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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?

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  • $\begingroup$ @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). $\endgroup$ – Chris Rackauckas Feb 20 '17 at 22:01
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What you're thinking about is the Lax Equivalence Theorem. It is a double implication (if and only if). You can prove generally using a Functional Analysis approach, though that might be more in depth and "further from practical methods" than an approach most classes would take, which is probably why only the forward implication was proved.

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  • $\begingroup$ the iff (i.e., Lax Equivalence) holds for (well-posed) linear PDE problems only. For nonlinear PDEs, stability and consistency do not necessarily imply convergence; conversely, convergence does not guarantee stability for this more general class of problems. $\endgroup$ – GoHokies Feb 22 '17 at 16:46
  • $\begingroup$ But you can extend it to the case where the non-linear part is Lipschitz, right? I think I remember something along those lines, or would have an idea on how to do so in my mind. But I don't know where such a result would be. $\endgroup$ – Chris Rackauckas Feb 22 '17 at 18:09
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    $\begingroup$ are you perhaps referring to nonlinear ODEs? there one can prove convergence for the usual schemes under the assumption that the RHS is Lipschitz-continuous, but for PDEs it is not that easy. There's a very nice discussion of this in Randall Leveque's Finite difference methods book (section 9.5.1). $\endgroup$ – GoHokies Feb 22 '17 at 20:05
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In general, I agree with the previous answer by Chris Rackauckas. However, to be more precise, the answer depends on the definition of stability and convergence.

In Stetter "Analysis of discretization methods for ordinary differential equations", in chapter 1, there are simple examples for schemes which are convergent, consistent but not stable or stable but not convergent. They are in a sense degenerate because a bad choice for the spaces is made.

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