lecture note on 3+1 formalism of numerical relativity -...
TRANSCRIPT
![Page 1: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/1.jpg)
1
Lecture note on 3+1
formalism of numerical
relativity
Masaru Shibata
(Yukawa Institute, Kyoto U)
2010/06/09
![Page 2: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/2.jpg)
2
Basis equations for Numerical Relativity
4
[ ]
8
0
0
4
0
l l
GG T
c
T
u
u Y Q
F j
F
fp f p S
p
Einstein equation
Matter fields:
Next time if any
![Page 3: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/3.jpg)
3
• Imposing gauge conditions
• Extracting gravitational waves
• Finding black holes
(finding apparent horizon)
• Adaptive mesh refinement
Others in Numerical Relativity
![Page 4: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/4.jpg)
4
Contents
1. Structure of Einstein‟s equation (briefly)
2. 3+1 formalism of Einstein‟s equation
3. BSSN formalism
4. Gauge conditions
5. Initial value problem
6. Implementation of finite differencing
7. Extracting gravitational waves
8. Adaptive mesh refinement
9. Apparent horizon finder
![Page 5: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/5.jpg)
5
Solution of Einstein’s equation for
dynamical phenomena
• We have to solve Einstein‟s equations as an initial value problem
• Einstein‟s equations, G = 8Gc4T , are equations for space and time Space and time coordinates appear in a mixed way; “time coordinate” does not always have the property of time.
E.g., Schwarzschild coordinates; t is time for r > 2M, but is not for r < 2M
Special formalism is necessary to follow dynamics of a variety of spacetimes
![Page 6: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/6.jpg)
6
Schwarzschild spacetime
t is the time
coordinate.
t is a radial
coordinate
1
2 2 2 2 2 2 2 2
2
2 21 sin
GM GMds c dt dr r d d
r rc
Time has to be always “time”
in numerical relativity
t = const
Coordinate
singularity
![Page 7: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/7.jpg)
7
Several formulations
1. 3+1 (N+1) formalism
2. Formulation based on a special
(harmonic) coordinates
(e.g., Pretorius formalism; also often used
in Post-Newtonian theory)
3. Hyperbolic formalism
(e.g., Kidder-Scheel-Teukolsky formalism)
• Others …..
In this lecture, I focus on the first one
![Page 8: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/8.jpg)
8
Einstein’s equation = hyperbolic equations
4
: Einstein tensor
: Ricci tensor
1 8
2
1
2
: P
G
R
GG R g R T
c
R
g g g g
g G g g g g g g t
t
2
2
seudo tensor of Landau-Lifshitz
de-Donder Gauge: 0
O g
g g
G g g g g O g
Wave equation
![Page 9: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/9.jpg)
9
Einstein’s equation is similar to
scalar wave equation
2
or
t
t
t
t i i
i
t i
3+1 way
Post-Newtonian way
Hyperbolic way
, Kij , gij
①
②
③
Similar: However, spacetime is not
a priori given in general relativity
![Page 10: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/10.jpg)
10
Section I: 3+1 (ADM) formalism
Concept
1. Foliate spacetime by spacelike surfaces
2. “Choose” time coordinates for a direction of time at each location
3. Follow dynamics of spacelike surfaces
Spacelike
hypersurfacesTime axes
![Page 11: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/11.jpg)
11
2 Draw timelike
unit normal wrt
spacelike surface,
and choose lapse
1 Choose a
spacelike surface
& give space-metric
3+1 decomposition (N+1)
4 Evolve gij & Kij
tn
Time
Space
t
t + ti
t
3 Choose time axes
at each point
,
~ : Talk later
ij ij
ij ij
K
K
g
g
nn = -1
![Page 12: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/12.jpg)
12
Definition of variables
gij = space metric
Kij= extrinsic curvature
= lapse function
i = shift vector
: 0
1
2
1
2
; 0
i
ij i j n ij
t ij i j j i
n
K n
D D
g n n
n t n
g
g g g
g
g
L
Lie derivative
g (gij, , k)
Dynamics
Gauge
![Page 13: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/13.jpg)
13
Line element
t
n
Time
Space
t
t + t
i
i
x
t
d
t
22 i i j j
ijds dt dx dt dx dt g
![Page 14: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/14.jpg)
14
Structure of variables
2 2 2
2
1, , , 0 ; cf. 1, 0
: lapse function, : shift vector;
, 1/ , /,
* , * , /
,Cf.
i
i j
i ij
k i
k i
ij i jij
k
k i
n n t
g g
g
g g
g
0 ,0
, * , * ,
, 0 ,0 ,
* , * ,
ijij
j j j
ij ij
ij
ij
K KK K
K K
gg g
![Page 15: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/15.jpg)
15
Geometric meaning of Kij
0u n
n
u
0i i
ij j iu K u n
g
Kij denotes the “curved” degree
of chosen spatial hyper-surfaces
If space-like hyper-surface is curved,
n is not parallel-transported.
