For the equation : $u_t + u_x = 0$
If I discritize with finite difference I get :
$$u^{m+1} - u^m + \frac{\Delta t}{2 \Delta x} (u^m_{n+1}-u^m_{n-1}) = 0$$
Where $m$ = time, $n$ is the $x$ axis, $u^0$ is known initial condition, assume boundaries known but I won't discuss them for simplicity.
I know how to solve this with time steps for finite difference, but what if I want to solve this with Newton-Raphson Method? Do I consider $u^m$ when calculating the Jacobean, or just $u^{m+1}$?
if I only consider $u^{m+1}$ then I think the Jacobean becomes an identity matrix and the update step becomes : $$\vec{u^{k + 1}}=\vec{u^k}- \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}^{-1} ( \vec{u^k} -\vec{u^m}+ \frac{\Delta t}{2 \Delta x}(\vec{u^m_{n + 1}} - \vec{\vec{u^m_{n - 1}}}) )$$
Where $\vec{u^k}$ is the current guess for $u^{m+1}$
Ignoring boundary conditions, is this correct? Seems to easy with