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Time step

As we have just discussed, to have sufficient accuracy in computation, we should have the proper \(\Delta x\). Once this is chosen we still have to decide how to set \(\Delta t\). Can we freely choose \(\Delta t\)? Obvious not. Theoretical study has shown that, if a simulation is stable, it must abide by Courant-Friedrichs-Lewy(CFL or Courant) condition:

\begin{displaymath}\alpha\equiv\frac{v\Delta t}{\Delta x}\leq 1
\end{displaymath} (60)

where \(\alpha\) is called Courant number and v is the sum of sound speed and fluid speed. From CFL condition we can see that, the smaller the space step, the smaller the time step to keep the program stable. After having defined a certain \(\alpha\leq1\) and having determined the spatial step, we can decide on the \(\Delta t\) to use.