Dive into a comprehensive lecture on numerical relativity and the mathematical formulation of simulating coalescing binary black holes. Explore the fundamental concepts of gravitational waves, the history of numerical relativity, and its key components. Learn about the goals of numerical relativity, differential geometry, extrinsic curvature, and adopted coordinates. Examine the decomposition of curvature, partial differential equations, constraint solving techniques, and the BSSN equations. Gain insights into black hole momentum and spin, puncture initial data for multi-black hole systems, and evolution strategies. Engage with Q&A sessions and discussions on constraint damping, providing a thorough understanding of this complex field in gravitational wave astrophysics.