Finite Difference Methods

Learning Objectives

Approximate derivatives using the Finite Difference Method

Finite Difference Approximation

For a differentiable function $f : R \to R$ , the derivative is defined as

f^{'} (x) = lim_{h \to 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^{″} (ξ) \frac{h^{2}}{2}

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

f^{'} (x) = \frac{f (x + h) - f (x)}{h} - f^{″} (ξ) \frac{h}{2}

Therefore, the truncation error of the finite difference approximation is bounded by $M h / 2$ , where $M$ is a bound on $| f^{″} (ξ) |$ for $ξ$ 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)}{2 h}

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