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

RecallShock 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).

RecallPhase Plane

t-x

Initial Conditions:u = exp[x]

The straight lines represent level curves.

RecallState 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.

State Space 2D t=.1

As the two pulses diverge (one going left, the other right), the curve billows out.

State Space 2D t=.5

The curve in state space continues to change…

State Space 2D t=1.1

The characteristics begin to overdefine the function…

t-x (u) t-x (v)

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.

Kinds of Waves

Constant Solutions

2D surfaces

Simple Waves

One Dimensional image in state space

The Riemann Problem

Given an initial state and a final state, can simple waves connect them?

RarefractionsCompressionsShocks

The curves found by integrating the eigenvectors represent a locus of states connected to the initial conditions.

Ahat-System:

If U=(u,v,w) and Ahat is a 2x2 matrix

Ut+Ahat(U)Ux=0

wt+f(w)wx=0

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)

Euler’s Full Gas Equations(holy grail)

pt + (pv)x = 0

(pv)t + (pv^2+P)x = 0

Et + (v(E+P))x = 0

The Elastic StringUt+Vx=0

Vt+T(U)x=0

Plane Solutions to Maxwell’s Equations

Lie Brackets of Eigenvectors

Definition:

[X,Y] = D[Y]X-D[X]Y

Frobeneous:

If [X,Y] = 0 Then the vectors define a surface

Find Eigensystems

Integrate Eigenvectors

Lie Bracket the Eigenvectors

To do…