The scheme is -contracting

5.4 Propagation of Numerical Discontinuities
Our goal in this section is to examine how numerical discontinuities propagate. Since almost all numerical schemes involve numerical diffusion in order to guarantee convergence to the desired physical solution, we will study the solution of a modified equation. The simplest of these is the convection-diffusion equation. For more discussion related to the ideas in this section, see [87, 113, 142].
We will consider the convection-diffusion equation with Riemann-problem initial data

The analytical solution for is

and the analytical solution for and is

The difference between the analytical solutions with and without diffusion is

Thus if a physical problem involves diffusion, we expect the initial discontinuity to spread a distance proportional to the square root of the product of the diffusion constant and time.
Numerical schemes typically involve numerical diffusion. For example, recall that in example 3.3.1 we saw that the modified equation analysis for the Lax-Friedrichs scheme produces

Decreasing the timestep or computing to large time will increase the spreading of the discontinuity.
No scheme is perfect. Suppose that we have a numerical scheme that spreads a discontinuity over a fixed number of cells, and that the greatest contribution to the error is due to the resolution of the discontinuity. Then the error in the numerical solution is

where is the size of the jump in and is an upper bound for the cell width. This suggests that the error should be no better than first-order (see definition 5.1.2) accurate at discontinuities.
Order accuracy in the norm at a discontinuity would require that the numerical width of the discontinuity be . This would in turn require that the position of the discontinuity within the cell be accurately determined. In particular, this would require a model for the variation of the solution within a grid cell, such as in [65]. Currently, no known scheme can do this for general problems.