This table below summarizes the RMS error from each numerical method as a function of changing the time step size. This is for case of constant speed.
| \(time\ step\) | \(Explicit\ FTCS\) | \(Explicit\ LAX\) | \(Implicit\ FTCS\) | \(C-R\) |
| \(0.0001\) | \(0.0546\) | \(0.0543\) | \(0.0548\) | \(0.0544\) |
| \(0.001\) | \(0.01264\) | \(0.0057\) | \(0.00742\) | \(0.00575\) |
| \(0.0013\) | \(0.0494\) | \(0.01125\) | \(0.01245\) | \(0.00128\) |
| \(0.0015\) | \(0.15249\) | \(0.00056\) | \(0.009\) | \(0.0056\) |
| \(0.045\) | \(blows\ up\) | \(0.000162\) | \(0.1306\) | \(0.01028\) |
| \(0.05025\) | \(blows\ up\) | \(0.1006\) | \(0.1394\) | \(0.011\) |
| \(0.06\) | \(blows\ up\) | \(blows\ up\) | \(0.1531\) | \(0.01244\) |
| \(0.07\) | \(blows\ up\) | \(blows\ up\) | \(0.1653\) | \(0.01403\) |
Notice that the Lax method became more accurate when the time step was increased from \(0.0001\) to \(0.04\) seconds, then it starts to become less accurate as time step is increased. This is counter intuitive to what one can expect. It will be interesting to investigate this further to obtain a mathematical explanation for this strange phenomena.
The accuracy of the implicit FTCS, and C-R also increased slightly as the time step became larger from \(0.0001\) to \(0.0015\), then the implicit FTCS became worst in terms of accuracy as the time step increased.
C-R method accuracy did not deteriorate as much with increasing the time step. This shows the C-R scheme to be more robust.
This table below summarizes the RMS error from each numerical method as a function of changing the time step size. This is for case of changing speed.
| \(time\ step\) | \(Explicit\ FTCS\) | \(Explicit\ LAX\) | \(Implicit\ FTCS\) | \(C-R\) |
| \(0.0001\) | \(0.003\) | \(0.003\) | \(0.003\) | \(0.0030\) |
| \(0.001\) | \(0.00352\) | \(0.00329\) | \(0.0033\) | \(0.0033\) |
| \(0.0013\) | \(0.00365\) | \(0.00331\) | \(0.00346\) | \(0.0033\) |
| \(0.0015\) | \(0.0038\) | \(0.00336\) | \(0.0035\) | \(0.00337\) |
| \(0.045\) | \(blows\ up\) | \(0.01117\) | \(0.0386\) | \(0.0119\) |
| \(0.05025\) | \(blows\ up\) | \(0.04385\) | \(0.0428\) | \(0.01317\) |
| \(0.06\) | \(blows\ up\) | \(0.01389\) | \(0.0493\) | \(0.01525\) |
| \(0.07\) | \(blows\ up\) | \(0.01564\) | \(0.0557\) | \(0.0174\) |
The effect of having the speed defined as \(\mu =\frac {t}{20}\)is to delay instability for the explicit methods as time step is increased. Notice also here the case where the Lax method became more accurate as the time step is increased from \(0.0001\) to \(0.0015\).