Apply discontinous Galerkin method to Einstein equations Zhoujian Cao Department of Astronomy, BNU 2019-1-4 Xiamen-CUSTIPEN workshop on EOS in the era of GWA

Content Introduction about GW data analysis and NR Challenges about template construction for future detectors Related work about NR computation Summary

arXiv: 1811.12907

GW151226 ? 13, 5sigma

Matched filtering and template

GW template construction GW source? Solve Einstein equation!? NR

History of NR Hahn and Lindquist, first BBH simulation (1964) Unstable, Unstable, Unstable!!! F. Pretorius, PRL 95, 121101 (2005); M. Campanelli et al, PRL 96, 111101 (2006); J. Baker et al, PRL 96, 111102 (2006) Stable finite difference code is available! Caltech&Cornell group, PRD 79, 024003 (2009) Stable and extremely acurate spectral code is available! Only for BH Caltech&Cornell group, JCP 335, 84 (2017) Highly parallel efficient finite element code is available! Only for GR fluid

Future GW detectors Mass ratio: 1<q<1e-10 Eccentricity: 0<e<1 Tidal disruption and tidal deformation Neutron star right after merger ……beyond GR, unexpected sources 1712.07345

AMR within spectral method Divide the space to multi spectral domains One domain, one cpu Typically 60-80 cpus Most serious problem: hard to adjust domains for large mass ratio BBH

Parallel problem of AMR Cao 2009

FE numerical scheme Combine advantages: High convergence as spectra method + As high even more parallel ability as finite difference Brand new topic in NR!! Ji, Cai and Cao, 2018

From Harmonic to GH require Still unstable! denote

Move on to first order systems Unstable !

Linearly degenerate or not ( ) Principle matrix: Characteristic variables: Characteristic speed:

Linearly degenerate or not In order to make the system linearly degenerate, we need analytically But numerical error makes it fail to be linearly degenerate except Avoid numerical shocks, but Unstable!

Linearly degenerate or not In order to make the system linearly degenerate, we need Avoid numerical shocks, but Unstable! Simplified estimate:

Shock forms, but really stable!

Simplified estimate:

Cai, Cao, Fu, Ji, and Xia 2018

P0: leading order polynomial P1: sub leading order polynomial

Summary GW data analysis and NR Template for high mass ratio BBH; Post-merger waveform; …… [high efficiency] Finite element with DG method for Einstein equations