2.3.9 Truncation error

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

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

(95)

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

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

(96)

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

$\begin{displaymath} \underbrace{ \frac{\partial \rho}{\partial t} + v_{} \fr... ...a t, \Delta x^2 \right)}_{\mbox{truncation error}} \enspace . \end{displaymath}$

(97)

The order of the truncation error is $\bgroup\color{DEFcolor}$O \left( \Delta t, \Delta x^2 \right)$\egroup$ in this case.