9.3 Program RIEMANN9 Additional Information9.1 Algorithms to recover primitive

9.2 Spectral decomposition of the 3D SRHD equations 

The full spectral decomposition including the right and left eigenvectors of the Jacobian matrices associated to the SRHD system in 3D has been first derived by Donat et al. [43]. Previously, Martí et al. [107] obtained the spectral decomposition in 1D SRHD, and Eulderink [49] and Font et al. [58] the eigenvalues and right eigenvectors in 3D. The Jacobians are given by

  equation2868

where the state vector tex2html_wrap_inline5699 and the flux vector tex2html_wrap_inline5701 are defined in (6Popup Equation) and (7Popup Equation), respectively. In the following we explicitly give both the eigenvalues and the right and left eigenvectors of the Jacobi matrix tex2html_wrap_inline6647 only (the cases i = y, z are easily obtained by symmetry considerations).

The eigenvalues of matrix tex2html_wrap_inline6651 are

  equation2886

and

  equation2901

A complete set of right-eigenvectors is

equation2905

equation2913

equation2917

equation2921

where

equation2931

The corresponding complete set of left-eigenvectors is

equation2941

equation2946

equation2951

equation2956

where tex2html_wrap_inline6653 is the determinant of the matrix of right-eigenvectors, i.e.,

equation2995

For an ideal gas equation of state tex2html_wrap_inline6655, i.e., tex2html_wrap_inline6657, and hence tex2html_wrap_inline6659 for tex2html_wrap_inline6661 .



9.3 Program RIEMANN9 Additional Information9.1 Algorithms to recover primitive

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de