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3. Prove inequality (13.19).
[Hint: using the Cauchy-Schwarz and Young inequalities, prove ¬rst that
(u ’ v)u dx ≥ ’v ∀ u, v ∈ L2 (0, 1).
2 2
u ,
L2 (0,1) L2 (0,1)

Then, use (13.18). ]
4. Assume that the bilinear form a(·, ·) in problem (13.12) is continuous and
coercive over the function space V (see (12.54)-(12.55)) with continuity and
coercivity constants M and ±, respectively. Then, prove that the stability
inequalities (13.20) and (13.21) still hold provided that ν is replaced by ±.

5. Show that the methods (13.39), (13.40) and (13.41) can be written in the
form (13.42). Then, show that the corresponding expressions of the arti¬cial
viscosity K and arti¬cial di¬usion ¬‚ux hdif f are as in Table (13.1).

6. Determine the CFL condition for the upwind scheme.
7. Show that for the scheme (13.43) one has un+1 ∆,2 ¤ un for all
n ≥ 0.
[Hint: multiply equation (13.43) by un+1 , and notice that

(un+1 ’ un )un+1 ≥ |un+1 |2 ’ |un |2 .
j j
j j j
Then, sum on j the resulting inequalities, and note that

un+1 ’ un+1 un+1 = 0
2 j=’∞ j+1 j’1 j

since this sum is telescopic.]
8. Show how to ¬nd the values µ and ν in Table 13.2 for Lax-Friedrichs and
Lax-Wendro¬ methods.
9. Prove (13.67).
10. Prove (13.69) when f = 0.
[Hint: take ∀t > 0, vh = uh (t) in (13.68).]

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