n n
![Page 16: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/16.jpg)
16
Next step is to rewrite
Einstein’s equation by gij & Kij
As a first step, it is necessary to define
covariant derivative associated with gij
' ' ' ' ' ' '
' ' ' ' ' ' '
' ' ' '
' ' '
0 Required property
: A spacetime tenso
Define
r
Then,
i jk
ijk
lmn
ijk h i j k l m n i j k
h lmn h i j k l m n h l m n
i j k i
i jk i j k i j k i j
D
T
D T T
D
g
g g g g g g g
g g g g g g g
' '
' ' ' ' '
' ' '
' ' ' ' ' '
( )
= 0
j k
k i j k j k
i j k
i j k j i k k i j
g n n
n n n n
g
g g g
projection
OK
![Page 17: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/17.jpg)
17
3+1 definition of Kij
1
2
0
1
21
21
2
ij i j i j
i j i i j
i j i j
ij i j j i i j
j i i j ij
j i i j
K n n n
n g n n g n n n
n n n n
K n n n n n
n n n
n
g g g g
g g
g g g
g g
1
2
ij
t ij i j j i
n n
D D
g
g
![Page 18: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/18.jpg)
18
Usuful, often used relations
: : accelerat
l
ln ion
n
No
K n n n n n
n n D
D
te
n
We will use it for many times
![Page 19: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/19.jpg)
19
3+1 decomposition of Einstein’s equation
• First two eqs = constraint eqs
・・ no second derivative of spatial metric
• Last one = evolution eqs
・・ hyperbolic eqs of spatial metric
• No time derivative for & k
8 : Hamiltonian constraint
8 : Momentum constraint
8 : Evolution equation
k k
i j i j
G n n T n n
G n T n
G T
g g
g g g g
![Page 20: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/20.jpg)
20
Similar to Maxwell’s eqs
4
0
4
i
i e
i
i
ii i
t
ii
t
E
B
E B j
B E
Constraint eqs
Evolution eqs
Step 1: Give an initial condition which
satisfies constraints.
Step 2: Solve evolution equations.
![Page 21: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/21.jpg)
21
All eqs have to be written by (g, K)
• Method: Derive Gauss-Codacci equations
(3) : 3 vector
Using defintion of 3D covariant de
Start fr
rivative,
om definition of 3D Riemann tensor
l
i j j i k ijk l l
a b c l m
i j k i j k a b c l m
a b c a b m
i j k a b c i j k a
D D D D R D
D D
g g g g g
g g g g g g g
l
b l m
a l c m
i j k a c l m
a b c a m c
i j k a b c a m k ij ik j cn K K K
g g g g
g g g g
![Page 22: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/22.jpg)
22
(3)
where we used
ln ln
0
Then, gives
=
c a b b
i
b b b b b b
c a c a a c a a c c a
b b b
a c c ac c a
l m l m m a
j l m j m l m j a
i j j
j
i k
l
i
c a ij
jk l
g n n n n n n n n
n K n D n K n D
n n K n
D
n K
D D D
R
g g
g
g g
g
(3)
1( )
Note:
a b c d
ijkl i j k l ab
a b c l l l
i j k abc l l j ik l i jk
NN a b c d
ijkl i j k l a
cd il jk ik j
bcd il
l
jk ik jl
R K K K K
R R
R R K K K K
K K K K
g g g g
g g g
g g g g
Gauss
Codacci
eq.
![Page 23: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/23.jpg)
23
Gauss-Codacci equations I, continued
(3)
(3) 2
3
3) 2
)
(
(
Multiplying
Multiplying again
2
1
He 2 2 16re
6
,
a b c d
ijkl i j k l abcd il jk ik j
b d a c k
jl j l bd abcd jk l jl
a c
a
il
ad il
i
c a c a c
ad
l
l
ik
j
a b
i
b
l
l
a
R R R n n K K KK
R R R n n K K K
R K K K
R R n n G n
R R K K K
n n
K
T n
g g g g
g
g
g
g
Hamiltonian constraint
a c
abT n n
This will be used later
1 component
![Page 24: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/24.jpg)
24
• Derive Gauss-Codacci equations II
Start fr
om
l
k a b k c a b k d c e
i j i j c a b i j c a b e d
a b k c a b k d c
i j c a b i j c a b d
a b k c e
i j c a e b
a b k c k d c b k e
i j c a b c ij d j i e b
a b k c k
i j c a b
D D n D n n
n n
n
n K n n K n n
n D
g g g g g g g g
g g g g g g g
g g g g
g g g g g
g g g
n
where ln , 0
ij
d c c e
d e b
K
n n D n n
![Page 25: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/25.jpg)
25
• Derive Gauss-Codacci equations II
c
ln
ln
ln
,
k a b k c k
i j i j c a b ij
i a b c i
i j c j a b ij
i a b c i
j i j c a b ij
i i a b c
i j j i c j a b b a
a b d
c j abd
i i
i j
D D n n D K
D D n n D K
D D n n D K
D D n D D n n
R n
D n K D n
g g g
g g
g g
g g
g g
c
8
Mo
,
mentum constraint
i b d b d
i j j bd
i a b d b d
j c j abd bd j
j bd j
K R
D K D K
n n
R n n
R
T
g g g
g g
3 components
(i,k)
(j,k)
![Page 26: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/26.jpg)
26
Derive evolution equation
(3)
2
Start from contracted Gauss-Codacci eq. I
contains or
Let us calculate
ln
a c
abcd
a
b d i
jl j l bd il j jl
t ij t ij
d
abcd a b b a c
a b c a bc b c
c
c
bcd
a b
a
R R K K KK
K
R n n
n K
R n
n D
K
n
R n n
g g
g
ln ln
Remem
l
ber: ln
n
ab a b
b
c
b a c
c bc b cn K n
K n D D
n D
D
![Page 27: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/27.jpg)
27
From
ln ln ln
ln
d
a b b a c abc d
