3.5 Stability analysis of Lax

Replace the trial function from (2) in Lax formulation in (4) and obtain

\[ A^{n+1}e^{jkih}=\frac {1}{2}\left ( A^{n}e^{jk\left ( i+1\right ) h}+A^{n}e^{jk\left ( i-1\right ) h}\right ) -\frac {u\tau }{2h}\left ( A^{n}e^{jk\left ( i+1\right ) h}-A^{n}e^{jk\left ( i-1\right ) h}\right ) \]

Divide by \(A^{n}e^{jkih}\) , the magnification factor \(\zeta \) is obtained

\begin{align*} \zeta & =\frac {1}{2}\left ( e^{jkh}+e^{-jkh}\right ) -\frac {u\tau }{2h}\left ( e^{jkh}-e^{-jkh}\right ) \\ & =\cos \left ( kh\right ) -j\frac {u\tau }{h}\sin \left ( kh\right ) \end{align*}

Hence

\[ \left \vert \zeta \right \vert =\sqrt {\cos ^{2}\left ( kh\right ) +\left ( \frac {u\tau }{h}\right ) ^{2}\sin ^{2}\left ( kh\right ) }\]

Since \(\cos ^{2}\left ( kh\right ) \leq 1\) and \(\sin ^{2}\left ( kh\right ) \leq \) \(1\), then it is seen that \(\left \vert \zeta \right \vert \leq 1\) if \(\frac {u\tau }{h}\leq 1\)

Hence the following is the condition for stability

\[ \tau \leq \frac {h}{u}\]

As mentioned earlier, this is called the CFL condition.

The Lax method is stable for \(\tau \leq \frac {h}{u}\) however, a modified version of this method is more accurate, which is the Lax-Wendroff method.