The Riemann (or Sod) Shock-Tube Problem
Key References
- Sod, G. A. 1978, Journal of Computational Physics, 27,
1-31.
-
Hawley, J. F., Smarr, L. L., and Wilson, J. R. 1984,
The Astrophysical Journal, 277, 296-311.
"A Numerical Study of Nonspherical Black Hole Accretion. I.
Equations and Test Problems"
[See especially section IV, starting on page 300.]
-
Hawley, J. F., Smarr, L. L., and Wilson, J. R. 1984,
The Astrophysical Journal Supplement Series, 55, 296-311.
"A Numerical Study of Nonspherical Black Hole Accretion. II.
Finite-Differencing and Code Calibration"
-
Stone, J. M., and Norman, M. L. 1992,
The Astrophysical Journal Supplement Series, 80, 753-790.
"ZEUS-2D: A Radiation Magnetohydrodynamics Code for Astrophysical
Flows in Two Space Dimensions.
I. The Hydrodynamic Algorithms and Tests"
[See especially section 5.4, pp. 775-777.]
Adopted Initial Conditions
|
g = 1.4
|
to the
left of x0 |
to the
right of x0 |
r = 1 |
P = 1 |
v = 0 |
r = 0.125 |
P = 0.1 |
v = 0 |
csound = [
g P /
r ]1/2
= 1.183216
|
csound = [
g P /
r ]1/2
= 1.058301
|
Key Positions
Key Velocities
Relationship Between Key Positions and Velocities
at any time t greater than 0.
Shock jump conditions imply :
So, if we can figure out the values of vpost
and Ppost, we can determine vshock.
Determining Ppost
and vpost
The value of P that satisfies the following expression is the desired
value of Ppost.
Then,
For example, with g = 1.4, Pright = 0.1,
and rright = 0.125
(as used by Sod), I derive
Ppost = 0.3190.
Hence, vpost = 0.89095.
Determining vshock,
rpost
and
rmiddle
Hence, also,
Notice that, behind the shock (i.e., to the left of x4)
but to the right of the contact
discontinuity (x3), the specific entropy of the fluid has been
increased over its pre-shock value because of heating through the shock. Hence,
( rpost/rright )
= 2.1915 does not equal
( Ppost/Pright )1/g
=
2.290 .
But this adiabatic relationship between P and r must
be hold for the material that is to the left of the discontinuity
(x3), that is,
Hence, rmiddle = 0.44215.
Solution Between x1
and x2
Finally in the rarefaction wave,
Then, at all locations,