a c b d a c b
j l abcd j l a bc b ac
j l j l
a c b a b
j l a bc j bl a
a b b k
j bl a a kl j
a c b a c
j l b ac j l n ac ab c
n R n
R n n n K K
D D D D
n K K n
K K n D K K
n K K K
g g g g
g g g
g
g g g g
L
1
2
b b
bc a
k
a c b d k
j l abcd
n
n jl jk l j l
jl jk l
R n n K
n K n
K K K
K K D Dg g
L
L
![Page 28: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/28.jpg)
28
(3) 1
(3)
1
Thus, the G-C equation become
2
s
Her
2
e
,
b d
jl j l bd n jl j l
k
kl j
k k k
n i
jl
k
t ij ij i j ij ik
j t ij k ij ik j j
j
k i
R R K D D
K K KK
K
K R D D KK
K D K K D
K
K D
K
g g
L
L
N
ote: 82
k k k b d
k ij
ijb d b d
i j bd
ik j jk i i j
d
d
i j b
bD K K D K D
R T T
R
gg g
g
g
g
g
Evolution equations = 6 components
![Page 29: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/29.jpg)
29
(3) 2
(3)
16
8
2
1 8 ( )
2
2
ij
ij
i
i j j j
l
ij ij ij il j i j
l l l
il j jl i l ij
i j ij
ij ij i j j i
R K K K T n n
D K D K T n
K R KK K K D D
K D K D D K
T n n
K D D
g
g g g g
g
Summary of 3+ 1 formalism
Constraints
Evolution
, Gauge conditioni
![Page 30: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/30.jpg)
30
Constrained system
• ADM equations seem to have too many
components: Constraints seem to be
redundant equations, because gij & Kij are
determined by solving evolution equations
• Constraints are guaranteed to be satisfied
if evolution equations are solved correctly
No inconsistency
![Page 31: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/31.jpg)
31
Evolution of constraints
0
0
0 0
0
8
0, 0 H & M Constraints
0 Evolution eqs.
0
2
i i i i
i i ij
i
ij
l i i ij
t l i i ij
l
t l i i i k
A G T
H n n H n H n H
H H
H
A
H KH H D D H H K
H H D KH H
g g g g
,
k k
i k iD H
If constrains are zero at t = 0 and evolution
equations are satisfied for any t,
constraints are always satisfied.
![Page 32: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/32.jpg)
32
Nature of standard 3+1 formalism
• Evolution equations are wave equations of
6 components, but it is not simple one:
Many additional terms even in linear order
(3)
, ,
1~
2
1~
2
. Maxwell's equation
0
ij ij
ij ij
ij i
kl
ik lj jk li ij k
j
l
K
R
cf
A A
g
g
g g
g g g g
![Page 33: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/33.jpg)
33
Linearized equations
, , ,
,
,
Linearized Einstein equations
with =1 & 0
; | | 1
Evolution eq. :
Constraint H : 0
M :
i
ij ij ij ij
ij ij ik kj jk ki kk ij
ii ik ki
ij i
h h
h h h h h
h h
h h
g
, 0ii j
This causes a problem
![Page 34: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/34.jpg)
34
Stability analysis
TT
, ( , )
TT TT
, ,
,
TT
,
Decomposition:
2
, ,
definition: 0
, : scala
, 0
Trace 3 ,
Divergen
: vec
c
t : tor
e
ensorr
ij ij ij i j ij
i i
ij
ij j ii
ii
ij j i
iA C
h A C B h
B h h
h A C
h
h A
B
,
substitute
, , ,
, , ,
( )
0, 0
i i
ij ij ik kj jk ki kk ij
ii ik ki ij i ii j
C B
h h h h h
h h h h
![Page 35: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/35.jpg)
35
3+1 Equations
,
TT TT
( , )
H: 0Constraints :
M: 2 0
0Evolution eqs. :
t i i
ij ij
i j
A
A B
h h
B
A A
C A
Strange
forms
Wave
equation
![Page 36: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/36.jpg)
36
Solutions ITT1 Equations for =Wave equations
True degree of GWs: No problem
2 Constraint (H) : 0
& Evolution equation for = wave eq.
Violated constraint will propagate a
ijh
A
A
,
way.
3 Constraint (M) : 2
For 0,
Evolution equation for gives 0
& i
B
i
i i
i
i Bi
i
i i B
i
i Bi
A B F x
A B F x
B B F x
B xF x t F
Perhaps no problem
Numerical integration
of zero is zero
![Page 37: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/37.jpg)
37
Solutions II
21
,
is not constrainted by constraints,
but determined by ,
If constraint is violated and 0 initially,
( ) ( )
( ) ( )
lmlm
l m
lm
C
C A A A
A
f r t g r tA Y
r
f r t g r tC Y C
rC t
Constraint violation is serious in this case
Small error in A results in serious error in C
![Page 38: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/38.jpg)
38
To summarize
• In the original 3+1 (N+1) formalism,
if constraints are violated even slightly,
the error increases with time even in a
nearly flat spacetime with no limit
• Namely,
it is unsuitable for numerical relativity
• Source:
, , , ik kij j jk ki kk ijij h hh h h
First, realized by T. Nakamura (1987)
![Page 39: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/39.jpg)
39
Section II: BSSN formalism
Essence
• Need reformulate of 3+1 formalism
• At least, in the linear level, constraint
violation mode must not appear
, ,
,
,
Define new variables
and rewrite as
i ij j
ii
i i j j i ijj ij
F h
Fh F
h
h
Evolution eqs.
for Fi and ?
