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.
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