The scheme is -contracting

5.5.2 Monotonizations
Consider the Lax-Wendroff scheme for linear advection:

If we multiply this equation by , we obtain

where . Recall from section 3.5 that this scheme has second-order local truncation error. Since this scheme is linear and second-order, Godunov's theorem [96, p. 174] shows that it cannot be monotonic. However, since this scheme is a function of the smoothness monitor and the Courant number , we can consider monotonizing the Lax-Wendroff scheme as follows:

Thus, in order for this scheme to be monotonic we want

Lemma 5.5.3 If and the function satisfies the inequality

which is equivalent to the inequalities

then the function defined by (5.14) satisfies for all for all .
Proof Note that and , so is at the required bounds when is at either of its extreme values. It follows that we must have

In particular,

Note that

It follows that if both of the constraints (5.16) are satisfied, then for all for all . The requirements (5.16) can be simplified to , which can be rewritten in the form (5.15).
Note that produces the upwind scheme. Also, note that the LaxWendroff scheme is itself monotonic precisely when satisfies the inequalities (5.15), that is, when . Unfortunately, even if we monotonize the Lax-Wendroff method by means of , the resulting scheme is not conservative.
Next, consider the Beam-Warming scheme

which can be rewritten in the form

Again, since this scheme is linear and second-order, it cannot be monotonic. However, since this scheme is a function of the smoothness monitor and the Courant number , consider monotonizing the Beam-Warming scheme as follows:

Following the same approach as in Lemma 5.5.3, we can easily prove the following lemma.
Lemma 5.5.4 If and the function satisfies the inequalities

which can be rewritten in the form

then defined in (5.17) satisfies for all for all .
Note that produces the upwind scheme. Also, note that the BeamWarming scheme is itself monotonic precisely when satisfies the inequalities (5.15), that is, when . Unfortunately, even if we monotonize the Beam-Warming method by means of , the resulting scheme is not conservative.
The Fromm scheme [51] is the average of the Lax-Wendroff and BeamWarming schemes, when applied to linear advection. If we average the monotonized Lax-Wendroff and monotonized Beam-Warming schemes, the result will be monotonic:

This expression shows that in order for this scheme to be conservative, it suffices to find a function so that and .