Leapfrog finite-difference scheme for the diffusion equation
= Leapfrog finite-difference scheme for the diffusion equation
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.