Given the following Euler-Bernoulli equation:
$$ (s(x) w(x)'')''= q(x),\ \ x \in [0,1]$$
Could someone explain why the following discretization scheme may not be a good idea?
\begin{align*} (sw'')''(x_i) &\approx \frac { (sw'')({ x }_{ i+1 })-2(sw'')({ x }_{ i })+(sw'')({ x }_{ i-1 }) }{ { h }^{ 2 } } \\ &= \frac { s({ x }_{ i+1 })w''({ x }_{ i+1 })-2(s)({ x }_{ i })w''({ x }_{ i })+(s)({ x }_{ i-1 })w''({ x }_{ i-1 }) }{ { h }^{ 2 } } \\ w''(x_i) &\approx \frac { w(x_{ i+1 })-2w(x_{ i })+w(x_{ i-1 }) }{ { h }^{ 2 } } \end{align*}
Where $x_0 = 0 , x_n = 1$ and the spacing between two consecutive x is h.
Thank You.