Nils Deppe - icerm.brown.edu
Transcript of Nils Deppe - icerm.brown.edu
SpECTRE– towards high-order hydrodynamics and exascalenumerical relativity
Nils Deppe
ICERM Advances and Challenges in Computational Relativity
September 14, 2020
github.com/sxs-collaboration/spectre 1
Overview
1 Robust high-order finite-difference schemes
2 Discontinuous Galerkin-finite-difference hybrid method
3 Binary black hole inspiral using discontinuous Galerkin
github.com/sxs-collaboration/spectre 2
Simulation Goals
• Binary black hole mergers• Binary neutron star mergers• Accretion disks• Core-collapse supernova
explosions
Hinderer et al. 2018
github.com/sxs-collaboration/spectre 3
Simulation Goals
• Binary black hole mergers• Binary neutron star mergers• Accretion disks• Core-collapse supernova
explosions
Hinderer et al. 2018
github.com/sxs-collaboration/spectre 3
Binary Neutron Star Simulation Goals
• Resolve =⇒ high resolution• Resolve surface of stars• Resolve tidal deformability
• Can’t wait for bigger computers
Hinderer et al. 2018
github.com/sxs-collaboration/spectre 4
Binary Neutron Star Simulation Goals
• Resolve =⇒ high resolution• Resolve surface of stars• Resolve tidal deformability• Can’t wait for bigger computers
Hinderer et al. 2018
github.com/sxs-collaboration/spectre 4
Wish List—Physics Perspective
• Conserve mass, . . .• Physical solutions (positive pressure, density)• Constant flow preserved• Resolve large discontinuities (& 1020)
github.com/sxs-collaboration/spectre 5
Hydrodynamics: Finite Difference Methods
• Solution error ∼ 1/N2 • Cartesian grids
u(x)
x
github.com/sxs-collaboration/spectre 6
Vacuum Evolutions: Spectral Methods
• Solution error ∼ exp(−N) • Curved grids
github.com/sxs-collaboration/spectre 7
Current Hydrodynamics Codes
• Bad values overwritten• Some positivity preservation• 2nd order• Manual tuning
github.com/sxs-collaboration/spectre 8
Wish List—Numerical Perspective
• High-order• Robust, no tuning• Spectral when possible
github.com/sxs-collaboration/spectre 9
Wish List
• Conserve mass, . . .• Physical solutions (positive pressure, density)• Constant flow preserved• Resolve large discontinuities (& 1020)
• High-order• Robust, no tuning• Spectral when possible
github.com/sxs-collaboration/spectre 10
Building Blocks for Finite Difference Methods
u(x)
x
github.com/sxs-collaboration/spectre 11
Building Blocks: Positivity
u(x)
x
github.com/sxs-collaboration/spectre 12
Building Blocks: Positivity
First order everywhere =⇒ very inaccurate
u(x)
x
github.com/sxs-collaboration/spectre 13
Building Blocks: Positivity
Arbitrary minimum value =⇒ total nonsense
u(x)
x
github.com/sxs-collaboration/spectre 14
Building Blocks: Unphysical Oscillations
u(x)
x
github.com/sxs-collaboration/spectre 15
Building Blocks: Oscillations
Detecting oscillations:
u(x) ≈N∑i=0
ciPi(x),
github.com/sxs-collaboration/spectre 16
Building Blocks: Oscillations
i = 0
i = 1
i = 2
i = 3
i = 4
i = 5
Modes
x
u(x)
Amplitude
github.com/sxs-collaboration/spectre 17
Building Blocks: Oscillations
Detecting oscillations:
u(x) ≈N∑i=0
ciPi(x),
Check:
wN (cN )2∑Ni=0wi(ci)2
github.com/sxs-collaboration/spectre 18
Positivity-Preserving Adaptive-Order Methods
Attempt 5th orderreconstruction
FailedPassed Attempt 3rd orderreconstruction
FailedPassed Use 1st orderreconstruction
Compute adaptivespatial derivative
Take time step
Current solutionat cell center
github.com/sxs-collaboration/spectre 19
The Shu-Osher Oscillatory Shock Tube Problem
4.0 2.0 0.0 2.0 4.0x
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
t
log10(ρ)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
1e 1
4.0 2.0 0.0 2.0 4.0x
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Dens
ity
NumericalReference
github.com/sxs-collaboration/spectre 20
The Shu-Osher Oscillatory Shock Tube Problem
4.0 2.0 0.0 2.0 4.0x
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
t
log10(ρ)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
1e 1
4.0 2.0 0.0 2.0 4.0x
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
t
Order
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
github.com/sxs-collaboration/spectre 21
Discontinuous Galerkin Method
• Solution error ∼ exp(−N)
• Shock capturing
−2 20
1
x
u(x)
Ωk−1 Ωk Ωk+1
github.com/sxs-collaboration/spectre 22
DG-FD Hybrid Method
Check DG solution
FailedPassed
Retake time stepusing FD
Take FD time step
FailedPassed
Check if DGshould work
DG time step
github.com/sxs-collaboration/spectre 23
The Shu-Osher Oscillatory Shock Tube Problem
4 2 0 2 4x
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
ρ
DG-P9
WCNS3-ZReference
0.5 1.0 1.5 2.0 2.5x
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
ρ
DG-P9
WCNS3-ZReference
github.com/sxs-collaboration/spectre 24
The Cylindrical Sod Explosion
github.com/sxs-collaboration/spectre 25
Wish List—Achievement Unlocked
Discontinuous Galerkin-finite-difference hybrid method
" Conserve mass, . . .
" Physical solutions (positive pressure, density)
" Constant flow preserved
" Resolve large discontinuities (& 1020)
" High-order
" Robust, no tuning
" Spectral when possible
github.com/sxs-collaboration/spectre 26
Binary Black Hole Mergers
• Cosmic Explorer + Einstein Telescope• Higher χ and q• Longer waveforms• Followup simulations• Space-based detector(?)
github.com/sxs-collaboration/spectre 27
Binary Black Hole Mergers – Animation
• Led by Geoffrey Lovelace (CSU Fullerton) & Prayush Kumar (Cornell)• Animation of lapse• q = 1, χ = 0
github.com/sxs-collaboration/spectre 28
Missing Features
• Control system (Eamonn O’Shea (Cornell) & Mark Scheel (Caltech))• Horizon finding (Mark Scheel)• CCE with moving mesh (Jordan Moxon + Mark Scheel)• AMR (Larry Kidder (Cornell) + ND)• Local time stepping (Will Throwe (Cornell))
github.com/sxs-collaboration/spectre 29
Binary Black Hole Mergers – Load Balancing
0 50 100 150 200 250 300 350Simulation Time (M )
16
17
18
19
20
21
22
23
24
Sim
ulat
ion
Rate
(M/H
r)
No LB1st Attempt2nd AttemptCharm++ "Patch"
Led by François Hébert & Jordan Moxon (Caltech)Simulations by Geoffrey Lovelace & Sierra Thomas (CSU Fullerton)
github.com/sxs-collaboration/spectre 30
Recap
1 Robust high-order finite-difference schemes
2 Discontinuous Galerkin-finite-difference hybrid method
3 Binary black hole inspiral using discontinuous Galerkin
github.com/sxs-collaboration/spectre 31