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A Special Relativistic Module for the FLASH Code

A Special Relativistic Module for the FLASH Code. A. Mignone Flash Code Tutorial May 14, 2004. Motivations. A wide variety of astrophysical flows exhibit relativistic behavior: accretion around compact objects (NS, BH); jets in extragalactic radio sources; pulsar winds; gamma ray bursts;

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A Special Relativistic Module for the FLASH Code

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  1. A Special Relativistic Modulefor the FLASH Code A. Mignone Flash Code Tutorial May 14, 2004

  2. Motivations • A 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 – 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).

  3. Special Relativistic Hydrodynamics (RHD) • The motion of an ideal fluid in RHD is governed by particle number and energy-momentum conservation (Weinberg, 72): • the system is hyperbolic in nature (Anile, 89) • closure 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 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. 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 • sound speed  eigenvectors & eigenvalues • the Riemann solver • the mappers • All the information is coded inrhd_eos.F90 • Gravity (weak limit)

  6. 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. PPM Interpolation (rhd_state.F90) • Start with primitive states at tn, • Apply monotonicity constraints (see Mignone et al, 2004 submitted) • Additional (relativistic) constraint:

  8. t 2<0 x tn 3>0 1<0 Time Integration (rhd_state.F90) • Consider quasi-linear form: • Use Taylor expansion: • Characteristic tracing: i-1/2 i i+1/2

  9. 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.

  10. Applications 2-D Riemann Problem Relativistic Shock tube  =10 Jet  =6 Jet Jet through collapsars (GRB),   50

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