![Page 40: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/40.jpg)
40
Reformulation using constraint equations
,
, , ,
, ,
,
,
Momentum constraint: 0
: Evolution eq for
Trace of
2 2
Hamiltonian constrain
0
0 0 0
t: 0
i i
i j j i i
ij j j
j
i
ij i
j i
i
ij ij
i i
i i
j
F
h h
F
h h
F
F
F F
F
h h
No problem !!
![Page 41: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/41.jpg)
41
Reformulation increasing the variables
and using constraints appears to be
robust
• Similar definition of new variables
Fi & is possible in the non-
linear case
First, derived by T. Nakamura (1987)
Subsequently modified by Shibata (1995),
Baumgarate and Shapiro (1998)
![Page 42: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/42.jpg)
42
Original BSSN formalism (Shibata-Nakamura 1995)
2 2 2 4
4
First of all, write the line element
2
Here, det 1.
As conjugat
( corr
es for and , define
1= and
3
esponds to .)
i i i j
i i ij
ij
ij
ij ij ij
ds dt dx dt e dx dx
A e K K
g
g
g
g
trace ij
ij ijK K Kg
Up to here, we increase 2 variables (, K)
and two constraints, det 1 and 0ij ij ijAg g
![Page 43: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/43.jpg)
43
Then, the equations are
, , ,
,
4 4
2( ) 2
31
( )6
1 1( )
3 3
2 + 2
3
l l l l
t l ij ij il j jl i ij l
l l
t l l
l
t l ij ij ij i j ij
l l l
ij il j il j jl i
A
K
A e R R e D D
KA A A A A
g g g g
g g
,
4
2
1 8
3
1( ) 4
3
l
l ij
i j ij
l ij
t l ij
A
e T
K A A K T n n
g g g g
g
Not sufficient in this stage !
Used
H-constraint
![Page 44: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/44.jpg)
44
Note
• The linear analysis for the simple
conformally-decomposed formalism shows
“System is even more unstable”
• An exponentially growing mode appear
TT
, ,
, ,
ij ij
ij
ik kj jk ki
ij iji j
h h
h C B h
h h
C C
![Page 45: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/45.jpg)
45
Linear analysis shows that the problems
come from
: is Ricci tensor of
2 2 4
4 scalar part
ij
ij ij ij ij ij
k
ij i j ij k i j
k
ij k
R
R R R R
R D D D D D D
D D
g
g
g
,
,
, ,
,
1
2
kl
ij klij
kl l k
k l ij jk il
ik jl jk ilR g g g g
g
Already
rewritten
by See appendix
![Page 46: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/46.jpg)
46
, , ,
, ,
fla
,
t ,
,
,
as
Write formally
in the linear c
as
Define
e
ij ij ij
kl
ij kl ik jl jk il
kl
ij kl i
kl
ij ik jl jk
kl
i ik l
k j i
l
l
i
l jk
f
f
F
g
g g g g
g g g
g
g
g
g
flat , ,
flat , , =
kl
ij ik jl jk il
ij i j j iF F
g g g
g
Linear
nonlinear
![Page 47: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/47.jpg)
47
Next step: Derive equations for the
new variables using constraints
• As in the linear case, the equation for Fi
should be derived from momentum
constraint
6 6 6
, , ,
Momentum constraint:
28
3
2or 6 8
3
2Here, 2 ( )
3
i
i j j j
i i
i j i i j j j
l l l l
ij t l ij il j jl i ij l
D e A e D K J e
D A D D A D K J
A
g g g g
![Page 48: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/48.jpg)
48
Thus, a term appears and
ik ik l l
i t jk i t jk ij t kl ik t j
ik
i t jk
t
l
ik ik l l
i t jk ij t kl ik t jlj
D
f
D
F
g g g g g g
g g g g
g g
Namely, momentum constraint can be
regarded as the evolution equation for Fi
, , ,
Other terms with are rewritten using
2( ) 2
3
t jk
l l l l
t l ij ij il j jl i ij lA
g
g g g g
![Page 49: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/49.jpg)
49
Equation for Fi
, , , , ,
, , ,
, , ,
,
1 2 2 6
2 3
Note no nonlinear t
2
2 16
3
erm of
k
t k i
jk jk jk k
ij k k ij jk i k i i
jk jl k
k ij l ij k
k k k jl
ik j jk i ij k i
l
F
f A A A A K
A
J
g g
g
g g g
, ; ij ij ij ij ijh A h g
![Page 50: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/50.jpg)
50
Summary of BSSN formalism
• Definition of 3 additional variables
(5 components) (Fi, K, ) is essential.
• Conformal transformation is not essential
at all: Only with conformal transformation,
the resulting formalism does not work
“Conformal formalism” is
misunderstanding (stupid) naming.
