Timestep Control and the CFL Limit

Information Moves One Cell at a Time

An explicit update can move information at most one cell per step, so the Courant number $C = \dfrac{v,\Delta t}{\Delta x}$ must stay at or below one. That bound is the CFL condition.

What Happens Above the Limit

At or below one the front advances cleanly, smeared a little by numerical diffusion. Above one the scheme tries to move the front more than a cell per step, the error compounds, and the solution blows up into growing oscillations.

Why It Costs

A simulator that hits the CFL limit must shrink the timestep to stay stable. That is the cost an explicit (IMPES) scheme pays and a fully implicit one avoids.