9.3 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. [75]. Previously, Martí et al. [179] obtained the spectral decomposition in 1D SRHD, and Eulderink [83] and Font et al. [92] derived the eigenvalues and right eigenvectors in 3D. The Jacobians are given by
i ∂Fi(u ) ā„¬ = -------, (79 ) ∂u
where the state vector u and the flux vector i F are defined in Equations (6View Equation) and (7View Equation), respectively. In the following we explicitly give both the eigenvalues and the right and left eigenvectors of the Jacobi matrix ā„¬x only (the cases i = y, z are easily obtained by symmetry considerations).

The eigenvalues of matrix ā„¬x (u) are

( āˆ˜ ---------------------------------) ----1--- x 2 2 x x 2 x x 2 λ ± = 1 − v2c2s v (1 − cs) ± cs (1 − v )[1 − v v − (v − v v )cs] , (80 )
and
λ0 = vx (triple). (81 )
A complete set of right-eigenvectors is
( š’¦ š’¦ ) r0,1 = ----,vx,vy,vz,1 − ---- , (82 ) hW hW ( ) r0,2 = W vy,2hW 2vxvy,h(1 + 2W 2vyvy ),2hW 2vyvz, 2hW 2vy − W vy , (83 ) ( z 2 x z 2 y z 2 z z 2 z z) r0,3 = W v ,2hW v v ,2hW v v ,h(1 + 2W v v ),2hW v − W v , (84 ) r = (1,hW š’œ λ ,hW vy,hW vz,hW š’œ − 1), (85 ) ± ± ± ±
where
--&tidle;κ--- κ- 1 −-vxvx- š’¦ ≡ &tidle;κ − c2, &tidle;κ = ρ , š’œ ± ≡ 1 − vx λ . (86 ) s ±
The corresponding complete set of left-eigenvectors is
l = -W----(h − W, W vx,W vy,W vz,− W ), (87 ) 0,1 š’¦ − 1 1 y x y x x y l0,2 = -------x-x-(− v ,v v ,1 − v v ,0,− v ), (88 ) h(1 − v v ) l = -----1-----(− vz,vxvz,0,1 − vxvx, − vz), (89 ) 0,3 h(1 − vxvx) ( hW š’œ (vx− λ )− vx − W 2(v2− vxvx)(2š’¦ − 1)(vx − š’œ λ ) + š’¦ š’œ λ ) | ± ± ± ± ± ± | || 1 + W 2(v2 − vxvx)(2š’¦ − 1)(1 − š’œ ) − š’¦ š’œ || 2|| ± ± || lāˆ“ = (±1 )h--|| W 2vy(2š’¦ − 1 )š’œ ± (vx − λ ±) || , (90 ) Δ || || || W 2vz(2š’¦ − 1 )š’œ ± (vx − λ ±) || ( ) − vx − W 2(v2 − vxvx )(2š’¦ − 1)(vx − š’œ ±λ± ) + š’¦ š’œ ±λ ±
where Δ is the determinant of the matrix of right-eigenvectors, i.e.,
Δ = h3W (š’¦ − 1)(1 − vxvx)(š’œ λ − š’œ λ ). (91 ) + + − −
For an ideal gas equation of state š’¦ = h, i.e., š’¦ > 1, and hence Δ ā„= 0 for |vx| < 1.
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"Numerical Hydrodynamics in Special Relativity"
José Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-2003-7		 © Max Planck Society and the author(s)
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