Choosing the time step

The maximum time step one can take is set by the assumption that one can consider each cell only to interact with its direct neighbours: the domain of influence of cell $i$ should not reach cells $i-2$ and $i+2$! Since the maximum communication speed is $v-s$ or $v+s$, this gives a condition on the time step. This is known as the Courant-Friedrichs-Lewy (CFL) condition. If your time step exceeds this value, the method will become unstable. Since this maximum time step depends on $v$ and $s$, this means that the time step is not fixed, but depends on the conditions on the grid, and should be re-calculated every time step. It is normally wise to take a fraction between 1 and 0.5 of this maximum time step.