Accelerating the Next Frontier of Gravitational Waves: Advanced Computational Methods & Simulations

We are entering an exciting era in gravitational wave (GW) astrophysics, with upcoming space-based detectors like LISA set to explore the millihertz frequency band enabling the detection of low-frequency sources such as massive black hole (MBH) binaries and intemediate/extreme mass-ratio inspirals (I/EMRIs), offering unprecedented insights into the astrophysics of galaxy evolution and fundamental physics. Realizing the full potential of these observatories requires significant advances in computational modeling. This thesis develops and applies advanced numerical techniques, particularly fast multipole methods (FMM) in N-body simulations, to study the merger dynamics of massive black hole (MBH) binaries in nuclear star clusters (NSCs). I investigate how collisional relaxation and mass segregation within NSCs influence binary hardening timescales, and how NSCs affect the orbital evolution of MBH seeds in post-merger galaxies, where dynamical friction may be inefficient, leading to long delays in binary formation and merger. I also explore whether GWs can probe dark matter distributions by analyzing dephasing effects caused by dark matter spikes around intermediate-mass black holes. Finally, I present proof-of-concept improvements to Hamiltonian splitting integrators, assessing the performance of a fully GPU-resident N-body implementation as a step toward future high-performance FMM-based GW source modeling frameworks.