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Update

Update. By Rob Chase and Pat Dragon Supervised by Robin Young. Given initial conditions and a system of PDEs, what happens?. The decoupled case can be modeled (u,v)t+(u+v,u-v)x=0 (u,v)t+(u,v)x=0 Decoupling is equivalent to finding the eigensystem. Recall. Shock Profiles u-x

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Update

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  1. Update By Rob Chase and Pat Dragon Supervised by Robin Young

  2. Given initial conditions and a system of PDEs, what happens? The decoupled case can be modeled (u,v)t+(u+v,u-v)x=0 (u,v)t+(u,v)x=0 Decoupling is equivalent to finding the eigensystem

  3. Recall Shock Profiles u-x Initial Conditions: u = exp[-x^2] As time goes on, burgur’s (backward) flux function the pulse will move to the right (left).

  4. Recall Phase Plane t-x Initial Conditions: u = exp[x] The straight lines represent level curves.

  5. Recall State Space u-v Initial Conditions: u = exp[-x^2] v = exp[-x^2] The Curve is a parametric function of x with one bell curve superimposed on the other as shock profiles.

  6. State Space 2D t=.1 As the two pulses diverge (one going left, the other right), the curve billows out.

  7. State Space 2D t=.5 The curve in state space continues to change…

  8. State Space 2D t=1.1 t-x (u) t-x (v) The characteristics begin to overdefine the function…

  9. State Space 2D t=2 The shock “eats” information (whatever u,v symbolize) and very little is left over at the end. The horizontal and vertical lines are where the shock profile has become overdefined.

  10. Kinds of Waves Constant Solutions 2D surfaces

  11. Simple Waves One Dimensional image in state space

  12. The Riemann Problem Given an initial state and a final state, can simple waves connect them? Rarefractions Compressions Shocks The curves found by integrating the eigenvectors represent a locus of states connected to the initial conditions.

  13. Ahat-System: If U=(u,v,w) and Ahat is a 2x2 matrix Ut+Ahat(U)Ux=0 wt+f(w)wx=0

  14. P-System: The shock tube is immersed in water of constant temperature ut + a*v^(-g)*x = 0 a, g constants vt - ux = 0 0 p’(v) -1 0 +/- c = Sqrt[-p’(v)] (c,1) (-c,1)

  15. Euler’s Full Gas Equations(holy grail) The Elastic String Ut+Vx=0 Vt+T(U)x=0 pt + (pv)x = 0 (pv)t + (pv^2+P)x = 0 Et + (v(E+P))x = 0 Plane Solutions to Maxwell’s Equations

  16. Lie Brackets of Eigenvectors Definition: [X,Y] = D[Y]X-D[X]Y Frobeneous: If [X,Y] = 0 Then the vectors define a surface

  17. To do… Find Eigensystems Integrate Eigenvectors Lie Bracket the Eigenvectors

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