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Week 4

7/5 Week 4:

Systems II Euler Gas, Maxwell's Equations

The “Gas Dynamics” equations consist of three coupled state variables. They model gas in the shocktube. While any three of six widely known physical quantities can be used as state variables for a total of 6*5*4=120 possible ways of specifying the system, we choose to work in Eulerian Coordinates.

4) Euler's full gas dynamics: p (density) v (velocity) P (pressure) E (energy)

(p)t + (p)x = 0
(p v)t + (p v^2 + P)x = 0
(E)t + (v (E + P))x = 0

The equations describe a plane electromagnetic wave in the absence of matter.

5) Maxwell linear eqs: We take any derivative with respect to y or z to be zero.

(B1)t   + (E3)y - (E2)z = 0
(B2)t - (E3)x   + (E1)z = 0
(B3)t + (E2)x - (E1)y   = 0
(E1)t   + (B3)y - (B2)z = 0
(E2)t - (B3)x   + (B1)z = 0
(E3)t + (B2)x - (B1)y   = 0

Note that in both cases there are two 'flux matrices.' (one for x and one for t) When this happens we find the "generalized eigenvectors" of the system. To find the generalized eigenvectors, we left multiplied the space 'flux matrix' by the inverse of the time 'flux matrix' and took the eigensystem of the result.

Produced and maintained by Robert Chase, the Department of Mathematics and Statistics