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

$$\underbrace{ \underbrace{ \frac{\partial \rho}{\partial t}... ...Delta t^2, \Delta x^2 \right)}_{\mbox{rest error}} \enspace .$$ (122)

Now, the error is $O \! \left( \Delta t^2, \Delta x^2 \right)$ instead of $O \! \left( \Delta t, \Delta x\right)$.

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

$$\frac{\partial \rho}{\partial t} + v_{} \frac{\partial \... ...{} \frac{\partial^2 \rho}{{\partial x}^2} \Delta x = 0$$ (123)

better than the original PDE.