Distributions in the Physical and Engineering Sciences, Volume 2 by Alexander I. Saichev & Wojbor A. Woyczynski
Author:Alexander I. Saichev & Wojbor A. Woyczynski
Language: eng
Format: epub
Publisher: Springer New York, New York, NY
(14)
An example of critical parabolas determining locations and magnitudes of discontinuities (shocks) of the weak solutions of the Riemann equation is shown in Fig. 13.4.4.
Figure 13.4.4 Top: The initial potential s 0(y) and two critical parabolas. Bottom: Their centers determine locations of the discontinuities (shocks) of the weak solutions of the Riemann equation.
13.4.4 The Convex Hull
The global minimum principle discussed above can be reduced to a geometric procedure for finding the generalized Eulerian-to-Lagrangian mapping y w (x, t). In general, the mapping is discontinuous. For this reason, it is sometimes convenient to work with the inverse function X w (y, t), which is continuous everywhere. It can also be constructed via an elegant geometric algorithm. To describe the latter, let us resort once more to the auxiliary function (10):
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