2.3.14 Truncation error

A sufficiently smooth function can be expanded in a Taylor series:

$\begin{displaymath} \rho \! \left( x + \Delta x, t \right) = \sum_{l=0}^{\inf... ...rt _{x,t}} \Delta x + O \left( \Delta x^2 \right) \enspace . \end{displaymath}$

(105)

Solving for $\bgroup\color{DEFcolor}$ \frac{\partial \rho}{\partial x} $\egroup$ gives

$\begin{displaymath} \frac{\partial \rho}{\partial x} = \frac{\rho \! \left( ... ... t \right)}{\Delta x} + O \left( \Delta x\right) \enspace . \end{displaymath}$

(106)

Repeating this for the time derivative and applying it to an entire PDE (FTFS) gives

$\begin{displaymath} \underbrace{ \frac{\partial \rho}{\partial t} + v_{} \fr... ...elta t, \Delta x\right)}_{\mbox{truncation error}} \enspace . \end{displaymath}$

(107)

The order of the truncation error is $\bgroup\color{DEFcolor}$O \left( \Delta t, \Delta x\right)$\egroup$ in this case (FTFS).

A high order of the truncation error (both in $\bgroup\color{DEFcolor}$\Delta t$\egroup$ and $\bgroup\color{DEFcolor}$\Delta x$\egroup$ ) hints at good accuracy for smooth functions.