5.2 Extrapolated difference

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §5.2. Rewritten in original wording. Continues 5.1 Differentiation. Labs: 5.2b Extrapolated-difference labs.

Extrapolating the error away

A differentiation rule that keeps a finite Taylor chunk also predicts the leading leftover. Combine two central approximations so those leftovers cancel — Richardson extrapolation. Walk $D(h)$, $D(h/2)$, and $D_{\mathrm{ed}}$ — pick the matching equation at each step.

Caveat: high-order stencils on noisy data amplify noise. Smooth or fit first (Chapter 6), then differentiate.

Second derivatives (force from $y(t)$)

Newton: $F=ma=m\,d^{2}y/dt^{2}$. From a measured $y(t)$, approximate $y''$ by nesting central differences (the guide after the reference one-liners walks that stencil).

The compact form is neat but may worsen cancellation by forming the large sum $y(t+h)+y(t-h)$ before subtracting $2y(t)$. Second derivatives need larger optimal $h$ than first derivatives: round-off scales roughly like $\varepsilon_m/h^{2}$.

Truncation vs round-off (recap with numbers)

From Chapter 3, the sweet spot is where $\varepsilon_{\mathrm{app}}\simeq\varepsilon_{\mathrm{ro}}$. Rough models:

$$\varepsilon_{\mathrm{app}}^{\mathrm{fd}}\sim\frac{|y''|h}{2},\qquad \varepsilon_{\mathrm{app}}^{\mathrm{cd}}\sim\frac{|y'''|h^{2}}{24},\qquad \varepsilon_{\mathrm{ro}}\sim\frac{\varepsilon_m}{h}.$$

Equating and taking $|y''|\sim|y'''|\sim 1$ with $\varepsilon_m\sim 10^{-15}$ suggests $h_{\mathrm{fd}}\sim 10^{-8}$ and $h_{\mathrm{cd}}\sim 10^{-5}$, with central error orders of magnitude smaller — the “better” method optimally uses a larger $h$.

On a $\log_{10}|\mathcal{E}|$ vs $\log_{10}h$ plot: large $h$ → algorithmic slope ($\sim 1$ for fd, $\sim 2$ for cd, $\sim 4$ for ed); tiny $h$ → round-off rising as $h$ falls.

Reference one-liners

FD = (y(t+h) − y(t)) / h
CD = (y(t+h/2) − y(t−h/2)) / h
ED = (8(y(t+h/4)−y(t−h/4)) − (y(t+h/2)−y(t−h/2))) / (3h)
D2 = (y(t+h) − 2y(t) + y(t−h)) / h²

Quick check of $D_{\mathrm{ed}}$ and the $y''$ stencil — pick the matching equation at each step.

Quiz

Extrapolation and y″

1 / 3

D_ed = (4 D_cd(h/2) − D_cd(h))/3 is built to…

Pick an answer to see if you are right.

Next: 5.2b Extrapolated-difference labs. Then 5.3 Integration.