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A Relativistic Magnetohydrodynamic (RMHD) C ode Based on an Upwind S cheme

A Relativistic Magnetohydrodynamic (RMHD) C ode Based on an Upwind S cheme. for isothermal flows. Hanbyul Jang, D ongsu Ryu Chungnam National University, Korea. HEDLA 2012 April 30-May 4, 2012 .

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A Relativistic Magnetohydrodynamic (RMHD) C ode Based on an Upwind S cheme

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  1. A Relativistic Magnetohydrodynamic(RMHD) Code Based on an Upwind Scheme for isothermal flows Hanbyul Jang, DongsuRyu Chungnam National University, Korea HEDLA 2012 April 30-May 4, 2012

  2. Relativistic Magnetohydrodynamic Codes 1. Introduction Centaurus A (NGC 5128) GRB 031203 Relativistic flows are involved in many high energy astrophysical phenomena. Magnetic fields are commonly linked up to these. GRO J1655-40 HEDLA 2012 April 30-May 4, 2012

  3. Relativistic Magnetohydrodynamic Codes Building RMHD codes based on upwind schemes is a challenging project, 1. Introduction Because the eigenvalues and eigenvectors for RMHDs have not yet been analytically given. Previous studies Using fully upwind schemes Balsara(2001) – numerical calculations of eigenvalues and eigenvectors Anton et al (2010) - analytic formulae for eigenvectors, but numerical calculations of eigenvalues Most of recent codes are based on HLL, HLLC, HLLD... HEDLA 2012 April 30-May 4, 2012

  4. Isothermal fluids Same temperature? Constant sound speed! When the constituent particles are UR 2. In degenerate matters (Weinberg 1972) 1. Introduction fluid’s density dose not change ~ constant We can ignore the density of the fluids and make the problem simpler 1/3 A step to move toward the full adiabatic RMHD code HEDLA 2012 April 30-May 4, 2012

  5. Conservation equations for isothermal flows Momentum density Total energy density 2. Conservation Equations state vector (c = 1) flux vector Lorentz factor Mass conservation equation is dropped out Equation of state for isothermal fluids p : gas pressure e : mass-energy density cs : sound speed Bi: Magnetic field : fluid velocity HEDLA 2012 April 30-May 4, 2012

  6. Jacobian matrix Aj 2. Conservation Equations parameter vector HEDLA 2012 April 30-May 4, 2012

  7. Analytic forms for Eigenvalues Compressible modes Alfven modes 3. Eigen-Structure General solutions for quartic equations are too complicated to use in the code. Eigenvalues are correspond to characteristic speeds of the fluids, so they have to be real. Using this criteria, we have obtained analytic, usable forms for eigenvalues. HEDLA 2012 April 30-May 4, 2012

  8. Analytic forms for Eigenvalues Compressible modes Alfven modes 3. Eigen-Structure A = …, B= …, C= … HEDLA 2012 April 30-May 4, 2012

  9. Analytic forms of Eigenvalues Compressible modes Alfven modes 3. Eigen-Structure HEDLA 2012 April 30-May 4, 2012

  10. Analytic forms of Eigenvectors Right Eigenvectors 3. Eigen-Structure Left Eigenvectors HEDLA 2012 April 30-May 4, 2012

  11. Analytic forms of Eigenvectors Right Eigenvectors 3. Eigen-Structure Left Eigenvectors If some eigenvalues become same, all components of the eigenvectors go to zero • →Degeneracy issue HEDLA 2012 April 30-May 4, 2012

  12. Degeneracy Case1 When Bx =0, 4. Degeneracy issue Degeneracy Case2 When and (By = Bz = 0 in NR) Degeneracy Case3 When and , and (By = Bz = 0 and cs = cA in NR) HEDLA 2012 April 30-May 4, 2012

  13. Using the alaytic expressions of eigenvalues and eigenvectors, An one-dimensional RMHD code for isothermal fluids based on the total variation diminishing (TVD) scheme was built. 5. Numerical Tests One-dimensional shock tube test HEDLA 2012 April 30-May 4, 2012

  14. Bruno (2006) -adiabatic TVD -isothermal SR SS FR FS 5. Numerical Tests FR: fast rarefaction wave SR: slow rarefaction wave FS: fast shock SS: slow shock HEDLA 2012 April 30-May 4, 2012

  15. Bruno (2006) -adiabatic TVD -isothermal RD SS SS FR FS RD 5. Numerical Tests Generic Alfven test RD: rotational discontinuity HEDLA 2012 April 30-May 4, 2012

  16. SS FR FR 5. Numerical Tests Slow compound Brio&Wu (1988) test HEDLA 2012 April 30-May 4, 2012

  17. Building RMHD codes based on upwind schemes is a challenging project because of an absence of analytic formula for eigenvalues and eigenvectors. 6. Conclusions We have obtained analytic forms for the eigenvalues and eigenvectors. Using these analytic expressions we have successfully built a code based on the TVD scheme for the case of isothermal RMHD. Based on this study, we will go toward a full adiabatic RMHD code. HEDLA 2012 April 30-May 4, 2012

  18. Thank you

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