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

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

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

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

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

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

3. Eigen-Structure Analytic forms for Eigenvalues Compressible modes Alfven modes HEDLA 2012 April 30-May 4, 2012 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. General solutions for quartic equations are too complicated to use in the code.

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

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

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

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

4. Degeneracy issue Degeneracy Case1 When B x =0, Degeneracy Case2 Degeneracy Case3 HEDLA 2012 April 30-May 4, 2012 When and When and, and (By = Bz = 0 in NR) (By = Bz = 0 and cs = c A in NR)

5. Numerical Tests 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. vxvyvzByBzpnstep=8192 test1 L Bx=5 R tend=0.4 test2 L Bx=1 R tend=0.375 test3 L000101Bx=0.5 R tend=0.4 One-dimensional shock tube test HEDLA 2012 April 30-May 4, 2012

5. Numerical Tests vxvyvzByBzpnstep=8192 test1 L Bx=5 R tend=0.4 Bruno (2006) -adiabatic TVD -isothermal FR: fast rarefaction wave SR: slow rarefaction wave FS: fast shock SS: slow shock SR SS FS FR HEDLA 2012 April 30-May 4, 2012

5. Numerical Tests vxvyvzByBzBz pnstep=8192 test2 L Bx=1 R tend=0.375 Generic Alfven test Bruno (2006) -adiabatic TVD -isothermal RD: rotational discontinuity FR FS SS RD HEDLA 2012 April 30-May 4, 2012

5. Numerical Tests vxvyvzByBzpnstep=8192 test3 L000101Bx=0.5 R tend=0.4 Brio&Wu (1988) test FR SS Slow compound HEDLA 2012 April 30-May 4, 2012

6. Conclusions Building RMHD codes based on upwind schemes is a challenging project because of an absence of analytic formula for eigenvalues and eigenvectors. 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

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