• The increase of variables results in the
increase of new constraints: Now, 17
equations with 9 constraint equations.
![Page 51: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/51.jpg)
51
Alternative (Baumgarte-Shapiro ,1998)
,
, ,
,
,
,
,
,
Define instead of .
(In the linear level, both reduce to .)
22 6
3
2
3
=
ij j
k i i jk ij ij
t
i ij jk
k jk j j
j i i j jk i
j j
k
k
j i i
j
j
h
A K A
F
g
g g
g
,
, , ,
, , , , ,
1
3
16
1
2
1
2
ik j
jk
ij
j
kl k k
ij ij kl ik j jk i
kl kl k l k
j ik l jk l i ij k ik jl
J
R
g
g
g g g g
g g g g g
Slightly simpler
![Page 52: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/52.jpg)
52
Nine components of constraints
(3) 2
Hamiltonian constraint (1)
16
Momentum constraint (3)
8
Tracefree condition for (1)
0
Determinant=1 for (1)
det
ij
ij
i
i j j j
ij
ij
ij
ij
R K K K T n n
D K D K T n
A
A
g
g
g
g
, ,
1
Auxiliary variable (3)
or
ij
i ij
i ij j jF g g
![Page 53: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/53.jpg)
53
Puncture-BSSN(Campanelli et al. 2005)
2 4
2 2
4 2
2 2
Define (or ) instead of
to follow a black hole spacetime.
Schwarzschild spacetime in the isotropic coordinates:
1 / 21
1 / 2 2
M r Mds dt dx dy
M
e W
r
e
r
2
4
0 ln 12
Define : regular everywhere
BH spacetime can be numerically followed
wi
12
th no special
r
d
M
r
z
M
r
technique
![Page 54: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/54.jpg)
54
New conformal factor
,
4
4 2
,
4
4
( )3
or
1( )
6
2 2 2
2
l l
t l l
k
ij i j
l l
t l l
ij
k
i j ij i j ij k
ij k
WW K
R WD D W W W D W
e W e
K
R
e D D D D D D
D W
g g
g
No irregular term even for BH spacetime
No divergence
![Page 55: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/55.jpg)
55
Other prescriptions: Special gauge
, , ,
, ,
Linearized Einstein equations
with 1 & 0
; | | 1, 1 / 2; | | 1
Evolution eq. :
i
ij ij ij ij
ij ij ik kj
ij i j
jk ki kk ij
h h a a
h
a
h h h h
g
,
,
, ,
Constraint H : 0
M : 0
ii ik ki
ij i
i
i j
j
i
h h
h h
![Page 56: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/56.jpg)
56
,
,
, , , , , , , ,
, ,
Constraint (H)
,
2
00
0
Harmonic gauge
:
i
i
i ij j
ik kj jk ki kk ij ij i j j i kk ij ij
ij ij kk ij ij
ij ji kk
t kk kk
ag
h
h h h a h a
h h h a
a h a h
h a h a
Only wave equations appear No problem
This shows an evidence why Pretorius formulation works
![Page 57: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/57.jpg)
57
Extension to N+1 case (N > 3)
• ADM equations are unchanged
• BSSN formalism is slightly modified
because the dimension is different
(Yoshino-Shibata 09)
• In the following, spacetime dimension
is denoted by D = N+1
![Page 58: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/58.jpg)
58
2 2 2 1
, , ,
,
For 2
2( ) 2
1
2( )
1
1 1( )
1 1
k k i j
k k ij
l l l l
t l ij ij il j jl i ij l
l l
t l l
l
t l ij ij ij i j ij
ds dt dx dt dx dx
AD
KD
A R R D DD D
g
g g g g
g g
,
2
2 + 2
1
1 8
1
8( ) 3
1 2
l l l l
ij il j il j jl i l ij
i j ij
l ij
t l ij
KA A A A A AD
TD
KK A A T D n n
D D
g g g g
g
![Page 59: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/59.jpg)
59
, ,
, , , ,
2 12
1 2
2 3
1 1
16
3 1
2 2
l i i jk ij ij
t l jk j j
j i i j jk i ik j
j j jk jk
ij
j
ij ij ij
ij i i
D DA K A
D
D
D D
J
R R R
DR D D
g
g g
g
2
2
3
4
1
4
k
ij k i j
k
ij k
DD D D D
DD D
g
g
Robust for any dimension (at least up to 7D,
Shibata & Yoshino, „10)
![Page 60: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/60.jpg)
60
Section III: Gauge conditions
Any time slice and any time axis
can be chosen in numerical relativity
Normal
direction
Shift
Lapse
Time
axis
![Page 61: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/61.jpg)
61
Slice: Required properties
- Black hole singularities and coordinate
singularities have to be avoided
t Singularity
Or, effectively excised
![Page 62: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/62.jpg)
62
Well-known good slice
Drawback: Solving elliptic-type equation
is computationally expensive
Maximal slic :
0 , ,
e
ij ijK K S K T g
Estbrook et al. PRD 1973
![Page 63: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/63.jpg)
63
Frame dragging coordinates distort
Coordinate shear has to be suppressed.
