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

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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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