2.5.8 Linear stability analysis of naive FTCS scheme

Applying ansatz (135) to the naive FTCS scheme Eq. (92) gives

$$A e^{- ji k \Delta x} = e^{- ji k \Delta x} - \frac{\D... ... e^{- j\left ( i - 1 \right) k \Delta x} \right) \enspace .$$ (137)

Multiplying with $e^{ji k \Delta x}$ and using the Courant number

$$\alpha := \frac{\Delta t}{\Delta x} v_{}$$ (138)

we get

$$A = 1 - \alpha \frac{1}{2} \left( e^{- jk \Delta x} ... ...Delta x} \right) = 1 + j\alpha \sin k \Delta x \enspace ,$$ (139)

$$\mbox{abs} \! \left( A \right) = \left( 1 + \alpha^2 \sin^2 k \Delta x\right)^{\frac{1}{2}} \enspace ,$$ (140)

$${\color{HIGH1color} \mbox{abs} \! \left( A \right) > 1 ... ...0 < k \Delta x< \pi \enspace , \enspace \alpha > 0 \enspace .$$ (141)

All waves except the ones with smallest wavenumber grow exponentially in time:

The scheme is unconditionally unstable, independent of the time-step.