The scheme is -contracting

Exercises for 5.5
5.5.1 Program the MUSCL scheme for linear advection. Compare it to Lax-Wendroff, Beam-Warming and Fromm for the Zalesak initial data in Exercise 2.2.5 of Section 2.2. Which scheme produces the smallest error for a given mesh width? Which scheme is most efficient? What convergence rates do the schemes produce? For which initial data are any of the schemes second-order accurate?
5.5.2 VanLeer also suggested

Show that this corresponds to using harmonically averaged slopes when the numerical solution has no local extremum.
5.6 Discrete Entropy Conditions
In section 3.1 we discussed the usefulness of an entropy function for scalar conservation laws. In this section we would like to study the usefulness of an entropy function for understanding the behavior of numerical methods.
Suppose that we have an explicit conservative numerical method

that approximates the solution of the system of conservation laws

Also suppose that is a convex entropy function (definition 3.1.14) for the conservation law, with entropy flux ; in other words,

Suppose that is concave, and we can find a numerical entropy flux that is consistent with (definition 5.1.2) and satisfies

Then it is possible to modify the proof of the Lax-Wendroff Theorem 5.2.2 (see [96, section 12.5]) to show that if , then the total entropy increases in time. Similarly, if is concave and the inequality on is reversed, then the total entropy decreases in time.

380-bet
sxema L_1-qurilishi

Lemma 5.3.8 [96, sahifa 166]. Raqamli sxemani ko'rib chiqamiz

Agar raqamli yechim umumiy o'zgarishni kamaytirsa va agar raqamli yechim katta fazoviy indekslar uchun doimiy bo'lsa, demak