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


Simulation Goals

• Binary black hole mergers• Binary neutron star mergers• Accretion disks• Core-collapse supernova

explosions

Hinderer et al. 2018


Binary Neutron Star Simulation Goals

• Resolve =⇒ high resolution• Resolve surface of stars• Resolve tidal deformability

• Can’t wait for bigger computers



Binary Neutron Star Simulation Goals

• Resolve =⇒ high resolution• Resolve surface of stars• Resolve tidal deformability• Can’t wait for bigger computers



Wish List—Physics Perspective

• Conserve mass, . . .• Physical solutions (positive pressure, density)• Constant flow preserved• Resolve large discontinuities (& 1020)


Hydrodynamics: Finite Difference Methods

• Solution error ∼ 1/N2 • Cartesian grids

u(x)

x


Vacuum Evolutions: Spectral Methods

• Solution error ∼ exp(−N) • Curved grids


Current Hydrodynamics Codes

• Bad values overwritten• Some positivity preservation• 2nd order• Manual tuning


Wish List—Numerical Perspective

• High-order• Robust, no tuning• Spectral when possible


Wish List

• Conserve mass, . . .• Physical solutions (positive pressure, density)• Constant flow preserved• Resolve large discontinuities (& 1020)

• High-order• Robust, no tuning• Spectral when possible


Building Blocks for Finite Difference Methods

u(x)

x


Building Blocks: Positivity

u(x)

x



First order everywhere =⇒ very inaccurate

u(x)

x



Arbitrary minimum value =⇒ total nonsense

u(x)

x


Building Blocks: Unphysical Oscillations

u(x)

x


Building Blocks: Oscillations

Detecting oscillations:

u(x) ≈N∑i=0

ciPi(x),



i = 0

i = 1

i = 2

i = 3

i = 4

i = 5

Modes

x

u(x)

Amplitude



Detecting oscillations:

u(x) ≈N∑i=0

ciPi(x),

Check:

wN (cN )2∑Ni=0wi(ci)2


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


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



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


Discontinuous Galerkin Method

• Solution error ∼ exp(−N)

• Shock capturing

−2 20

1

x

u(x)

Ωk−1 Ωk Ωk+1


DG-FD Hybrid Method

Check DG solution

FailedPassed

Retake time stepusing FD

Take FD time step

FailedPassed

Check if DGshould work

DG time step



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


The Cylindrical Sod Explosion


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


Binary Black Hole Mergers

• Cosmic Explorer + Einstein Telescope• Higher χ and q• Longer waveforms• Followup simulations• Space-based detector(?)


Binary Black Hole Mergers – Animation

• Led by Geoffrey Lovelace (CSU Fullerton) & Prayush Kumar (Cornell)• Animation of lapse• q = 1, χ = 0


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))


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)


Recap

1 Robust high-order finite-difference schemes

2 Discontinuous Galerkin-finite-difference hybrid method

3 Binary black hole inspiral using discontinuous Galerkin


nils deppe - icerm.brown.edu

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