Spatial gauge; required property
![Page 64: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/64.jpg)
64
1/ 2
1/ 2
(Smarr-York, PRD1978) :
Minimize global distortion defined as
( )( )
0 0
1
3
Minimal distortion gauge
ik
k
jl
t ij t kl
k
t iki
k
jk j k jD
I dV
I
D R
D
g g g g g
g
g g
1/ 2 2 + ln
3
k
k
k l l k kl m
jl k m jD D D D S
g g g
Very physical & beautiful.
But, elliptic eqs. & expensive
![Page 65: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/65.jpg)
65
Puncture gauge; =0 at puncture
2
2 2
4, 12
;
2 for 4D & 1< 2 for N-D
1 for 4D and
1
2
1
2
is p
1 3
2 2 4
l
t l
klk
t l t l
D Vl k k k
t l
l k l k k
t l t l
or
V V
K
DV F t F
D
DV B B
D
B B
g
BH
referable for N-D
~1/ M (Alcubierre, Bruegman 03, and others)
![Page 66: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/66.jpg)
66
Properties
• Slice: Asymptotically approaches to
maximal slice-like slice;
slice freezes (,t = 0, and thus, K const)
• Shift: Similar to minimal distortion gauge !!
• Computational costs are very small
• Horizon sucking does not matter because
it happens inside horizon and hyperbolic
gauge is suitable for this “effective
excision” (Bruegmann et al., 2006)
![Page 67: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/67.jpg)
67
Section IV: Initial value problem;
How to impose constraints
(3) 2 16 16
8 8
ij
ij H
i
i j j j j
R K K K T n n
D K D K T n J
g
Only 4 components equations
for 12 component of g & K
These are not hyperbolic equations
Write in elliptic equations(Method of O‟Murchadha-York, 1973)
![Page 68: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/68.jpg)
68
Hamiltonian constraint
4
(3)
4
(
2
2
55
3)
1 8
16
Elliptic equation for a given set of
28 8
, ,
i
ij ij
ij
ij H
ij j H
j
H
i
ijR K K
R R
R K K K
K
K
g g
g
See Appendix
![Page 69: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/69.jpg)
69
Momentum constraint
4
6 6
6 TT
6
1,
32
03
Then, set
2
3
1 2
3
0
3
ij ij ij ij ij
j j
j i i j i i i jk
j j j j k j
i i i i k i
i TT j TT i
i
j j
i i j i j
i
K A K
D K D K D A D K D
A DW D W D W
W D D W
K
D
R W D
K K
g g g
g
g
6
Vector elliptic equation of
for a given set of , , ,
8
i
ij j
i iJ
W
K
K Jg
![Page 70: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/70.jpg)
70
Section V: Numerical implementation
• Discretize all variables;
g (x, y, z) g i, j, k (i, j, k = 1—n)
• And then, perform finite difference;
e.g.,
, , 1, , 1, ,
, , 2, , 2, , 1, , 1, ,
2nd order finite difference
1
2
4th order finite difference
18
12
x i j k i j k i j k
x i j k i j k i j k i j k i j k
g g g
g g g g g
![Page 71: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/71.jpg)
71
Basic equations
, , ,
, , ,
, , , , , ,
, , , ,
, , , , ,
i
ij ij
ij ij i
i i
A ij ijij ij
K ij ijk
t ki i i
ij ij
i i
i i
S A
S K
S A KA A
S A KK K
S A K
B B
g
g g g
g
g
g
,
,
, , , , , ,
i
i
i i
B ij ij
S K
S B
S A K
g
![Page 72: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/72.jpg)
72
Method for space finite difference
• Metric = smooth, no shock (different from
fluid) Every variable can be expand as
the Taylor series
2 3 4' '' ''' ''''
0
2' ''
0
3 4''' ''''
2! 3! 4!
42 2
2!
8 16
3! 4!
ij ij ij ij ij ij
ij ij ij ij
ij ij
x
x
g g g g g g
g g g g
g g
4 derivatives 5 values
4th order
scheme
![Page 73: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/73.jpg)
73
Need 1st and 2nd derivatives
, 2, , , 2, ,
, 1, , , 1. .
2
, 2, , , 2, ,2
, 1, , , 1, , , , ,
1
12
8
1
12
16 30
x ab ab j k l ab j k l
ab j k l ab j k l
x ab ab j k l ab j k l
ab j k l ab j k l ab j k l
g g g
g g
g g g
g g g
Write all the derivatives using these rules:
Currently, 4th-order is most popular
![Page 74: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/74.jpg)
74
Einstein’s equation is complicated
2
, 2, 2, , 2, 2, , 1, 2, , 1, 2,
, 2, 2, , 2, 2, , 1, 2, , 1, 2,
18
12
+ 8
+
x y ab ab i j k ab i j k ab i j k ab i j k
ab i j k ab i j k ab i j k ab i j k
g g g g g
g g g g
, 2, 1, , 2, 1, , 1, 1, , 1, 1,
, 2, 1, , 2, 1, , 1, 1, , 1, 1,
8 8
8 8 ]
ab i j k ab i j k ab i j k ab i j k
ab i j k ab i j k ab i j k ab i j k
g g g g
g g g g
Many points are
used
![Page 75: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/75.jpg)
75
Advection term: Upwind scheme I
1
1 1
Consider the simplest wave equation:
0
Simple Finite difference
unstable !!
von Neumann stability analysis
exp exp
1
t x
n n n n
j j j j
n n n
j j
f c f f F x ct
c tf f f f
x
u ikx ikj x
sin 1 c t
i k xx
j
Cf. Numerical
Recipe
(Press et al.)
