Capability and Validity of GRAstro_AMR

Mew-Bing Wan

E. Evans, S. Iyer, E. Schnetter, W.-M. Suen, J. Tao, R. Wolfmeyer, H.-M. Zhang, Phys. Rev. D 71

(2005)

Why GRAstro_AMR?• Needs:

a) resolution on the order of 0.1*baryonic mass of 1 neutron star with a typical EOS stable evolution

b) initial separation of 2 neutron stars in a binary inspiral on the order of 50*baryonic mass of 1 neutron star astrophysically-relevant

c) distance of computational domain boundary from system 0.5*gravitational wavelength of system artificial influence

d) size of computational domain 1*gravitational wavelength of system accurate extraction of gravitational waveform from system

e) binary inspiral evolution of neutron stars up to coalescence point within convergence regime

Capability of GRAstro_AMR• Solution of full Einstein field equations:

coupling between space-time and hydrodynamics

• Multiple-length scale resolution:

a) multiple levels of refinement

b) resolutions increasing with higher refinement levels

c) refinement criteria based on matter density and Hamiltonian constraint violation

Capability of GRAstro_AMR

• Merging of grid patches comoving with neutron stars in binary inspiral

• Equivalence to high-memory, long-term and high-resolution unigrid evolution:

inspiral run of binary neutron stars

AMR 4 levels of refinement 603 finest grid covering

each star maximum memory of 8GB

Unigrid

10253 grid minimum memory of 1.2TB

Merging of grid patches

t=0 t=61.2sm

t=122.4sm

Height field of lapse function of coalescing neutron stars

Not showing whole computational domain!

Equivalence to high-resolution unigrid run

Full computational domain of size (34R)3

R - proper radius of each star

Close-up on stars

Central density of coalescing neutron stars

t=0

t=418sm

Close-up on grid structure

Validity of GRAstro_AMR• of full EFEs:

TG 8

1623 KKKR

Hamiltonian constraint equation Momentum constraint equation

SSKKKKRN 2423

3+1 evolution equation

JKK 8

NKL Nn

3+1 split

Validity of GRAstro_AMR

We monitor the convergence of:

a) the Hamiltonian constraint violation:

ndxOKKKR 1623

n - order of convergence

dx - size of grid element

b) the momentum constraint violation

c) various physical quantities for e.g

• Our code carries out unconstrained evolution

Monitoring convergence of Hamiltonian constraint violation

• 3 kinds of convergence tests: a) comparison of Hamiltonian constraint violation (HCV)

between the same levels of refinement with different resolutions

b) comparison of HCV between different levels of refinement generated from the same base grid

c) comparison of HCV between the finest level of refinement and its unigrid equivalent

Monitoring HCV convergence• Example: single static neutron star

a) comparison of HCV between the with

Level 1 refinement Level 2 refinementx x

dx=1.2sm

dx=0.6sm

scaled

2nd-order convergencesame levels of refinement

different resolutions

Monitoring HCV convergenceb) comparison of HCV between generated from

the

Finest base gridx

different levels of refinementsame base grid 2nd-order convergence Level 1 refinement

Level 2 refinement

Monitoring HCV convergence• Example: single boosted neutron star

a) comparison of HCV between the with

Level 2 refinement

dx=2.88sm

dx=1.2sm

same levels of refinementdifferent resolutions 1st-order convergence High-Resolution

Shock Capturing (HRSC) Total-

Variation- Diminishing (TVD) scheme in evolving

the hydrodynamics! 1st-order HCV at

isolated points which propagate to

other points during evolution

Monitoring HCV convergenceb) comparison of HCV between generated from

the

Finest base grid

x

different levels of refinementsame base grid 1st-order convergence

Level 2 refinement

Level 1 refinement

Monitoring HCV convergencec) comparison of HCV between the and

its

Finest base grid

finest level of refinement

unigrid equivalent

Equivalent to base grid

Equivalent to Level 2 refinement

dx=1.2sm

Summary• We have carried out various convergence tests of

GRAstro_AMR:

a) convergence order of HCV: from 1st-order to 2nd-order

b) order of convergence proven valid for the simplest non-trivial case of the boosted neutron star and various configurations involving boosted NS’s

• Development of further computational tools and convergence of physical quantities will be shown in later talks on the usage of GRAstro_AMR in physical problems

• We invite researchers to utilize GRAstro_AMR