Linearization of Partial Differential Equation

Question

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?

Answer 1

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.