Numerical methods for non-linear diffusion

Question

I have the following non-linear diffusion equation, for $\ z(x,t)$:

$\ z_t = -C(\sin(\omega t))^m x^{hm}(hm x^{-1}(z_x)^n + n z_{xx} (z_x)^{n-1}) $

Any advice for numerical (or analytical) solutions?

Answer 1

In fact, your equation is a non-linear advection-diffusion. Due to the fact that your problem is time-dependent, it could be easily solved by finite-difference:

$$\frac{\partial z}{\partial t} = -C (\sin(\omega t))^{m} x^{hm} n (\frac{\partial z}{\partial x})^{n-1} \frac{\partial^{2} z}{\partial x^{2}} -C (\sin(\omega t))^{m} x^{hm} h m x^{-1} (\frac{\partial z}{\partial x})^{n-1} \frac{\partial z}{\partial x}$$

$$D(x,t,\frac{\partial z}{\partial x}) = C (\sin(\omega t))^{m} x^{hm} n (\frac{\partial z}{\partial x})^{n-1}$$

$$u(x,t,\frac{\partial z}{\partial x}) = -C (\sin(\omega t))^{m} x^{hm} h m x^{-1} (\frac{\partial z}{\partial x})^{n-1}$$

$$\frac{\partial z}{\partial t} = -D(x,t,\frac{\partial z}{\partial x}) \frac{\partial^{2} z}{\partial x^{2}} + u(x,t,\frac{\partial z}{\partial x}) \frac{\partial z}{\partial x}$$

$$\frac{z^{t+\delta t}_{x} - z^{t}_{x}}{\delta t} = -D(x,t,\frac{z^{t}_{x+\delta x}-z^{t}_{x-\delta x}}{2\delta x}) \frac{z^{t}_{x+\delta x}+z^{t}_{x-\delta x}-2 z^{t}_{x}}{\delta x^{2}} + u(x,t,\frac{z^{t}_{x+\delta x}-z^{t}_{x-\delta x}}{2\delta x})\frac{z^{t}_{x+\delta x}-z^{t}_{x-\delta x}}{2\delta x}$$

Or finally your update equation is:

$$z^{t+\delta t}_{x} = z^{t}_{x} + \delta t (-D(x,t,\frac{z^{t}_{x+\delta x}-z^{t}_{x-\delta x}}{2\delta x}) \frac{z^{t}_{x+\delta x}+z^{t}_{x-\delta x}-2 z^{t}_{x}}{\delta x^{2}} + u(x,t,\frac{z^{t}_{x+\delta x}-z^{t}_{x-\delta x}}{2\delta x})\frac{z^{t}_{x+\delta x}-z^{t}_{x-\delta x}}{2\delta x})$$

You can close your equation by specifying appropriate boundary conditions and play around with different $\delta x$ and $\delta t$ for your spatial and temporal discretization to ensure stability. If you define a local Peclet number as: $Pe = \frac{u x}{D}$, the Peclet number of your non-linear advection-diffusion equation would be calculated as: $$Pe = \frac{u(x,t,\frac{\partial z}{\partial x}) x}{D(x,t,\frac{\partial z}{\partial x})} = -\frac{hm}{n}$$

As a result, the proportion of advection over diffusion is constant and is equal to $-\frac{hm}{n}$, which might be useful for identifying the stable regions.