MIT OpenCourseWarehttp://ocw.mit.edu 18.306 Advanced Partial Differential Equations with ApplicationsFall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Lecture 21 2009 11 23 MON TOPICS. Gas dynamics in 1-D. Characteristics, simple waves, Riemann Invariants, rarefaction waves, shocks and shock conditions. Riemann problem. Generalizations to N by N systems. Example: Gas Dynamics in 1-D. Isentropic % --------------------- Formulation in terms of mass Lagrangian coordinates. Riemann Invariants and simple waves. Wave breaking. Shock conditions (Rankine-Hugoniot ) for systems. Lax entropy: explain how it works for causality. Shocks in the p-v plane. Right and left shocks. Lax entropy equivalent to compressive shocks. Shock curve: for a fixed ``right'' state on a ``right'' shock, states in phase space (u, v) that can be reached by a shock. Similar curve exists for left shocks, starting from left state. Rarefaction curve: Same idea s for the shock curve. Write rarefactions using characteristic form, in particular: Riemann Invariants. RIEMANN PROBLEM: Show how to solve using the shock/rarefaction curves as a sort of coordinate system in phase space. Describe how solution looks in space-time. General systems: there are N shock curves and N rarefaction curves. At least locally they can be used to solve the Riemann problem. In general not always clear as the states on the right and left in a Riemann problem get further
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