Central Difference

Topics: Numerical Methods

Definition

Central Difference is a second-order, finite difference method to numerically approximate the derivative of a function or a dataset. It estimates the derivative using the previous point and the next point as
$\frac{df}{dt} \approx \frac{f(t+\Delta t) - f(t-\Delta t)}{2\Delta t}$

Derivation

Starting from the Taylor series expansion of $f(t+\Delta t)$ and $f(t-\Delta t)$:
$f(t+\Delta t) = f(t) + \Delta t \; \frac{df(t)}{dt} + \frac{\Delta t ^2}{2!} \; \frac{d^2f(t)}{dt^2} + \frac{\Delta t ^3}{3!} \; \frac{d^3 f(t)}{dt^3} + \mathcal{O}(\Delta t ^4)$
$f(t-\Delta t) = f(t) - \Delta t \; \frac{df(t)}{dt} + \frac{\Delta t ^2}{2!} \; \frac{d^2f(t)}{dt^2} - \frac{\Delta t ^3}{3!} \; \frac{d^3 f(t)}{dt^3} + \mathcal{O}(\Delta t ^4)$
Subtracting $f(t-\Delta t)$ from $f(t+\Delta t)$ and dividing by $2 \Delta t$ gives:
$\frac{f(t+\Delta t) - f(t-\Delta t)}{2\Delta t} = \frac{df}{dt} + \frac{\Delta t ^2}{3!} \; \frac{d^3 f(t)}{dt^3} + \mathcal{O}(\Delta t ^3)$
The terms after $\frac{df(t)}{dt}$ form the error of the Central Difference method. This error is $\mathcal{O}(\Delta t^2)$, i.e., it is proportional to $\Delta t^2$. This is why it is a second-order method.

References

1. https://www.youtube.com/watch?v=9fGaTU1-f-0&pp=ygUgc3RldmUgYnJ1bnRvbiBmb3J3YXJkIGRpZmZlcmVuY2U%3D