18.306 Lecture 20 - Thu 2021 05 06 - Virtual
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Finish with hyperbolic equations.
SKIP #046, #047
#048 Simple waves and breaking.
     Genuine nonlinearity and linear degeneracy.
#049 Shocks and Entropy
     When are shocks appropriate and when they are not.
     Conservation forms and Rankine-Hugoniot jump conditions.
     Lax entropy and characteristic geometry.
General Outlook:
  How to solve such equations. Godunov and Riemann solvers.
  n-D  Definition of hyperbolic for 1st order systems.
  n-D  How to solve. Godunov and 1-D problems.
  Back to 1-D
       Domain of dependence. Characteristic method.

This is a 1-lecture outlook on something that would take 3 courses to
cover properly.
Nice introductory book for the numerical stuff
  Numerical methods for conservation laws
  by Randall J. LeVeque
Introductory book for the nitty gritty of the physics
  Introduction to Wave Propagation in Nonlinear Fluids and Solids
  by D.S. Drumheller

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 % -------------------------------------------------------------------- #048
 Y_t + A(Y)*Y_x = 0; hyperbolic [A real diagonalizable]
 Simple waves (from simple eigenvalues and eigenfunctions).
 Wave breaking occurs.
 As in the scalar case, characteristics cross.
 Note simple waves also provide rarefaction wave solutions.
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 % -------------------------------------------------------------------- #049
 Breakdown of solutions: need to input appropriate physics.
 An example is when shocks apply.
 Shocks and shock conditions for systems of conservation laws.
 Rankine Hugoniot conditions.
 Derivation of the Lax entropy conditions as needed for causality.
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