= Solution
For the test equation $y'=\lambda y$, write one numerical step as
$$
y_{n+1}=R(h\lambda)y_n.
$$
The <linear stability domain> is
$$
\mathcal S=\{z\in\mathbb C:|R(z)|\leq1\}.
$$
A method is A-stable when $\{z:\Re z\leq0\}\subseteq\mathcal S$.
Forward Euler has $R(z)=1+z$, so
$$
\boxed{\mathcal S_{\rm FE}=\{z:|1+z|\leq1\}}.
$$
This disk does not contain the whole left half-plane, so forward Euler is not A-stable. Backward Euler has $R(z)=(1-z)^{-1}$, so
$$
\boxed{\mathcal S_{\rm BE}=\{z:|1-z|\geq1\}}.
$$
It contains the left half-plane, and backward Euler is A-stable.
Back to article page
