1} $$

where $z = h \lambda \in \mathbb{C} $. It follows that z must lie inside a disk of radius 1 centred 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.