Implicit Euler Scheme

Question

The evolution equation is given by

$$u_t=u-u^2+u_{xx}$$

on interval $[0,10]$ with the initial condition:

$$u(x,0)=\frac{t}{5}-1$$

I want the form for fully implicit Euler scheme to generate approximate values of $u(x,t)$.

Anyone can help me please?

Answer 1

For the Euler step you have to solve $$ u(x,t-h)=u(t,x)-h\Bigl(u(x,t)-u(x,t)^2+u_{xx}(x,t)\Bigr) $$ as an ODE with boundary conditions for the function $[x\mapsto u(x,t)]$ for fixed $t$ and $h$.

Usually one works with a fixed discretization in $x$ direction, so that this BVP transforms into a collection of coupled quadratic equations. You can also try shooting or multiple shooting using $u(x,t-h)$ to initialize it, as $u(x,t)$ should only have a distance $O(h)$ from it.