Order Form - FENOMEC

24 CHAPTER I. FUNDAMENTAL CONCEPTS AND EXAMPLES
to a limit which satisfies the conservation laws (1.1) and the entropy inequality
(3.8). Mild oscillations and spikes are visible near jump discontinuities, only.
Interestingly enough, the limiting solution strongly depends on the parameter
γ. That is, for the same initial data different values of γ lead to different
shock wave solutions !
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Figure I-1 : Numerical solution for
|δ| > ε 2 , respectively.
From this discussion we conclude that no “universal” admissibility criterion can
be postulated for nonlinear hyperbolic systems. Instead, some additional information
should be sought and an admissibility condition should be formulated for each problem
(or rather each class of problems) of interest. Before closing this section let us
introduce a few more properties and definitions. First of all, for the entropy inequality
(3.8) we have the obvious analogue of the Rankine-Hugoniot relation derived earlier
in Section 2.
Theorem 5.1. (Jump relation for the entropy inequality.) We use the same notation
as in Theorem 2.3. The piecewise smooth function (2.7) satisfies the entropy inequality
(3.8) if and only if
−ϕ ′ (t) ( U(u + (t)) − U(u − (t)) ) + F (u + (t)) − F (u − (t)) ≤ 0. (5.2)
In particular, given a shock wave (2.9) connecting two constant states u − and u + and
associated with the speed λ, the entropy inequality reads
−λ ( U(u + ) − U(u − ) ) + F (u + ) − F (u − ) ≤ 0. (5.3)
When dealing with nonclassical solutions, the Rankine-Hugoniot relations (2.10)
and the entropy inequality (5.3) will be supplemented with the following additional
jump condition:
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