The scheme is -contracting

5.5.3 MUSCL Scheme
With this function in the Fromm monotonization, Lemma 5.5.3shows that the Lax-Wendroff scheme will be monotonic if

and Lemma 5.5.4 shows that the Beam-Warming scheme will be monotonic if

By solving these inequalities, we see that for the average scheme to be monotonic

Fig. 5.2 MUSCL curves: Monotonizers vs. smoothness monitor; is thick curve
we must require

Next, because is a function of and is a function of , we require that . This is condition is satisfied for , where has already been defined. As a consequence of this choice, . Finally, we would like to minimize the value of the monotonizer, so that we differ from the original Fromm scheme as little as possible. As a result, we take for and for . Similarly, we take for . In summary, our monotonizer is

This function is illustrated in Figure 5.2.
An equivalent formulation of this method has numerical flux


(a) Beam-Warming

(d) Upwind

(b) Lax-Wendroff

(e) MUSCL

(c) Fromm
Fig. 5.3 Schemes for Linear Advection exact, blue numerical solution)
where is the Courant number and

is the MUSCL slope. If we define the side-centered slope

then we can write

Here "MUSCL" is an acronym for "Monotone Upstream-centered Scheme for Conservation Laws." Note that we can interpret as the flux function evaluated at the state . In Section 5.9, we will generalize the MUSCL scheme to nonlinear scalar conservation laws.
Figure 5.3 shows the computational results with various first-order and secondorder schemes. This figure was generated by Program 5.5-63: GUILinearAdvectionMain.C. The MUSCL scheme has been implemented in Program 5.5-64: muscl.f. Students can test and compare the upwind, Beam-Warming, Lax-Wendroff, Fromm and MUSCL schemes for the Zalesak linear advection test problems (see exercise 2.2.5 of section 2.2) by clicking on Executable 5.5-26: guilinearad. Selecting a positive number of grid cells causes the program to run the first method selected on the
chosen initial data, while selecting zero grid cells causes the program to perform mesh refinement and efficiency studies on all selected programs.