1. 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)

2. 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

3. 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

4. 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

5. Merging of grid patches t=0t=61.2sm t=122.4sm Height field of lapse function of coalescing neutron stars Not showing whole computational domain!

6. 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

7. Validity of GRAstro_AMR of full EFEs: Hamiltonian constraint equationMomentum constraint equation 3+1 evolution equation 3+1 split

8. Validity of GRAstro_AMR  We monitor the convergence of: a) the Hamiltonian constraint violation: 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

9. 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

10. Monitoring HCV convergence Example: single static neutron star a) comparison of HCV between the with Level 1 refinement Level 2 refinement xx dx=1.2sm dx=0.6sm scaled  2nd-order convergence same levels of refinement different resolutions

11. Monitoring HCV convergence b) comparison of HCV between generated from the Finest base grid x different levels of refinement same base grid  2nd-order convergence Level 1 refinement Level 2 refinement

12. 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 refinement different 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

13. Monitoring HCV convergence b) comparison of HCV between generated from the Finest base grid x different levels of refinement same base grid  1st-order convergence Level 2 refinement Level 1 refinement

14. Monitoring HCV convergence c) 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

15. 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