4 different numerical finite difference schemes are examined for CPU time, stability and accuracy in solving the advection PDE for constant speed and for a speed which is a function of time.
For accuracy, an interesting result is observed. The Lax scheme is the most accurate for Courant number close to unity. This means as the time step is increased, the Lax become more accurate of the 4 methods. But beyond the CFL condition, Both explicit methods (FTCS and Lax) became less accurate. Explicit FTCS became unstable sooner than Lax, while the implicit methods remained stable.
The implicit FTCS was less accurate than the C-R method. This implies that one should use the Lax method if one can be satisfied with a time step such that the courant number is close to a unit.
For stability, Crank-Nicolson was the most stable of all methods. Stability by itself is not sufficient condition to use to select a numerical scheme. It must also be accurate. The C-R method has both these properties for the range of the time steps considered. But as mentioned above, there is a range of time steps in which the Lax method is more accurate than all the other methods.
For CPU usage, the explicit methods used less CPU time when the time step was small, up to \(0.0015\sec .\) This can be explained as follows: for small step size, the number of time to solve \(Ax=b\,\) is large. Hence the implicit methods will be slower. As the time step is increased to the range of \(0.045\sec \) and over, the implicit methods actually became more CPU efficient due to the fact that the number of times to solve \(Ax=b\) is less because the number of steps is less.
In conclusion, the selection of a finite difference scheme depends on many factors. Stability and accuracy being the most important. The time step size plays a critical rule. For Courant number close to a unity, the Lax method is the most attractive. For larger time steps, the C-R method should be considered.