For the same Fourier mode, the <leapfrog finite-difference scheme for the diffusion equation> gives
$$
r^{n+1}=r^{n-1}-2qr^n,
\qquad
p(r)=r^2+2qr-1.
$$
The two amplification roots are
$$
r_\pm=-q\pm\sqrt{1+q^2}.
$$
For every nonzero mode $q>0$, one has
$$
|r_-|=q+\sqrt{1+q^2}>1.
$$
Thus every $\mu>0$ admits a growing Fourier mode, and
$$
\boxed{\text{the leapfrog scheme is unstable for every }\mu>0.}
$$

Solution

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

Applying a centered leapfrog step in time and the centered second difference in space to $u_t=u_{xx}$ gives
$$
u_m^{n+1}=u_m^{n-1}+2\mu(u_{m-1}^n-2u_m^n+u_{m+1}^n).
$$
For every $\mu>0$, each nonconstant Fourier mode has an amplification root of modulus greater than one, so the scheme is unconditionally unstable.


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