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This figure shows the effects of the courant number on the stability of the numerical scheme.

First-order upwind scheme stability

When the first-order upwind scheme is applied to the one-dimensional time-dependent advection equation, the courant number is formed to indicate the scheme's stability. The scheme is stable if the courant number is less than or equal to one. When the courant number is one, the exact solution is formed. In the figure above, the exact solution is a square pulse with no diffusion. If the courant number is less than one, numerical diffusion will occur, not the actual diffusion that the solution experiences. The figure above shows that the two graphs on the left have some spreading when the actual solution is a square pulse. When the courant number is larger than one, the scheme will create large oscillations and grow unstable, as seen with the graph on the far right.