Evaluating the inequality in Eq. (7.12), the two possible situations which must hold simultaneously are

Thus,

Since is always positive, this condition always holds.

The other condition is

Thus,

For the above condition to hold

(7.13)

Eq. (7.13) gives the stability requirement for which the solution of the difference Eq. (3.4) will be stable.

It can be said that for a given the allowed value of must be small enough to satisfy Eq. (7.13). For the error will not grow in subsequent time marching steps in t, and the numerical solution will proceed in a stable manner. On the contrary, if then the error will progressively become larger and the calculation will be useless.

The above mentioned analysis using Fourier series is called the Von Neumann stability analysis.

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