confusion with a manipulation in a equation

Question

In line(5), I dont understand why the $O(k^2)$ and $O(h^2)$ became linear in the lhs. Is it because

$$ k C + O(k^2) = O(k) $$

where $C = \frac{u_{tt}}{2} $ ??

Answer 1

You have first order terms in $k$ and $h$. Provided enough regularity, equation (5) shows that $$ \tau_j^n = O(k) + O(h) + O(k^2) + O(h^2) = O(k)+O(h). $$