The Center for Astrophysical Thermonuclear Flashes
A Special Relativistic Module
for the FLASH Code
A. Mignone
Flash Code Tutorial
May 14, 2004
An Advanced Simulation & Computing (ASC)
Academic Strategic Alliances Program (ASAP) Center
at The University of Chicago
Motivations
 A wide variety of astrophysical flows exhibit relativistic behavior:
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accretion around compact objects (NS, BH);
jets in extragalactic radio sources;
pulsar winds;
gamma ray bursts;
 (special) relativistic effects are twofold:
 kinematical,
v  c ( = 1/(1 – v2)1/2 >> 1)
 thermodynamical, cs  c
 Relativistic flows with  > (3/2)1/2 are always supersonic, and
therefore shock-capturing methods are essential (Martí and Müller,
2003).
The ASC/Alliances Center for Astrophysical Thermonuclear Flashes
The University of Chicago
Special Relativistic Hydrodynamics (RHD)
 The motion of an ideal fluid in RHD is governed by particle number and energymomentum conservation (Weinberg, 72):
 = proper rest density
U = four-velocity
h = specific enthalpy
p = pressure
D = Lab Density
m = momentum density
v = velocity
E = energy density
 the system is hyperbolic in nature (Anile, 89)
 closure is provided by an equation of state (Eos)
The ASC/Alliances Center for Astrophysical Thermonuclear Flashes
The University of Chicago
The FLASH RHD module
The flash RHD module is based on the following 3 steps:
- interpolation (PPM or TVD)
- characteristic tracing
- Riemann Solver  Final update
The ASC/Alliances Center for Astrophysical Thermonuclear Flashes
The University of Chicago
Current Status
 RHD module is currently working with FLASH 2.3 (official release 2.4):
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Cartesian Geometry
1, 2 and 3D
Ideal EoS
Under current development:
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Extension to arbitrary geometry
More EoS:
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The module is designed for a general EoS, best given as h = h(p, ) (Mignone et al, 2004)
A few algebraic relations are required by
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o
o
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sound speed  eigenvectors & eigenvalues
the Riemann solver
the mappers
All the information is coded in rhd_eos.F90
Gravity (weak limit)
The ASC/Alliances Center for Astrophysical Thermonuclear Flashes
The University of Chicago
Conservative/Primitive Mappers
 Two sets of variables, conservative U and primitive V:
 Conversion is handled by the C2P and P2C mappers
 C2P requires solving a non-linear equation to findpressure,  time
consuming
The ASC/Alliances Center for Astrophysical Thermonuclear Flashes
The University of Chicago
PPM Interpolation (rhd_state.F90)
 Start with primitive states at tn,
 Apply monotonicity constraints (see Mignone et al, 2004 submitted)
 Additional (relativistic) constraint:
The ASC/Alliances Center for Astrophysical Thermonuclear Flashes
The University of Chicago
Time Integration (rhd_state.F90)
 Consider quasi-linear form:
 Use Taylor expansion:
 Characteristic tracing:
t
2<0
x
tn
3>0
1<0
i-1/2
i
i+1/2
The ASC/Alliances Center for Astrophysical Thermonuclear Flashes
The University of Chicago
Riemann Solver (rhd_riemann.F90)
 Solve Riemann problem given left and right states UL and UR. ;
 Riemann problem: evolution of a discontinuity separating two constant
hydrodynamical states;
 As in the Newtonian case, the solution is self-similar, i.e. function of x/t;
 Two-shock approximation  rarefaction waves are treated as shocks
Rankine-Hugoniot jump conditions for
a relativistic flows (Marti, 1994)
Pressure and velocity (p*,v*) are continuous across the contact
discontinuity. The same is true in the Newtonian limit.
The ASC/Alliances Center for Astrophysical Thermonuclear Flashes
The University of Chicago
Applications
2-D Riemann Problem
Relativistic
Shock tube
 =10 Jet
 =6 Jet
Jet through collapsars
(GRB),   50
The ASC/Alliances Center for Astrophysical Thermonuclear Flashes
The University of Chicago