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I've been trying to find a finite difference scheme for the 1D partial differential equation as follows:

$frac{partial F}{partial t}=frac{partial}{partial x} (( frac{partial F}{partial x})^k)$

However I have not found any material on how to construct one for a non-linear function such as this one. I have tried a few methods however without knowledge of how the steps in space occur in a function such as this I am unsure of their accuracy. Any comment on the stability of such a scheme would also be very useful.

Thanks!

This is how I would try to solve it numerically (mind, I haven't worked with problems like that).

Differentiate w.r.t. $x$:

$$frac{partial}{partial x}frac{partial F}{partial t}=frac{partial^2}{partial x^2} left(left( frac{partial F}{partial x}right)^k right)$$

Change the order of the derivatives on the l.h.s. and introduce a new function:

$$frac{partial F}{partial x}=G$$

Now we are solving:

$$frac{partial G}{partial t}=frac{partial^2}{partial x^2} G^k$$

This could be done with the usual second order finite difference scheme.

Suppose we get a numerical solution $G(x,t) approx G_{nm}$, where the indices are for time and space grid points.

Now we need to solve:

$$frac{partial F}{partial x}=G$$

This is again, a very standard problem for any first order finite difference scheme. In an explicit way, for example:

$$frac{F_{n,m+1}-F_{n,m}}{Delta x}=G_{n,m}$$

Important note: the boundary/initial conditions might change this scheme a little. We should be careful not to lose some solutions or introduce spurious ones, because we have taken an extra $x$ derivative.