Yin-Yang grid in Numerical Relativity
Wasilij Barsukow1,2
Pedro Montero1, Ewald Muller1, Thomas W. Baumgarte3
1Max-Planck-Institute for Astrophysics Garching2University of Heidelberg, 3Bowdoin College
June 30, 2014
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Introduction
Astrophysical applications
# Astrophysical objects often have roughly spherical shapes with a largevariety of different length scales
# In certain situations a cartesian grid induces artefacts that spoil theresults of the simulation
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Introduction
RelativityBaumgarte et al. (2012)
γij := e−4φγij conformal metric
Aij := e−4φ(Kij −
1
3γijK
)conformally rescaled traceless extr. curv.
Γijk background connection (spherical polar!)
∆Γijk := Γijk − Γijk , Λi := γjk∆Γijk
∂⊥ := ∂t − Lβ
% := nanbTab
Si := −γianbTab
Sij := γiaγjbTab
S := γijSij
with na = (−α, 0, 0, 0)
∂⊥γij = −2
3γij Dkβ
k − 2αAij
∂⊥Aij = −2
3Aij Dkβ
k − 2αAik Akj + αAijK + e4φ
[−2αDi Djφ + 4αDiφDjφ + 4D(iαDj)φ− Di Djα + α(Rij − 8πSij )
]TF
∂⊥φ =1
6Dkβ
k −1
6αK
∂⊥K =α
3K2 + αAij A
ij − e−4φ(D2α + 2D i
αDiφ) + 4πα(% + S)
∂⊥Λi = γjk Dj Dkβ
i +2
3Λi Djβ
j +1
3D i Djβ
j − 2Ajk (δi j∂kα− 6αδi j∂kφ− α∆Γijk )−4
3αγ
ij∂jK − 16παγ ijSj
Rij = −1
2γkl Dk D`γij + γk(i Dj)Λk + Λk∆Γ(ij)k + γ
k`(2∆Γmk(i∆Γj)m` + ∆Γmik∆Γmjl )
∂tα = −2αK
∂tβi = B i
∂tBi =
3
4∂t Λi
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Introduction
RelativityBaumgarte et al. (2012)
γij := e−4φγij conformal metric
Aij := e−4φ(Kij −
1
3γijK
)conformally rescaled traceless extr. curv.
Γijk background connection (spherical polar!)
∆Γijk := Γijk − Γijk , Λi := γjk∆Γijk
∂⊥ := ∂t − Lβ
% := nanbTab
Si := −γianbTab
Sij := γiaγjbTab
S := γijSij
with na = (−α, 0, 0, 0)
∂⊥γij = −2
3γij Dkβ
k − 2αAij
∂⊥Aij = −2
3Aij Dkβ
k − 2αAik Akj + αAijK + e4φ
[−2αDi Djφ + 4αDiφDjφ + 4D(iαDj)φ− Di Djα + α(Rij − 8πSij )
]TF
∂⊥φ =1
6Dkβ
k −1
6αK
∂⊥K =α
3K2 + αAij A
ij − e−4φ(D2α + 2D i
αDiφ) + 4πα(% + S)
∂⊥Λi = γjk Dj Dkβ
i +2
3Λi Djβ
j +1
3D i Djβ
j − 2Ajk (δi j∂kα− 6αδi j∂kφ− α∆Γijk )−4
3αγ
ij∂jK − 16παγ ijSj
Rij = −1
2γkl Dk D`γij + γk(i Dj)Λk + Λk∆Γ(ij)k + γ
k`(2∆Γmk(i∆Γj)m` + ∆Γmik∆Γmjl )
∂tα = −2αK
∂tβi = B i
∂tBi =
3
4∂t Λi
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Introduction
Relativistic Hydrodynamics
Montero et al. (2013)h := 1 + ε +
p
%0
enthalpy
W := −naua = αut Lorentz factor
D := W%0 density seen by normal observer
Si := αTit = W 2
%0hvi momentum density seen by normal observer
τ := W 2%0h − p − D internal energy seen by normal observer
∂t
(e6φ
√γ
γD
)+ ∂j (fD )j = −(fD )j Γkjk f
jD
= αe6φ
√γ
γD
(v i −
βi
α
)
∂t
(e6φ
√γ
γSi
)+ ∂j (fS )i
j = (sS )i + (fS )kj Γkji − (fS )i
k Γjkj
(fS )ij = αe6φ
√γ
γ(W 2
%0hvi
(v j −
βj
α
)+ pδi
j )
(sS )i = αe6φ
√γ
γ
(−T 00
α∂iα + T 0k Diβ
k +
1
2(T 00
βjβk + 2T 0j
βk + T jk )Diγjk
)
∂t
(e6φ
√γ
γτ
)+ ∂j (fτ )j = sτ − (fτ )k Γ
jjk
(fτ )i = αe6φ
√γ
γ
(τ
(v j −
βj
α
)+ pv j
)
sτ = αe6φ
√γ
γ
(T 00(βi
βjKij − β
i∂iα)+
T 0i (2βjKij − ∂iα) + T ijKij
)6 / 273
Spherical polar grids
Polar is different
# Relevant length in the CFL-condition ∼ min(∆r , ∆r∆ϑ, ∆r∆ϑ∆φ)
# Coordinate singularities: e. g. in two dimensions
vr = vx cosφ+ vy sinφ
vφ = −vx sinφ+ vy cosφ
are not defined at the origin.
