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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!

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    $\begingroup$ What are the boundary/initial conditions? $\endgroup$ – Yuriy S Nov 14 '18 at 17:52
  • $\begingroup$ Any Neumann, Dirichlet or other boundary conditions are good! They would all be useful in understanding how it works. $\endgroup$ – user536082 Nov 14 '18 at 17:57
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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.

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  • $\begingroup$ Thanks that is useful, I can see how I would copmute a forwards step in space using this scheme, however how would I obtain values for F in future times? $\endgroup$ – user536082 Nov 14 '18 at 19:08
  • $\begingroup$ @user536082, but you already know them from solving for $G$? Notice that the index $n$ doesn't change... $\endgroup$ – Yuriy S Nov 14 '18 at 19:09
  • $\begingroup$ Ah I think I understand, just to make sure the n index is the time correct? $\endgroup$ – user536082 Nov 14 '18 at 19:13
  • $\begingroup$ @user536082, yes, it's the time $\endgroup$ – Yuriy S Nov 14 '18 at 19:14
  • $\begingroup$ I'm sorry I thought I understood, but I am still stuck, could you outline a little more how I could get an expression for $$F_{n+1,m}$$ using only information from the timeperiod n. I may be missing something very obvious as I can't see how this is possible based on the previous. $\endgroup$ – user536082 Nov 14 '18 at 20:00

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