1.2

The plots show the results for varying timesteps \(\Delta t\). Since the spatial step \(\Delta x\) and wave speed \(a\) are constant, the CFL number is solely dependent on \(\Delta t\). In the first two plots, the CFL \(<\) 1, meaning the wave is propagating faster than the solution (i.e. the wave is out ahead of the time of the solution). The time marching solution shows the wave dissipating with the magnitude of the higher frequency more damped than the lower frequency. The third plot (CFL = 1) shows the wave propagating at the same speed as the solution, so the calculated solution matches the exact (i.e. the time marching solution is computed exactly where the wave is in space at each time step). The fourth plot (CFL \(>\) 1) shows the solution propagating in time faster than the wave speed (i.e. the solution is out ahead of the wave). In this case, the information (the wave) has not reached the point in time where the solution is being computed, and therefor, it is unstable and diverges from the exact solution.


HW1_2.pdf (240.79 KB)