Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 39A / iii / Solution

The leapfrog finite-difference scheme for the diffusion equation uses centered time and space differences:A Fourier mode with amplification factor $z$ givesWriting $b=4\mu {\sin }^2(\theta /2)$, the roots are $z_{\pm }=-b\pm \sqrt{1+b^2}$. For any $\mu >0$ and any nonzero spatial frequency, $b>0$ andThusAlthough the other root decays, the growing parasitic root is excited by generic starting data or perturbations, so it cannot be discarded in a stability test.