I'm trying to develop my knowledge about numerical methods. When studying finite difference methods in the boundary value problem I try to realize my code solving problem I came up with.
I start thinking about easy problem $u''=1, u(0)=0, u'(0)=0, x\in [0,1]$. I took $h=0.1$, and used the second finite difference $$\frac{u_{i-1} -2u_i+u_{i+1}}{h^2}=1, u_0 = 0, \frac{u_1-u_0}{h}=0$$
Having solved the finite difference system on the grid with 10 nodes I noticed that my solution differs with analytical one sharply. I realized then that I actually should have solved initial value problem instead of boundary value problem. Then I used RK4 for solving this problem and it gave good results.
I wonder why finite difference method produce this result with my first formulation of the problem? What is the problem I solved at the first time?
If somebody is interested this question was fully investigated already. My matrix for finite difference method was badly organized as Eliad supposed, so it produced wrong result.
The finite difference method works for this problem well, it converges to the actual solution. Full answer is here.