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Gmunu : Multigrid methods for solving Einstein Gmunu : Toward multigrid based Einstein field equations CHEONG, Chi-Kit field equations solver for (Patrick) Tjonnie G. F. LI, general-relativistic hydrodynamics L. M. LIN simulations

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Gmunu: Toward multigrid based Einstein field equations solver for general-relativistic hydrodynamics simulations

TAUP2019 CHEONG, Chi-Kit (Patrick) Tjonnie G. F . LI, L. M. LIN

Department of Physics The Chinese University of Hong Kong

Sep 12 2019

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Outline

Introduction Methods Results Covergence properties Code test Conclusions Backup slides

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Introduction

◮ Implementation details of Gmunu (P . C. K. CHEONG et al., in prep.) ◮ Application of Gmunu (H. Y. NG et al.,Talk on Tuesday, in prep.) ◮ Why numerical simulations ◮ Introduction of CCSNe and its gravitational waves ◮ Develop a code for:

◮ Core-collapse supernovae ◮ Isolated Neutron stars

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Introduction

1 2 3 4 5 6 7 8 9

f [kHz]

Relativistic (Case GR) Newtonian with modified TOV potential and rotation corrections (Case Arot)

F H1 H2 H2 H1 F

  • B. Muller, et al. 2008

General relativistic simulations is needed!

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Introduction

Different ways to solve Einstein equations ◮ “free-evolution” formulations ◮ constrained formulation

◮ elliptic sector and hyperbolic sector

Conformally fatness approximation (CFC) is applied successfully in various astrophysical problems even in spherical-polar coordinate However... ◮ high computational cost for elliptic equations ◮ on the fly simulation?

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Conformally flatness approximation

CFC approximation gµν =     −α2 + βiβi β1 β2 β3 β1 ϕ4 β2 ϕ4r2 β3 ϕ4r2 sin2 θ    

A1B3G3 ρmax (1014 g cm−3) 2 3 4 5 A2B4G1 ρmax (1013 g cm−3) 2 4 6 full GR CFC A1B3G5 ρmax (1014 g cm−3) t - tbounce (ms)

  • 5

5 10 15 2 3 4 5 A1B3G3 h+ R (cm)

  • 200
  • 150
  • 100
  • 50

50 100 A2B4G1 h+ R (cm)

  • 150
  • 100
  • 50

50 full GR CFC A1B3G5 h+ R (cm) t - tbounce (ms)

  • 5

5 10 15

  • 20
  • 10

10 20 30

Ott, et al. 2006

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

xCFC scheme

Cordero-Carrion et al. (2009) ˜ ∆Xi + 1 3 ˜ ∇i ˜ ∇jXj = 8π ˜ Si ˜ Aij ≈ ˜ ∇iXj + ˜ ∇jXi − 2 3 ˜ ∇kXkfij ˜ ∆ψ = −2π ˜ Eψ−1 − 1 8fikfjl ˜ Akl ˜ Aijψ−7 ˜ ∆(αψ) = (αψ)

  • 2π
  • ˜

E + 2 ˜ S

  • ψ−2 + 7

8fikfjl ˜ Akl ˜ Aijψ−8

  • ˜

∆βi + 1 3 ˜ ∇i ˜ ∇jβj = 16παψ−6fij ˜ Si + 2 ˜ Aij ˜ ∇j

  • αψ−6
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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Boundary condiditon

Robin B. C. ∂ψ ∂r

  • rmax = 1 − ψ

r ∂α ∂r

  • rmax = 1 − α

r βi

  • rmax = 0
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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Code features

Gmunu (General-relativistic multigrid numerical Einstein solver) ◮ 3+1 in Spherical polar coordinates (1,2,3-D) ◮ Hydro: high-resolution shock-capturing (HRSC) methods

◮ Reconstruction method: Piecewise-Constant (PC), TVD, (5-th order)WENO, MP5 ◮ Riemann solver: HLL, HLLE, Marquina ◮ Time update: RK3 methods ◮ Conserved variables to Primitive variables: Regula-Falsi method

