 Diffusion Courant number

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.