As expected, CPU time usage will be less as the time step is increased. There is an anomaly cased noticed where the CPU time increased for the Lax method when the time step is increased from \(0.05025\) to \(0.06\,,\) This needs further investigation.
This table below summarizes the CPU time in seconds used by each method for the case of constant speed as time step is increased.
| \(\tau \ \ \ \sec \) | \(Explicit\ FTCS\) | \(Explicit\ LAX\) | \(Implicit\ FTCS\) | \(C-R\) |
| \(0.0001\) | \(20\) | \(31\) | \(45\) | \(49\) |
| \(0.001\) | \(2.42\) | \(3.48\) | \(4.7\) | \(4.9\) |
| \(0.0013\) | \(1.9\) | \(2.78\) | \(3.7\) | \(3.9\) |
| \(0.0015\) | \(1.7\) | \(2.56\) | \(3.34\) | \(3.45\) |
| \(0.045\) | \(0.73\) | \(0.281\) | \(0.43\) | \(0.4\) |
| \(0.05025\) | \(0.7\) | \(0.25\) | \(0.5\) | \(0.468\) |
| \(0.06\) | \(0.65\) | \(0.9\) | \(0.4\) | \(0.41\) |
| \(0.07\) | \(0.5\) | \(0.89\) | \(0.45\) | \(0.36\) |
This is the plot of the above table
This table below summarizes the CPU time in seconds used by each method for the case of varying speed as time step is increased.
| \(\tau \ \ \ \sec \) | \(Explicit\ FTCS\) | \(Explicit\ LAX\) | \(Implicit\ FTCS\) | \(C-R\) |
| \(0.0001\) | \(21\) | \(31\) | \(67\) | \(69\) |
| \(0.001\) | \(2.5\) | \(3.5\) | \(7\) | \(7.5\) |
| \(0.0013\) | \(2\) | \(2.9\) | \(5.56\) | \(6\) |
| \(0.0015\) | \(1.8\) | \(2.53\) | \(4.73\) | \(5\) |
| \(0.045\) | \(0.28\) | \(0.54\) | \(0.45\) | \(0.45\) |
| \(0.05025\) | \(0.28\) | \(0.31\) | \(0.45\) | \(0.56\) |
| \(0.06\) | \(0.265\) | \(0.29\) | \(0.36\) | \(0.36\) |
| \(0.07\) | \(0.23\) | \(0.22\) | \(0.33\) | \(0.31\) |
This is the plot of the above table
This plot below compares the CPU time for each method when the speed is constant vs. when the speed was changing with time.