Can the method of lines be used to discretize all PDEs?

Question

I have found that the method of lines is a very natural way to think about the discretization of PDE's. Therefore I always default to that mindset when presented with a new set of equations. I have never seen a PDE where this would not work.

What I am wondering is if there are discretization methods (or types of PDEs) which can not be formulated through method of lines. I expect that any PDE where the time derivative is implicit in the equation and can't be solved for would be one such case (although I know of no actual example of this). I am looking for reasoning as to why the method of lines is always applicable or a counter example.

Answer 1

One situation in which the usual method-of-lines approach cannot be used in a straightforward way is with equations that have mixed space-time derivatives.. By "usual method-of-lines approach", I mean discretization of spatial derivatives followed by application of a Runge-Kutta or linear multistep method. This usually applies only to systems of first-order (in time) evolution PDEs.

An example of equations with such mixed derivatives is Eq. (2.1) of http://epubs.siam.org/doi/pdf/10.1137/060676064.

In at least some cases, it is possible to rewrite such equations as first-order systems of evolution PDEs, but I don't immediately see a way to do it here. There may be other tricks to apply the method of lines to such equations, but I don't know of them.