I would like to solve the equation $$\partial_t u = C(t)\cdot u-u^2$$ using the Crank-Nicolson approach. That resulted in the equations $$\begin{align} \frac{u_1-u_0}{\Delta t}&=0.5\left((C_1u_1-u_1^2)-(C_0u_0-u_0^2)\right)\\ u_1\left(1-\frac{\Delta t}{2}C_1+\frac{\Delta t}{2}u_1\right)&=u_0\left(1+\frac{\Delta t}{2}C_0-\frac{\Delta t}{2}u_0\right) \end{align}$$ Without the square part the solution would be easy, but I am lost with the $u_1$ in the left bracket. How can I fix that? Or is the Crank-Nicolson-Approach not usable for that problem?
Crank-Nicolson for quadratic PDE
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pde
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0You have a non-linear ODE, using an implicit scheme will result in a set of _non-linear_ algebraic equations. – 2017-02-14
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0Which I then have to solve, but how (especially in my case here)? – 2017-02-14
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0You can either $1.$ solve for the roots of your quadratic in $u_{1}$ and determine which root is the true solution, $2.$ use a Newton iteration at each step to solve for the root, or $3.$ you can scrap the Crank-Nicholson method and use an IMEX approach (discretise the linear terms implicitly, the non-linear terms explicitly) instead. – 2017-02-14