# In numerical methods, eg, finite differencing approaches, does there exist convergent schemes that are not both consistent and stable?

In a book that our course is following this semester, the theorem given is only in one direction: if the scheme is both consistent and stable, then the scheme is convergent.

However, since this isn't a double implication theorem, then are there convergent schemes that are either consistent but not stable, or stable but not consistent?

• @KyleW it's all about the limiting situation of small h, in which case Euler is stable (and thus the Lax Equivalence Theorem does show convergence for h sufficiently small). – Chris Rackauckas Feb 20 '17 at 22:01