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.