◮ Conformally flatness condition(CFC) metric evolution:

◮ extended CFC (xCFC) scheme (Cordero-Carrion et

  • al. (2009))

◮ Multigrid solver for the elliptic non-linear coupled equations

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Key features of our MG solver

◮ Nonlinear FAS (Full Approximation Storage) algorithm ◮ Cycles options: V, W and F ◮ Cell-Centered discretization (no grid communication is needed) ◮ Smoother: Red-Black Gauss-Seidel relaxation ◮ Prolongation: bi-linear ◮ Restriction: piecewise constant

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Models

Table: Γ = 2, K = 100,(c = G = Msun = 1).

Model ρc [10−3] Ω [10−2] BU0 1.28 0.000 SU 8.00 0.000 BU2 1.28 1.509 BU8 1.28 2.633

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Convergence properties

◮ solving the α of BU8 with initial guess: α = 1 ◮ nr × nθ = 640 × 64 ◮ Gauss-Seidel (∼ 5.3 × 105) to MG (∼ 50) ◮ O(minus) → O(ms)

10 20 30 40 50 60 Number of iterations 10−8 10−6 10−4 10−2 100 102 L1 norm V 1 V 2 V 3 V 4 V 5 V 6

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Model: BU0

◮ nr × nθ = 640 × 64 ◮ r = [0, 30], θ =

  • 0, π

2

  • 2

4 6 8 10 12 t (ms) −10.0 −7.5 −5.0 −2.5 0.0 2.5 5.0 [ρc(t)/ρc(0) − 1] × 104 1000 2000 3000 4000 5000 6000 7000 8000 f (Hz) 0.00 0.02 0.04 0.06 FFT of vr(t) and vθ(t) FFT(vr) FFT(vθ) F H1 H2 H3

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Model: SU

Migration test: unstable -> stable NS ◮ nr × nθ = 1024 × 1 (1D simulation) ◮ r = [0, 34]

2 4 6 8 t (ms) 0.0 0.2 0.4 0.6 0.8 1.0 ρc(t)/ρc(0)

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Model: BU2

◮ nr × nθ = 640 × 64 ◮ r = [0, 30], θ =

  • 0, π

2

  • 0.0

2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 t (ms) −10.0 −7.5 −5.0 −2.5 0.0 2.5 5.0 [ρc(t)/ρc(0) − 1] × 104 1000 2000 3000 4000 5000 6000 f (Hz) 0.00 0.02 0.04 0.06 0.08 FFT of vr(t) and vθ(t) FFT(vr) FFT(vθ) F H1

2f 2p1

i2

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Model: BU2

◮ nr × nθ = 640 × 64 ◮ r = [0, 30], θ =

  • 0, π

2

  • 5

10 15 20 r (km) 0.0 0.2 0.4 0.6 0.8 1.0 ρ/ρc

Tmax = 20 ms

equatorial initial equatorial pole initial pole

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Model: BU2

◮ nr × nθ = 640 × 64 ◮ r = [0, 30], θ =

  • 0, π

2

  • 5

10 15 20 r (km) 0.0 0.2 0.4 0.6 0.8 1.0

  • vφvφ/max
  • vφvφ
  • Tmax = 20 ms

equatorial initial equatorial

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Model: BU8

◮ nr × nθ = 640 × 64 ◮ r = [0, 30], θ =

  • 0, π

2

  • 0.0

2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 t (ms) −10 −8 −6 −4 −2 2 [ρc(t)/ρc(0) − 1] × 104 1000 2000 3000 4000 5000 6000 f (Hz) 0.0 0.1 0.2 0.3 0.4 FFT of vr(t) and vθ(t) FFT(vr) FFT(vθ) F H1

2f

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Model: BU8

◮ nr × nθ = 640 × 64 ◮ r = [0, 30], θ =

  • 0, π

2

  • 5

10 15 20 r (km) 0.0 0.2 0.4 0.6 0.8 1.0 ρ/ρc

Tmax = 20 ms

equatorial initial equatorial pole initial pole

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Model: BU8

◮ nr × nθ = 640 × 64 ◮ r = [0, 30], θ =

  • 0, π

2

  • 5

10 15 20 r (km) 0.0 0.2 0.4 0.6 0.8 1.0

  • vφvφ/max
  • vφvφ
  • Tmax = 20 ms

equatorial initial equatorial

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Conclusions

◮ We presented our Multigrid metirc solver ◮ Fast and accurate!