![Page 76: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/76.jpg)
76
11
1
Simplest stable method
= First-order upwind scheme
0
0
von Neumann stability analy 1sis
n n n
j j jn
j
n n n
j j j
c tf f f c
xf
c tf f f c
x
Advection term: Upwind scheme II
j
c>0
j
c<0
![Page 77: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/77.jpg)
77
Advection term: Upwind scheme III
'
0 0 0
0 0
'
0 0 0
Advection term:
Need to use upwind scheme
For 0,
13 6 2 18
12
10 3
For 0,
13 6 2 18
12
l
l ij
l
ij ij ij ij
ij ij
l
ij ij ij ij
x x x
x x
x x x
g
g g g g
g g
g g g g
0 0 10 3ij ijx xg g
4th-order scheme
![Page 78: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/78.jpg)
78
Time evolution: Standard one
• Use Runge-Kutta method
• Simple 2nd order method Unstable
• 3rd or 4th order (popular) Stable
1 0
2 0 1
3 0 2
4 0 3
51 2 3 40 0
/ 2
/ 2
2 2
6
t a a
a
a
a
a
a a
Q F Q
k tF Q t
k tF Q t k
k tF Q t k
k tF Q t k
k k k kQ t t Q t O t
![Page 79: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/79.jpg)
79
Boundary conditions
• Geometry obeys wave equations
2
Outgoing boundary condition
Usually, only the leading order
is tak
/1/
: characteristic angular frequency
Location of boundary should
en into account
be
.
a
a
F t r cQ O r
r
1r
![Page 80: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/80.jpg)
80
Numerical implementation of
outgoing boundary condition
tr
/
/
( ) /
a a
a
a
rQ F t r c
rQ t r c
r r Q t t r r c
t
r
Determine by
interpolation
![Page 81: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/81.jpg)
81
Section: Extracting gravitational
waves by complex Weyl scalar
4
'
4
4
Using a complex Weyl scalar
: 4D Riemann tensor
, , , : Null tetrad
2 ' '' ''
1,
2
t t
R n m n m
R
n l m m
h ih dt dt t
r h ih
g g
g
![Page 82: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/82.jpg)
82
Weyl scalar for gravitational waves I
, , , : Null tetrad
0, 1
1 : Ingoing Null
2
: Complex orthnormal to &
: Unit timelike normal as before
n l m m
n n l l m m n l m m
n n r
m n l
g n l l n m m m
n
m
: Unit radial vector, 0r r n
![Page 83: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/83.jpg)
83
Weyl scalar for gravitational waves II
(4) (4)
4
(4)
(4)
(4)
(4)
1( 2
2
)
i j i j
i j k i j k
i
ijkl ik jl
k
ij i jk i
il j
j ij
j ik k ij
k
ijk
R K K KK E
D K D K
R n m n m R n m r m
R r m r m
R n m n m m m m m
R n m r m m r m m r m
R r m r m R K m
B
K K K r
g g g g
g g
g g
g g
g g
Note in 3D,
2
j k l
ijkl ik jl il jk jk il jl ik ik jl il jk
r m
RR R R R Rg g g g g g g g
cf, derivation of Gauss-Codacci eqs.
![Page 84: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/84.jpg)
84
Weyl scalar for gravitational waves III
2 0 (vacuum, Hamiltonian constr.
,
2
then
)
wher
e
ijkl ijkl ik jl il jk
l
ik ik ik il k ik
ijkl ik jl il jk jk il jl ik ik jl
i j k l j l
ijkl jl
il jk
j i
i jE R K K K
r m r m E m
R K K K K
R K K K K E
EE
m
E E Eg g g g g g g g
R
R
R
R&
4
0, 0, 1
Finally,
j l j l j l
jl jl
i k i j k
i
j
k ij
l
k
m m r m r r
E m m B m r m
g g g
Quite simple
![Page 85: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/85.jpg)
85
Section: Adaptive Mesh Refinement (AMR)
L l ~ 4GM/c2
Why is it required ?
