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6.2 Characteristic fields in the 3+1 conservative Eulerian formulation of Section 2.1.3

This section collects all information concerning the characteristic structure of the general-relativistic hydrodynamics equations in the Eulerian formulation of Section 2.1.3. As explained in Section 3.1.2, this information is necessary in order to implement approximate Riemann solvers in HRSC finite-difference schemes.

We present only the characteristic speeds and fields corresponding to the x-direction. Equivalent expressions for the other two directions can be obtained easily from symmetry considerations. The characteristic speeds (eigenvalues) of the system are given by:

λ0 = αvx − βx (triple), (106 ) α { ∘ ------------------------------------} λ± = ------22 vx (1 − c2s) ± cs (1 − v2)[γxx(1 − v2c2s) − vxvx (1 − c2s)] 1 − v cs − βx, (107 )
where cs denotes the local sound speed, which can be obtained from 2 -p hcs = χ + ρ2κ, with ∂p χ ≡ ∂ρ and κ ≡ ∂∂pɛ. We note that the Minkowskian limit of these expressions is recovered properly (see [101Jump To The Next Citation Point]) as well as the Newtonian one (λ0 = vx, λ± = vx ± cs).

A complete set of right eigenvectors is given by (superscript T denotes transpose):

( )T r0,1 = -𝒦--,vx,vy,vz,1 − -𝒦-- , (108 ) hW hW
( r0,2 = W vy,h(γxy + 2W 2vxvy),h(γyy + 2W 2vyvy),h (γzy + 2W 2vzvy), T W vy(2hW − 1)) , (109 )
( 2 2 2 r0,3 = W vz,h(γxz + 2W vxvz),h(γyz + 2W vyvz),h (γzz + 2W vzvz), T W vz(2hW − 1)) , (110 )
r ± = (1,hW 𝒞x,hW vy,hW vz,hW 𝒜&tidle;x − 1)T , (111 ) ± ±
where the following auxiliary quantities are used:
&tidle;κ x x 𝒦 ≡ -----2, &tidle;κ ≡ κ∕ρ, 𝒞± ≡ vx − 𝒱 ±, (112 ) &tidle;κ − cs
vx − Λx γxx − vxvx 𝒱x± ≡ -xx----x±-x- 𝒜&tidle;x± ≡ -xx----x--x, (113 ) γ − v Λ ± γ − v Λ ±
Λi± ≡ &tidle;λ± + &tidle;βi, λ&tidle;≡ λ∕α, &tidle;βi ≡ βi∕α. (114 )
Finally, a complete set of left eigenvectors is given by:
W x y z T l0,1 = ------(h − W, W v ,W v ,W v ,− W ) , (115 ) 𝒦 − 1
⌊ − γzzvy + γyzvz ⌋ | | || x || || v (γzzvy − γyzvz) || 1 || || l0,2 = ---|| γzz(1 − vxvx) + γxzvzvx || , (116 ) hξ || || ||− γ (1 − v vx) − γ v vx|| || yz x xz y || ⌈ ⌉ − γzzvy + γyzvz ⌊ − γyyvz + γzyvy ⌋ | | || x || || v (γyyvz − γzyvy) || 1 || || l0,3 = ---||− γzy(1 − vxvx) − γxyvzvx|| , (117 ) hξ || || || γ (1 − v vx) + γ v vx || || yy x xy y || ⌈ ⌉ − γyyvz + γzyvy ⌊ hW 𝒱x ξ + Δ-l(5) ⌋ || ± h2 ∓ || || x x 2 x x || || Γ xx(1 − 𝒦 𝒜&tidle;±) + (2𝒦 − 1)𝒱 ±(W v ξ − Γ xxv )|| 2|| || l∓ = ± h-|| Γ xy(1 − 𝒦 𝒜&tidle;x ) + (2𝒦 − 1)𝒱x (W 2vyξ − Γ xyvx )|| , (118 ) Δ || ± ± || || x x 2 z x || || Γ xz(1 − 𝒦 𝒜&tidle;±) + (2𝒦 − 1)𝒱 ±(W v ξ − Γ xzv )|| ⌈ ⌉ (1 − 𝒦 )[− γvx + 𝒱x±(W 2ξ − Γ xx)] − 𝒦W 2𝒱x±ξ
where the following relations and auxiliary quantities have been used:
&tidle;x x x &tidle;x &tidle; x x x x 1 − 𝒜 ± = v 𝒱±, 𝒜 ± − 𝒜 ∓ = v (𝒞± − 𝒞∓ ), (119 )
(𝒞x± − 𝒞x± ) + (𝒜&tidle;x∓ 𝒱x± − 𝒜&tidle;x±ð’±x∓) = 0, (120 )
3 x x x x Δ ≡ h W (𝒦 − 1)(𝒞 + − 𝒞− )ξ ξ ≡ Γ xx − γv v , (121 )
γ ≡ detγ = γ Γ + γ Γ + γ Γ , (122 ) ij xx xx xy xy xz xz
Γ xx = γyyγzz − γ2, Γ xy = γyzγxz − γxyγzz, Γ xz = γxyγyz − γxzγyy, (123 ) yz
Γ xxvx + Γ xyvy + Γ xzvz = γvx. (124 )
These two sets of eigenfields reduce to the corresponding ones in the Minkowskian limit [101].


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