I have a differntial equation $du/dx=f(x,u)$, where
$$f(x,u)=\frac{d\psi}{dx}+p(u-\psi(x))^4,$$
where $\psi$ is a given function of $x$ independent of $u$.
The backward Euler Method written for step $i$ is
$$ u_{i+1} = u_i+hf(x_{i+1},u_{i+1}). $$
So, at every step we have a 4th degree polynomial to solve for $u_{i+1}$, which can have more than one root. I am told to employ a root-finding algorithm. But I don't understand how to find the one correct root and reject all others. Bisection, Newton's and other methods seem to only work when there is one root in an interval. Would appreciate any help!
Best Answer
If $h$ is small and $f$ is Lipschitz you will have that $u_{i+1}$ is close to $u_i$. Hence, your first guess for the Newton algorithm ($u_i$ itself) will be always "close enough" to the fixed point for Newton's method to work in practice.