Front Tracking for Hyperbolic Conservation Laws by Helge Holden & Nils Henrik Risebro

Author:Helge Holden & Nils Henrik Risebro
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

(5.120)

and we easily see that is strictly decreasing, unbounded, and starting at . Using (5.49) and (5.106), we find that reads

(5.121)

which is strictly increasing, unbounded, with minimum . Thus we conclude that the Riemann problem for shallow water has a unique solution in the region where

(5.122)

To obtain explicit equations for the middle state u m we have to make case distinctions, depending on the type of wave curves that intersect, i.e., rarefaction waves or shock curves. This gives rise to four regions, denoted by . See Fig. 5.7. For completeness we give the equations for the middle state u m in all cases.

Assume first that . We will determine a unique intermediate state such that . These requirements give the following equations to be solved for such that :

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