Derive a second order difference approximation

Question

Derive a second order difference approximation to $y(a)$ using the values $$y(a + h/2), \space y(a + h) \space,\space y(a + 2h)$$ Verify the order of your approximation.

Have no idea how to tackle this one. Any help or hints are appreciated.

Answer 1

Have you learned about the method of undetermined coefficients? It's a good place to start:

Roughly speaking, this method looks for constants $\alpha$, $\beta$, and $\gamma$, such that $$ y^{\prime\prime}(a)\approx\alpha y(a+h/2)+\beta y(a+h)+\gamma y(a+2h) $$ Begin by assuming $y$ is smooth (i.e. it has derivatives of all orders) and plug in the power series around $a$ for $y(a+h/2)$, $y(a+h)$, and $y(a+2h)$. Pick $\alpha$, $\beta$, and $\gamma$ such that the zero-th and first equal to $y^{\prime\prime}(a)$ plus higher order derivatives.

For a more complete introduction, see Section 1.2 of http://www.siam.org/books/ot98/sample/OT98Chapter1.pdf.