I'm practicing with numerical schemes, and derived the linear spatial FEM scheme of the ODE $u_{x} \left( x \right) = f \left( x \right)$ on $a \le x \le b$ given $u \left( a \right) = U_{0}$. The scheme was linearized and outputs all interpolants $U_{0}, U_{1}, \cdots , U_{n}$. The first row of the scheme is as follows (trivially given):
$$U_{0}* \left( 1 \right) =U_{0}$$
The next rows $n-1$ rows follow the same pattern. For $i \in \{ 1, 2, \cdots , n-1 \}$ the scheme is as follows (on a uniform mesh for simplicity):
$$U_{i-1}*\left( -\frac{1}{2} \right) + U_{i+1}* \left( \frac{1}{2} \right) = \int^{x_{i}}_{x_{i-1}} f \left( x \right) * \frac{x-x_{i-1}}{h} dx \ + \int^{x_{i+1}}_{x_{i}} f \left( x \right) * \frac{x_{i+1}-x}{h} dx$$
For the last row, I applied a scheme transformation using $U_{n}ā\frac{U_{n+1}+U_{n-1}}{2}$ to get:
$$U_{n-1}* \left( -1 \right) + U_{n} * \left( 1 \right) = \int^{x_{n}}_{x_{n-1}} f \left( x \right) * \frac{x-x_{n-1}}{h} dx$$
I tested the scheme and got convergence order $pā1.5$ from $p=LOG_{2} \left( \frac{error_{h}}{error_{\frac{h}{2}}} \right)$ with explicit values $f \left( x \right) = \sin \left( x \right)$, $a = -\pi$, $b = \pi$, $U_{0}=1$, and $n \in \{ 100, 200, 400, 800 \}$ of course implying $h = \frac{b-a}{n}$.
Regarding stability and consistency analyses assuming I don't have the true solution to obtain errors from, how would I go about showing my scheme is stable and consistent if it even is?
In trying to answer this myself I also came across stability analysis for spatial versus temporal ODEs is performed differently. Would it be too much to ask on some more details on that as well?
