Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 38E / b / Solution

For the Fourier mode $e^{i(i\theta +j\varphi )}$, the Forward Euler method has amplification factorIt is real and ranges from $1-4\mu (1+c)$ to $1$. The two-dimensional Fourier stability analysis therefore requires its minimum to be at least $-1$. Thus the exact stable range isThe endpoint is included; the highest-frequency mode then has amplification $-1$ and bounded norm.