Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 40E / a / Solution

For the discrete Fourier mode $u_m^n=G^n{e}^{im\theta }$, the centred second difference has multiplier $-4{\sin }^2(\theta /2)$. Hence the amplification factor isFor every $\mu >0$, $|G(\theta )|\le 1$. The discrete Parseval identity then shows directly that the discrete $L^2$ norm cannot increase at any time step, proving unconditional stability for the Cauchy problem.