# 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 $$M$$ 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