Abstract:
The two breakthroughs in physics in recent times: direct detection of gravitational waves and the first images of black hole shadows demand high accuracy in theoretical modelling in order to test our understanding of fundamental physics and probe regimes which are otherwise inaccessible in terrestrial laboratories. Since analytical solutions to the non-linear Einstein field equations and highly coupled magneto-hydrodynamic processes are rare and often require simplifying assumptions, large-scale Numerical Relativity (NR) and General Relativistic Magneto-Hydrodynamic (GRMHD) simulations are unavoidable for interpreting these observations. This thesis explores two systems and numerical challenges in their modeling. First, accurate gravitational waveform extraction is limited by finite computational domains introducing extraction errors. The computational domain can be extended all the way to future null infinity, where the gravitational wave extraction is free from gauge ambiguities and near field effects, by recasting the equations on hyperboloidal slices. In the first part of the thesis we implement this for solving a class of linear wave equations on Minkowski spacetime posed on hyperboloidal slices. We also introduce three dimensional Summation-By-Parts (SBP) scheme to discretize the continuum system of equations in a way that preserves the continuum energy dynamics even at the discrete level. We emply second order finite difference methods for numerical differentiation while noting that the scheme can be easily extended to higher order differentiation schemes and spectral methods as well. Second, modeling accreting black holes and their electromagnetic signatures requires numerical frameworks capable of resolving turbulent plasmas and instabilities, such as the Magneto-Rotational Instability (MRI). The second part of this thesis focuses on the validation of the GRMHD framework in SpECTRE, a numerical relativity code. SpECTRE employs element-by-element hybrid Discontinuous Galerkin and Finite Difference (DG-FD) scheme, allowing it to efficiently resolve smooth flows while also capturing shocks.
