Lecture 12, 18.300, Tue Mar 15, 2022.
Summary
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#036 Finish. Riemann Problem.
Describe solution to Riemann Problem for Traffic Flow, Flood Waves, etc.
         Two cases: shock wave or rarefaction wave.
Describe Godunov's method and how the Riemann Problem is critical.
         Important: explain role of finite speed of propagation.
         Preview of domain of dependence and domain of influence
         ... to be seen in more detail later.
Describe how to get higher order Godunov, using piece-wise linear and
         higher order approximants. Important to keep conservation though.
         Generalized Riemann Problems [briefly mention].
         Explain need for switches, to prevent undesirable oscillations.
Method easy to extend to systems; e.g.: Gas Dynamics.
         At students prompting tried to explain how the Riemann problem
         works for Gas Dynamics.
         For example, in isentropic Gas Dynamics [or Shallow Water] there
         are two sets of characteristics, thus two types of shocks and
         two types of rarefactions. This leads to 4 cases, instead of two:
         shock-shock; shock-rarefaction; rarefaction-shock, and
         rarefaction-rarefaction.
         Note: derived characteristic equations for Isentropic Gas Dynamics.
         Hopefully this will all become clearer later in the semester.
Extension to multi-D. Pointed out need only the 1-D Riemann problems!

Note#1. Points that are recaps of things covered earlier are sometimes
        skipped. Students should read these points and ask the lecturer
[during lecture or office hours] to fill in any details they do not
understand.
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