Courant number

The Courant number is the dimensionless ratio of physical propagation during one time step to one spatial grid spacing, commonly $\mu =c\Delta t/\Delta x$.

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.

A Courant–Friedrichs–Lewy condition requires the numerical domain of dependence to contain the differential equation's physical domain of dependence. It commonly bounds a ratio such as $c\Delta t/\Delta x$ for an explicit hyperbolic scheme.