Accuracy

Scheme (3.18) employs only one difference operator, a second-order accurate approximation $\delta _{tt}$ to the operator $\frac {d^2}{dt^2}$. While it might be tempting to conclude that the scheme itself will generate a solution which converges to the solution of (3.1) with an error that depends on $k^2$ (which is in fact true in this case), the analysis of accuracy of an entire scheme is slightly more subtle than that applied to a single operator in isolation. It is again useful to consider the action of the operator $\delta _{tt}$ on a continuous function $u(t)$. In this case, scheme (3.18) may be rewritten as

$$\left(\delta_{tt}+\omega_{0}^2\right)u(t) = 0$$ (3.32)

or, using (2.6),

$$\left(\frac{d^2}{dt^2}+\omega_{0}^2\right)u(t) = O(k^2)$$ (3.33)

The left hand side of the above equation would equal 0 for $u(t)$ solving (3.1), but for approximation (3.18), there is a residual error on the order of the square of the time step $k$.

In general, the accuracy of a given scheme will be at least that of the constituent operators; an interesting case study is presented in §3.3.4.