18.306 Lecture 19 - Tue 2021 05 04 - Virtual
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Finish #044  First order quasi-linear 1-D systems of equations.
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 First order quasi-linear 1-D systems of equations. Classification.
 Hyperbolic systems and characteristics.
 Domains of dependence and influence. Examples.

 First order quasilinear systems of equations
            u_t + A*u_x = F(x, t, u), where A = A(u, x, t).

 Characteristics and characteristic form of the equations.
 Example: linear, constant coefficients, no sources, case.

 Hyperbolic if A is real diagonalizable.
 Example: general solution for a hyperbolic system where A is
          constant and F = 0.
 In general, F \neq 0, characteristics couple.

 Domains of dependence and influence.

 Examples:
   Linear Gas Dynamics (acoustics). Sound waves, general solution.
   Wave equation. Reduce to form above.
   Klein Gordon equation. Characteristic form.
      Domains of dependence and influence.
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 Examples of first order 1-D hyperbolic systems. Linear acoustics.
 Example:  Linear acoustics in 1-D.
           Exact solution by characteristics.
           System equivalent to wave equation.
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 Systems of the form Y_t + A*Y_x = 0.
 Show ill-posed if complex eigenvalue exists.

 Motivation and physical meaning:
 When A is not a constant (particularly if A = A(u)) the system can
 change type, switching from well posed to ill posed.

 Example: stratified flows.
 --- Interpretation of the Richardson # criteria as a change of
     type (not an instability issue, as often portrayed).
 --- Physical reason: mixing occurs, stratification assumption fails.
     Ad-hoc fix: add vertical diffusivity in mass equation (mixing).
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 Example:  Wave equation.
 Solution of the initial value problem: D'Alembert.
 Domains of dependence and influence (Note that they involve the full
 wedge, not just the characteristics at the edge, for data u and u_t).
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 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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 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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