Lax Equivalence Theorem

Suppose we have a numerical method

where is the spatial step, while superscripts denote time steps. Then this converges in norm if and only if it is:

1. Consistent
2. Stable

Proof ()

Fix a norm (any norm) and Then since:

we get that the method is stable if and only if

Suppose the method is also consistent. Then, assuming , we have

for some constant and local truncation errors Since , we get , which shows convergence.