# Boundary conditions: can involve values not at the physical boundariesof the domain
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Spherical polar grids
Polar is different
# Relevant length in the CFL-condition ∼ min(∆r , ∆r∆ϑ, ∆r∆ϑ∆φ)
# Coordinate singularities: e. g. in two dimensions
vr = vx cosφ+ vy sinφ
vφ = −vx sinφ+ vy cosφ
are not defined at the origin.
# Boundary conditions: can involve values not at the physical boundariesof the domain
# Equations (i.p. flux functions) explicitly space-dependent
# Source terms
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Spherical polar grids
Free-stream in polar coordinates
2D hydrodynamics:
∂t% + r−1∂r (r%vr ) + r−1∂φ(%vφ) = 0
∂t(%vr ) + r−1∂r (r(%v2r + P)) + r−1∂φ(%vφvr ) = r−1(%v2
φ + P)
∂t(%vφ) + r−1∂r (r(%vrvφ) + r−1∂φ(%v2φ + P) = −r−1%vrvφ
∂te + r−1∂r (r(e + P)vr ) + r−1∂φ((e + P)vφ) = 0
Taking for simplicity ~v ≡ 0 and all other quantities constant:
∂t% = 0
∂t(%vr ) + r−1∂r (r)P = r−1P
∂φP = 0
∂te = 0
In 3D sinϑ factors lead to similar issues.
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Spherical polar grids
Free-stream in polar coordinates
Momentum density vanishes: S i = 0 = βi , all quantities are functions of ronly. Take γ to be the flat metric in spherical polar coordinates:diag(1, r2, r2 sin2 ϑ). Momentum equation in ϑ-direction
∂t
(e6φ
√γ
γSϑ
)+ ∂j(fS)ϑ
j = (sS)ϑ + (fS)kj Γk
jϑ − (fS)ϑk Γj
kj
contains
∂j(fS)ϑj = ∂j
(αe6φ
√γ
γpδi
j
)In the original Valencia formulation γ = 1, thus
∂j(fS)ϑj = αe6φpr2∂ϑ(sinϑ)
which is cancelled analytically (but not numerically) by
(sS)ϑ = αe6φr2 sinϑpγjk∂ϑγjk
2
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Spherical polar grids
Free-stream in polar coordinates
Looking at this term again with γ being the flat metric in spherical polarcoordinates yields
∂j(fS)ϑj = ∂ϑ
(αe6φp
)which vanishes by itself, as does the source term.
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Yin-Yang grid
Yin-Yang gridoriginal publication: Kageyama, Sato (2004)
hydrodynamics: Wongwathanarat, Hammer, Muller (2010)
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Yin-Yang grid
Yin-Yang gridoriginal publication: Kageyama, Sato (2004)
hydrodynamics: Wongwathanarat, Hammer, Muller (2010)
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Yin-Yang grid
Yin-Yang gridoriginal publication: Kageyama, Sato (2004)
hydrodynamics: Wongwathanarat, Hammer, Muller (2010)
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Yin-Yang grid
Yin-Yang gridoriginal publication: Kageyama, Sato (2004)
hydrodynamics: Wongwathanarat, Hammer, Muller (2010)
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Yin-Yang grid
Yin-Yang grid
xyz
7→ −1
11
xyz
cos ϑ = sinφ sinϑ tan φ =cosϑ
− cosφ sinϑ
(vϑvφ
)=
(− sinφ sin φ − cosφ/ sin ϑ
cosφ/ sin ϑ − sin φ sinφ
)(vϑvφ
)
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Yin-Yang grid
Initial data
Two coordinate systems: two choices of coordinates!
Two examples:
1): Switch from orthonormal to coordinate basis: factors (sinϑ) involvecoordinates of patch.2): Setup of initial data: any functions of ϑ or φ involve only the angles ofthe usual spherical polar coordinate system.
# Note that as the trafo is orthogonal, determinants are unaltered.
# Non-scalar quantities need to be transformed during initialization.
# Axisymmetric setup not possible with a Yin-Yang grid!
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Yin-Yang grid
Initial data
Two coordinate systems: two choices of coordinates!
Two examples:1): Switch from orthonormal to coordinate basis: factors (sinϑ) involvecoordinates of patch.
2): Setup of initial data: any functions of ϑ or φ involve only the angles ofthe usual spherical polar coordinate system.
