B1 - representing continuous functions and derivatives

considering a continuous function, $f (x)$
the domain can be discretised to a grid of $N$ points, separated by $Δ x$ , such that $x_{k} = k Δ x$ , and $f_{k} = f (x_{k})$ , where $k \in [0, N - 1]$

image: B Hnat, lecture notes

the derivative can be approximated as the slope of a line passing through two neighbouring points, given a smooth function and a fine grid
in the above image, the derivative is well represented on the left, but not on the right as the grid is not fine enough for the rapidly varying function
the derivative is defined as:

\frac{d f}{d x} |_{x = x_{k}} = \frac{f_{k + 1} - f_{k}}{Δ x}

considering a 1D initial value problem with the differential equation: $\frac{d f}{d x} = F (x)$ , and $f (x_{0}) = X$

\begin{matrix} F (x_{k}) = \frac{f_{k + 1} - f_{k}}{Δ x} \\ ⟹ f (x_{k + 1}) = f (x_{k}) + Δ x F (x_{k}) \end{matrix}

eg: $d_{x} f = \cos x$ , $f (0) = 0$

image: B Hnat, lecture notes

as $Δ x$ increases, the accuracy deteriorates
the expression $f (x_{k + 1}) = f (x_{k}) + Δ x F (x_{k})$ is the simplest example of the finite difference method