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2.5.3 Modified equation for the FTFS scheme I

Keeping one more term in Eq. (105) we get instead of Eq. (107) the equation

\underbrace{ \frac{\partial \rho}{\partial t}...
...Delta t^2, \Delta x^2 \right)}_{\mbox{rest error}}
\enspace .
\end{displaymath} (122)
Now, the error is \bgroup\color{DEFcolor}$O \! \left( \Delta t^2, \Delta x^2 \right)$\egroup instead of \bgroup\color{DEFcolor}$O \! \left( \Delta t, \Delta x\right)$\egroup.

I.e., the numerical scheme describes the modified equation

\frac{\partial \rho}{\partial t} + v_{}   \frac{\partial \...
...{}   \frac{\partial^2 \rho}{{\partial x}^2}   \Delta x
\end{displaymath} (123)
better than the original PDE.