The scheme is -contracting

Exercises for 5.4
5.4.1 Consider the linear advection problem with the initial data described in this section. Define the front width to be the distance (in ) between the points where and . Run the explicit upwind, Lax-Friedrichs and Lax-Wendroff schemes for various values of CFL and . Plot front width versus time and explain your results.
5.4.2 Consider the linear advection problem with initial data described in this section. For each of the explicit upwind, Lax-Friedrichs and Lax-Wendroff schemes, plot the logarithm of the error versus the logarithm of the mesh width . The results should be computed at CFL 0.9 for the time at which the discontinuity has crossed of the grid. Plot the results for . What rate of convergence do you observe from this plot?
5.5 Monotonic Schemes
In this section, we will follow a line of development due to van Leer [168, 169, 170, 171]. Since a linear monotonicity-preserving scheme is at best first-order accurate, a monotonicity-preserving higher-order scheme must be nonlinear. We will develop a nonlinear monotonicity-preserving scheme that is designed to obtain second-order accuracy as much as possible.
5.5.1 Smoothness Monitor
We begin with a definition.
Definition 5.5.1 If , a scheme for linear advection is monotonic if and only if lies between and . An equivalent requirement is that lies between 0 and . This can be rewritten in the form

Thus the van Leer smoothness monitor for the scheme is defined to be

Note that a monotonic scheme is monotonicity-preserving.
Example 5.5.2 The explicit upwind difference can be rewritten

It follows that explicit upwind differencing for linear advection is monotonic if and only the Courant numbers satisfy

for all timesteps.
There are several reasons why the smoothness monitor is useful. Note that implies a local extremum in the numerical solution, possibly due to a numerical oscillation. On the other hand, implies monotonic behavior in the numerical solution, either monotonically increasing or decreasing. Furthermore, implies smooth behavior in the numerical solution, and or indicates a numerical discontinuity.