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

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

4. arXiv:

6. GW151226 ?
13, 5sigma

7. Matched filtering and template

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

9. History of NR Hahn and Lindquist, first BBH simulation (1964)
Unstable, Unstable, Unstable!!!
F. Pretorius, PRL 95, (2005); M. Campanelli et al, PRL 96, (2006); J. Baker et al, PRL 96, (2006)
Stable finite difference code is available!
Caltech&Cornell group, PRD 79, (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

10. 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

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

12. Parallel problem of AMR
Cao 2009

13. 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

15. From Harmonic to GH
require
Still unstable!
denote

16. Move on to first order systems
Unstable !

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

18. 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!

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

21. Shock forms, but really stable!

22. Simplified estimate:

23. Cai, Cao, Fu, Ji, and Xia 2018

24. P0: leading order polynomial
P1: sub leading order polynomial

25. 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