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2.5.7 Linear stability analysis: use

Linear stability analysis tells about stability - of linear schemes for linear equations.

Linear stability \bgroup\color{DEFcolor}$\Rightarrow$\egroup convergence for consistent schemes (Sect. 2.3.15).

Ansatz: for

\rho_i^n = e^{- ji k \Delta x}
\end{displaymath} (135)
with \bgroup\color{DEFcolor}$k \le k_0=\frac{\pi}{\Delta x}$\egroup we search \bgroup\color{DEFcolor}$A \in \mathbb{C}$\egroup with
\rho_i^{n+1} = A   \rho_i^n = A   e^{- ji k \Delta x}
\enspace .
\end{displaymath} (136)

Amount \bgroup\color{DEFcolor}$\mbox{abs} \! \left( A \right)$\egroup of \bgroup\color{DEFcolor}$A$\egroup \bgroup\color{DEFcolor}$\Rightarrow$\egroup damping (diffusion) or growth (instability) of waves.

Phase of \bgroup\color{DEFcolor}$A$\egroup \bgroup\color{DEFcolor}$\Rightarrow$\egroup wave speed and dispersion.

Ideally \bgroup\color{DEFcolor}$A = e^{ j\omega \Delta t} \enspace \mbox{with} \enspace \omega = v_{} k$\egroup.