Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 41E (Numerical Analysis)

Let ${\delta }_x^2{u}_m=u_{m-1}-2u_m+u_{m+1}$. For an exact smooth solution of the diffusion equation, the second-order central difference givesSince the Courant number $\mu =k/h^2$ is fixed, $h^2=O(k)$. Taylor expansion in time givesUsing $u_t=u_{xx}$ and $u_{tt}=u_{xxt}$ cancels the leading terms. Because $k{h}^2=O(k^2)$, the residual, and hence the local truncation error in the convention of the question, is

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

For the same Fourier mode, the leapfrog finite-difference scheme for the diffusion equation givesThe two amplification roots areFor every nonzero mode $q>0$, one hasThus every $\mu >0$ admits a growing Fourier mode, and