Numerical Solution of non-linear diffusion equation using Finite Differencing

Question

I'm trying to solve the following non-linear diffusion equation: $$ \frac{\partial}{\partial t} u(x,t)= \frac{\partial^{2}}{\partial x^{2}}u(x,t)^{3}$$ $$ -1\leq x \leq1, t \geq 0 $$ with the boundary conditions: $$u(1,t)=-u(-1,t)=1$$ and the initial conditions: $$u(x,0)=x$$ I'm trying to implement a Crank-Nicolson scheme but I'm a little stuck. Here's what I have so far: $$\frac{u_{i}^{n+1} - u_{i}^{n}}{\Delta t} = \frac{(u^{3})_{i-1}^{n+1} - 2(u^{3})_{i}^{n+1} + (u^{3})_{i+1}^{n+1} + (u^{3})_{i-1}^{n} - 2(u^{3})_{i}^{n} + (u^{3})_{i+1}^{n}}{2\Delta x^2}$$ I'm not sure how to evaluate $$(u^3)_i^{n+1}$$ How should I proceed?

Edit: Forgot to mention that the analytical solution is: $$u(x,\infty) = x^{\frac{1}{3}}$$

Answer 1

You should rearrange the terms so that all of the $n+1$ terms are together on one side of the equals sign and all of the $n$ terms are on them other. Then you will have a system of non-linear equations like:

$$ A(u^{n+1}_i) u^{n+1}_i=b $$

Where $A$ is a non-linear matrix. Then you can use something like Newton-Raphson to linearize the system and your favorite linear solver to solve the resulting linear systems at each Newton step.