First and second order finite difference schemes

A finite difference scheme is of order $p$ if its truncation error (the terms dropped from the Taylor series) is proportional to $\Delta x^p$ (or $\Delta t^p$). Higher order means the error falls faster when the grid is refined.

First-order accurate schemes

From Taylor's series,

$$\frac{f_{i+1}-f_i}{\Delta x}=f'_i+\frac{\Delta x}{2}f''_i+\dots\ \Rightarrow\ f'_i=\frac{f_{i+1}-f_i}{\Delta x}+O(\Delta x)$$

The forward difference $(f_{i+1}-f_i)/\Delta x$ and the backward difference $(f_i-f_{i-1})/\Delta x$ are first-order accurate. Applied to $\partial Q/\partial t+c\,\partial Q/\partial x=0$ with forward time and backward space:

$$\frac{Q_i^{n+1}-Q_i^n}{\Delta t}+c\frac{Q_i^n-Q_{i-1}^n}{\Delta x}=0,\qquad \text{error } O(\Delta t,\Delta x)$$

Second-order accurate schemes

Subtract the two Taylor series for $f_{i+1}$ and $f_{i-1}$; the $f''$ terms cancel:

$$\frac{f_{i+1}-f_{i-1}}{2\Delta x}=f'_i+\frac{\Delta x^2}{6}f_i^{(3)}+\dots\ \Rightarrow\ f'_i=\frac{f_{i+1}-f_{i-1}}{2\Delta x}+O(\Delta x^2)$$

This central difference is second-order accurate. Likewise $f''_i=\dfrac{f_{i+1}-2f_i+f_{i-1}}{\Delta x^2}+O(\Delta x^2)$. A second-order scheme in both variables is the four-point (box) scheme centred at $(i+\tfrac12,\ n+\tfrac12)$:

$$\frac{Q_i^{n+1}+Q_{i+1}^{n+1}-Q_i^n-Q_{i+1}^n}{2\Delta t}+c\,\frac{Q_{i+1}^{n+1}+Q_{i+1}^n-Q_i^{n+1}-Q_i^n}{2\Delta x}=0$$

Graphical meaning

 f
 |                     . f(i+1)
 |                 . '
 |            f(i)
 |         .'
 |   f(i-1)
 +----|---------|---------|----> x
     i-1        i        i+1

 forward : chord i  -> i+1
 backward: chord i-1-> i
 central : chord i-1-> i+1 (closest to tangent at i)

The slope of the chord approximates the slope of the tangent at $i$. The forward and backward chords are tilted either side of the tangent (error proportional to $\Delta x$); the central chord is almost parallel to it (error proportional to $\Delta x^2$).

Grid view of the two PDE schemes

 t
 n+1 :   o--------o            o   o
         |        |              \ |
 n   :   o        x         x-----x-----x   
        (i-1)    (i)       (i-1)  (i)  (i+1)
   first order (forward t,      second order (central
   backward x): 2 known pts     in x): 3 known points

Comparison