2.5.4 Modified equation for the FTFS scheme II

To replace $\frac{\partial^2 \rho}{{\partial t}^2}$ in the modified equation (123) we write

$$\frac{\partial^2 \rho}{{\partial t}^2} \Delta t+ v_{} \frac{\partial^2 \rho}{{\partial x}^2} \Delta x = \frac{\partial }{\partial t} \left( - v_{} \frac{\partial \rho... ...l^2 \rho}{{\partial x}^2} \Delta x + O \! \left( \Delta t^2, \Delta x^2 \right)$$ (124)

$$= - v_{} \frac{\partial }{\partial x} \left( \frac{\partial \rho... ...l^2 \rho}{{\partial x}^2} \Delta x + O \! \left( \Delta t^2, \Delta x^2 \right)$$ (125)

$$= v_{}^2 \frac{\partial^2 \rho}{{\partial x}^2} \Delta t + v_{} ... ...l^2 \rho}{{\partial x}^2} \Delta x + O \! \left( \Delta t^2, \Delta x^2 \right)$$ (126)

$$= v_{} \Delta x \left( \frac{v_{} \Delta t}{\Delta x} + 1 \right) \frac{\partial^2 \rho}{{\partial x}^2} + O \! \left( \Delta t^2, \Delta x^2 \right)$$ (127)

and get for the modified equation for the FTFS scheme

$$\frac{\partial \rho}{\partial t} + v_{} \frac{\partial \... ...ht) \frac{\partial^2 \rho}{{\partial x}^2} = 0 \enspace .$$ (128)

The coefficient of the additional diffusion term is positive: it describes anti-diffusion.