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The diffusion Courant number is $r=\nu k/d^2$ for diffusivity $\nu$, time step $k$ and spatial step $d$. Keeping $r$ fixed is a parabolic mesh refinement. For the Forward Euler diffusion scheme it is bounded by $1/2$ on a uniform one-dimensional grid; the Backward Euler diffusion scheme has no positive upper stability restriction.

For a smooth solution of the heat equation, backward Euler with the centered second difference has residual divided by $k$ equal to $-(k/2+d^2/12)u_{xxxx}+O(k^2+k{d}^2+d^4)$. Both leading terms have the same sign. Under parabolic mesh refinement their coefficient is $-d^2(r/2+1/12)$, which cannot vanish for a positive diffusion Courant number. Consistency of an update $U^{n+1}-U^n=\alpha (r){\delta }^2{U}^{n+1}$ forces $\alpha (r)=r$ for every fixed positive $r$.