• More than 2 different length scales
• For binary, gravitational wavelength
and radius of compact star
Resolve stars while extracting waves
![Page 86: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/86.jpg)
86
Typical length scales
1 2
2
2
3/ 23
3
2 2
Angular velocity o
Radius of
f binary
black hole
Radius of neutron
,
Gravit
star
ational wavelength
5~8
10010 /
GM
GmR
c
M m mr
c
GmR
c
r r GMc R
GM GM c c
2 is orbital radius ~5
GMr
c
![Page 87: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/87.jpg)
87
• Prepare N=5—10 levels of different
resolutions and domain size:
Domain Dl, Size Ll, Grid spacing l
where l=1—N
• For the simple case, we set 2Ll= Ll-1 ,
2l=l-1 (same grid number for l=1—N)
and L1 > , N << Gm/c2
AMR grid in numerical relativity
Ratio=2:1 N
N-1 N-2
![Page 88: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/88.jpg)
88
• For the finer grid spacing, the time step is
smaller: 2tl=tl-1
AMR time step
For each evolution,
4th-order Runge-Kutta method is used
Refinement
boundary
123
4
56
7
![Page 89: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/89.jpg)
89
Key: Interpolation and treatment of
buffer zone
• Grid in each domain is composed of main
and buffer zones
c
Main BufferBuffer
Main : Solve equations with no prescription
Buffer: Interpolation is necessary
![Page 90: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/90.jpg)
90
A method for interpolation:
Example I
• Before the first Runge-Kutta step
Quantity in the buffer zone is
determined by Lagrange interpolation,
e.g., 5th-order interpolation (using 6 points)
c
For 3-dimension, 63=216 grid points are used
Main
![Page 91: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/91.jpg)
91
A method for interpolation II
• The Runge-Kutta evolution step
• Inner 3 buffer points are solved in the
same manner as in main region
• 4th one: The same as above but for
advection term; mixture of 2nd and 4th
order upwind schemes
• 5th and 6th one: Interpolation in space and
time: Time = 2nd-order interpolation
c
Bruegmann et al., Yamamoto et al., PRD2008
![Page 92: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/92.jpg)
92
• Concept of time interpolation
AMR time interpolation
12
3
Runge-Kutta
middle steps
![Page 93: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/93.jpg)
93
Moving domain
• Small-size domains (finer domains) which
cover compact objects move with them
• Large-size domains (coarser domains) do
not move (fixed grid)
• Finer domains are moved when the center
of black hole or neutron star moves
For black holes, solve
0 at black hole center, =0
in the puncture gauge
For neutron stars, find the location of
ii
k
t k
dx
dt
max density
![Page 94: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/94.jpg)
94
Section: Finding a black hole
• Popular method = Finding apparent horizon
• Apparent horizon = surface for which
expansion of outgoing null ray is zero
• Important property (Hawking-Ellis, 1973):
If an apparent horizon exists for a globally
hyperbolic spacetime, event horizon of a
black hole always exists
Apparent horizon is formed
= A black hole is formed
![Page 95: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/95.jpg)
95
: Time like unit normal
: Radial unit normal, 0.
is perpendicular to AH sphere.
, : Unit normal on sphere
g
l n r
n
n n r r a a b b
r r n
r
a b
Basic equation for AH I
• Define outgoing null vectorr
![Page 96: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/96.jpg)
96
Apparent horizon: 0
0
k i j
k ij
k i j
k ij
a a b b l
r r n r
K D r r r K
D r r r K K
g
Basic equation for AH II
• Expansion = change rate of area
r
r
Outside AH
Inside AH
![Page 97: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/97.jpg)
97
2 2
2
1, ,
is determined by 1
1
1 cot ....
sin
0 2D elliptic type equation
i
i
i
k k
k k
r C h h
C rr
D r r
C h h h
gg
Basic equation for AH III
• AH=two sphere, r=h(, )
![Page 98: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/98.jpg)
98
Basic equation for AH IV
• Boundary condition
0
0
2
Periodic
0
, 0
,
Well defined Standard solver is OK
![Page 99: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/99.jpg)
99
Appendix: Conformal decomposition I
4 , 0, 0
1
2
1
2
2
2
+
=
=
ij ij i jk i jk
i il
jk j kl k jl l jk
il
j kl k jl l jk
il
kl j jl k jk l
i
jk
i i i
k j j k j
i
k
i
jk jk
D D
D D D
C
g g g g
g g g g
g g g g
g g g
g
g
![Page 100: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/100.jpg)
100
Conformal decomposition II
i i i l i l
jk i jk j ik jk il jl ik
i i i l i l
i jk j ik jk il jl ik
i i i l i l
i jk j ik jk il jl ik
i l i l i l i l
jk il jl ik jk il jl ik
j
i i i l
i jk j ik jk ilk D C D C
R
C C C C C C
C C C C
C C CR
2
1 6 2
2
i i i l i
i l
j
l
jk i jk j ik jk il jl ik
l
j k jk l
j k jk
l ik
R D C D C C C C C
D D D D
D D
C
g
g
![Page 101: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/101.jpg)
101
Methods for code validation
• Confirm to reproduce exact solutions:
-- Black hole solutions (check area etc)
-- Propagation of linear gravitational waves
• Check violation of constraints is small:
Monitor Hamiltonian and momentum
constraints
• Check convergence:
If 4th-order scheme is used, quantities
converges at 4th order; Q = Q0 + Q4 x4
![Page 102: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/102.jpg)
102
Some technical issues I
• Imposing some constraints within O(4):
1/3
1/3
1/ 6
det 1 and 0
det
1det Tr
3
det
Tr
, are unchanged
ij ij ij
ij ij ij
ij ij ij ij ij
ij
ij
ij ij
A
A A A
W W
K K A
K
g g
g g g
g g
g
g
![Page 103: Lecture note on 3+1 formalism of numerical relativity - KEKbridge.kek.jp/lecture/01-shibata/slide_100609.pdf · formalism of numerical relativity ... ki n n t gg PP P PQ PQ PQ E D](https://reader035.vdocuments.us/reader035/viewer/2022062413/5aa1b64d7f8b9a80378c025f/html5/thumbnails/103.jpg)
103
Some technical issues II
• Kreiss-Oliger-type dissipation > O(4):
6 (6)
(6)
is a constant of O 0.1
is the sum of sixth derivatives
Q Q x Q
Q
Purpose is to suppress
high-frequency numerical noise