# Note that as the trafo is orthogonal, determinants are unaltered.
# Non-scalar quantities need to be transformed during initialization.
# Axisymmetric setup not possible with a Yin-Yang grid!
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Yin-Yang grid
Initial data
Two coordinate systems: two choices of coordinates!
Two examples:1): Switch from orthonormal to coordinate basis: factors (sinϑ) involvecoordinates of patch.2): Setup of initial data: any functions of ϑ or φ involve only the angles ofthe usual spherical polar coordinate system.
# Note that as the trafo is orthogonal, determinants are unaltered.
# Non-scalar quantities need to be transformed during initialization.
# Axisymmetric setup not possible with a Yin-Yang grid!
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Yin-Yang grid
Initial data
Two coordinate systems: two choices of coordinates!
Two examples:1): Switch from orthonormal to coordinate basis: factors (sinϑ) involvecoordinates of patch.2): Setup of initial data: any functions of ϑ or φ involve only the angles ofthe usual spherical polar coordinate system.
# Note that as the trafo is orthogonal, determinants are unaltered.
# Non-scalar quantities need to be transformed during initialization.
# Axisymmetric setup not possible with a Yin-Yang grid!
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Yin-Yang grid
Boundary and parity conditions
# no angular parity conditions
# ghostcells are filled via interpolationfrom other patch
# non-scalar quantities need to betransformed onto the right patch
# procedure exactly the same for both
patches: −1
11
2
= idR3
# radial boundary / parity unaltered:singularity at r = 0: treated as inBaumgarte at al. (2012) using PIRKschemes
# interpolation may also be needed withradial ghost cells!
But: This is a point-value interpolation!
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Yin-Yang grid
Boundary and parity conditions
# no angular parity conditions
# ghostcells are filled via interpolationfrom other patch
# non-scalar quantities need to betransformed onto the right patch
# procedure exactly the same for both
patches: −1
11
2
= idR3
# radial boundary / parity unaltered:singularity at r = 0: treated as inBaumgarte at al. (2012) using PIRKschemes
# interpolation may also be needed withradial ghost cells!
But: This is a point-value interpolation!18 / 273
Yin-Yang grid
Yin-Yang grid
+ Easy implementation into existing spherical polar coordinates
+ No artificial boundary/parity conditions in ϑ and φ direction
+ Relevant length for CFL-condition in spherical polar coordinates:
∆`CFL ∼ ∆r∆φ∆ϑ
Now:
∆`(YY)CFL ∼ ∆r∆φ sinϑmin ∼ ∆r∆φ
⇒ expected speedup ∝ Nϑ (this is optimistic of course)
+ Interpolation found not to introduce any numerical instability
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Yin-Yang grid
Yin-Yang grid
− Overlap zones involve interpolations, and for non-scalar quantities alsotransformations
− Overlaps can still not be fully handled without violatingconservativeness of the algorithm
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Test problems
Quadrupole gravitational waves Teukolsky (1982)textbook Baumgarte, Shapiro (9.1.2)
m = 022 / 273
Test problems
Quadrupole gravitational waves Teukolsky (1982)textbook Baumgarte, Shapiro (9.1.2)
m = 223 / 273
Conclusion
Last slide
Yin-Yang grid
# allows for a reduction of the time step and avoidance of axialsingularities
# needs interpolations and transformations, but is largely based on usualspherical polar coordinates
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Alternative form
∂t(%vr ) + r−1∂r (r(%v2r + P)) + r−1∂φ(%vφvr ) =
%v2φ
r+
P
r
∂t(%vr ) + r−1∂r (r%v2r ) + r−1∂φ(%vφvr ) =
%v2φ
r− ∂rP
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Spherical polar coordinates
∂t% +1
r2∂r (r2
%vr ) +1
r sinϑ∂ϑ(sinϑ%vϑ) +
1
r sinϑ∂φ(%vφ) = 0
∂t (%vr ) +1
r2∂r (r2(%v2
r + P)) +1
r sinϑ∂ϑ(sinϑ%vϑvr ) +
1
r sinϑ∂φ(%vφvr ) =
2P + %(v2φ + v2
ϑ)
r
∂t (%vϑ) +1
r2∂r (r2
%vr vϑ) +1
r sinϑ∂ϑ(sinϑ(%v2
ϑ + P)) +1
r sinϑ∂φ(%vφvϑ) =
cosϑ(P + %v2φ)
r sinϑ−%vϑvr
r
∂t (%vφ) +1
r2∂r (r2
%vr vφ) +1
r sinϑ∂ϑ(sinϑ%vϑvφ) +
1
r sinϑ∂φ(%v2
φ + P) = −%vϑvφ cosϑ
r sinϑ−%vr vφ
r
∂t e +1
r2∂r (r2vr (e + P)) +
1
r sinϑ∂ϑ(sinϑvϑ(e + P)) +
1
r sinϑ∂φ(vφ(e + P)) = 0
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