I have this beautiful Non-linear PDE $$ \frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2} $$

Where C is a function of (x,t) It comes from the diffusion equation where D is concentration depending, and has the linear form $D=k \cdot C$ and is furthermore made dimensionless.

To solve this numerically I would like to first Linearize the PDE (if this is even possible), because i know how to solve linear PDE's, but how can I do this, is there any methods, articles or books which shows some procedures? or is linearization unnecessary and the PDE can be solved directly?


You can linearize it when you perform time discretisation. If you denote $C_n = C(\cdot, t_n)$, you can consider approximations like

$$ \left(\frac{\partial C_{n+1}}{\partial x}\right)^2 \approx \frac{\partial C_n}{\partial x} \frac{\partial C_{n+1}}{\partial x}, \quad C_{n+1} \frac{\partial^2C_{n+1}}{\partial x^2}\approx C_n \frac{\partial^2C_{n+1}}{\partial x^2}. $$

This will lead to the solution of a linear problem in each time step.

  • $\begingroup$ Thank you, is there any popular theories on linearization of partial differential equation? $\endgroup$ – Peter Panduro Jørgensen May 13 at 10:43

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