1. An Advanced Simulation & Computing (ASC) Academic Strategic Alliances Program (ASAP) Center at The University of Chicago The Center for Astrophysical Thermonuclear Flashes A Special Relativistic Module for the FLASH Code A. Mignone Flash Code Tutorial May 14, 2004

2. The ASC/Alliances Center for Astrophysical Thermonuclear Flashes The University of Chicago Motivations qA wide variety of astrophysical flows exhibit relativistic behavior:  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 – v 2 ) 1/2 >> 1)  thermodynamical, c s  c  Relativistic flows with  > (3/2) 1/2 are always supersonic, and therefore shock-capturing methods are essential ( Martí and Müller, 2003 ).

3. The ASC/Alliances Center for Astrophysical Thermonuclear Flashes The University of Chicago Special Relativistic Hydrodynamics (RHD) qThe motion of an ideal fluid in RHD is governed by particle number and energy- momentum conservation (Weinberg, 72): qthe system is hyperbolic in nature (Anile, 89) qclosure is provided by an equation of state (Eos) D = Lab Density m = momentum density v = velocity E = energy density  = proper rest density U = four-velocity h = specific enthalpy p = pressure

4. 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

5. 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): Cartesian Geometry 1, 2 and 3D Ideal EoS  Under current development:  Extension to arbitrary geometry  More EoS:  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 o sound speed  eigenvectors & eigenvalues o the Riemann solver o the mappers  All the information is coded in rhd_eos.F90  Gravity (weak limit)

6. 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

7. The ASC/Alliances Center for Astrophysical Thermonuclear Flashes The University of Chicago PPM Interpolation (rhd_state.F90) qStart with primitive states at t n, qApply monotonicity constraints (see Mignone et al, 2004 submitted) qAdditional (relativistic) constraint:

8. The ASC/Alliances Center for Astrophysical Thermonuclear Flashes The University of Chicago Time Integration (rhd_state.F90) qConsider quasi-linear form: qUse Taylor expansion: qCharacteristic tracing: i i+1/2 i-1/2 x t tntn 3 >0 1 <0 2 <0

9. The ASC/Alliances Center for Astrophysical Thermonuclear Flashes The University of Chicago Riemann Solver (rhd_riemann.F90) qSolve Riemann problem given left and right states U L and U R. ; qRiemann problem: evolution of a discontinuity separating two constant hydrodynamical states; qAs 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.

10. The ASC/Alliances Center for Astrophysical Thermonuclear Flashes The University of Chicago Applications Relativistic Shock tube  =10 Jet Jet through collapsars (GRB),   50 2-D Riemann Problem  =6 Jet