Finite Difference Methods

Learning Objectives

Approximate derivatives using the Finite Difference Method

Finite Difference Approximation

For a differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ , the derivative is defined as

$f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

Let’s consider the forward finite difference approximation to the first derivative as

$f'(x) \approx \frac{f(x+h)-f(x)}{h}$

where $h$ is often called a “perturbation”, i.e., a “small” change to the variable $x$ (small when compared to the magnitude of $x$ ). By the Taylor’s theorem, we can write

$f(x+h) = f(x) + f'(x)\, h + f''(\xi)\, \frac{h^2}{2}$

for some $\xi \in [x,x+h]$ . Rearranging the above we get

$f'(x) = \frac{f(x+h)-f(x)}{h} - f''(\xi)\, \frac{h}{2}$

Therefore, the truncation error of the finite difference approximation is bounded by $M\,h/2$ , where is a bound on $\vert f''(\xi) \vert$ for $\xi$ near $x$ .

Using a similar approach, we can summarize the following finite difference approximations:

Forward Finite Difference Method

In addition to the computation of $f(x)$ , this method requires one function evaluation for a given perturbation, and has truncation order $O(h)$ .

$f'(x) = \frac{f(x+h)-f(x)}{h}$

Backward Finite Difference Method

In addition to the computation of $f(x)$ , this method requires one function evaluation for a given perturbation, and has truncation order $O(h)$ .

$f'(x) = \frac{f(x)-f(x-h)}{h}$

Central Finite Difference Method

This method requires two function evaluations for a given perturbation ( $f(x+h)$ and $f(x-h)$ ), and has truncation order $O(h^2)$ .

$f'(x) = \frac{f(x+h)-f(x-h)}{2h}$

Reference text: “Scientific Computing: an introductory survey” by Michael Heath

Review Questions

See this review link

ChangeLog

2021-01-20 Mariana Silva mfsilva@illinois.edu: Moved FD content from Taylor to this new section