◮ for ǫ ∼ 10−7 ◮ O(minus) → O(ms) per equation ◮ solving the metric ∼ 5 to 10 hydro steps ◮ no need to solve the metric at every time steps

◮ suitable for dynamical simulations Future direction ◮ extend to FCF (fully-constrained formalism) ◮ apply to study astrophysical problems

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Q&A Thank you for your attention !!

Q & A

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Backup Slides

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Hydro

The 3+1 "Valencia" formulation ∂t(√γ U) + ∂i(√γ F i) = S where U =   D Sj τ   =   ρW ρhW 2vj ρhW 2 − p − D   , F i =   D

  • vi − βi/α
  • Sj
  • vi − βi/α
  • + δi

jD

τ

  • vi − βi/α
  • + Pvi

  , Q =    T µν ∂gνj

∂xµ − Γλ µνgλj

  • α
  • T µ0 ∂ ln α

∂xµ − T µνΓ0 µν

  Plus EOS

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

CFC approximation

gµν =     −α2 + βiβi β1 β2 β3 β1 ψ4 β2 ψ4r2 β3 ψ4r2 sin2 θ     Dimmelmeier et al. (2005) Cordero-Carrion et al. (2009)

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

xCFC scheme

Cordero-Carrion et al. (2009) ˜ ∆Xi + 1 3 ˜ ∇i ˜ ∇jXj = 8π ˜ Si ˜ Aij ≈ ˜ ∇iXj + ˜ ∇jXi − 2 3 ˜ ∇kXkfij ˜ ∆ψ = −2π ˜ Eψ−1 − 1 8fikfjl ˜ Akl ˜ Aijψ−7 ˜ ∆(αψ) = (αψ)

  • 2π
  • ˜

E + 2 ˜ S

  • ψ−2 + 7

8fikfjl ˜ Akl ˜ Aijψ−8

  • ˜

∆βi + 1 3 ˜ ∇i ˜ ∇jβj = 16παψ−6fij ˜ Si + 2 ˜ Aij ˜ ∇j

  • αψ−6
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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Elliptic solvers

Table: Elliptic solvers

Method Complexity (accuracy ǫ) Gauss Elimination O(N2) Jacobi iteration O(N2) log(ǫ) Gauss-Seidel iteration O(N2) log(ǫ) SOR O(N3/2) log(ǫ) Conjugate Gradient O(N3/2) log(ǫ) ADI O(N log(N)) log(ǫ) FFT O(N log(N)) Multigrid (iterative) O(N) log(ǫ) Multigrid (FMG) O(N)

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Elliptic solvers

credit: Klaus Stuben

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Elliptic solvers

◮ LORENE using spectral methods for CFC ◮ require accurate grid communication

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Multigrid Method

◮ A strategy, not a single method ◮ various possible implementation ◮ fast and efficient ◮ suitable for non-linear problems ◮ no grid communication is needed

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Multigrid Method

credit: Jaysaval et al. (2016)

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Multigrid Method

cell-centered grid Dimmelmeier et al. (2002)

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Multigrid Method

credit: Hooshyar (2014)

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Gmunu: Multigrid methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI,

  • L. M. LIN

Introduction Methods Results

Covergence properties Code test

Conclusions Backup slides

Multigrid Method

◮ Restriction (Left): piecewise constant ◮ Prolongation (Right): bi-linear credit: F . W. Yang et al. (2015)