Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 41E / b / Solution

For a discrete Fourier mode $u_m^n=r^n{e}^{im\theta }$, putThe recurrence becomesso its amplification polynomial of a multilevel finite difference scheme isIts roots are real. Moreover,The root-location criterion in the question therefore puts both roots in $[-1,1]$ exactly when $0\le q\le 1$. Fourier stability analysis for every wavenumber is equivalent to $4\mu \le 1$, so