Delve into the second lecture of the Numerical Relativity course, focusing on the mathematical formulation presented by Thomas Baumgarte from Bowdoin College. Explore advanced concepts in solving Einstein's field equations using computational methods, essential for modeling complex astrophysical phenomena such as black hole mergers, neutron star collisions, and stellar collapses. Gain insights into the mathematical framework underpinning numerical simulations crucial for interpreting gravitational wave signals and multi-messenger astronomy observations. This lecture is part of a comprehensive summer school on Gravitational-Wave Astronomy, designed for graduate students and postdocs with a strong background in General Relativity and numerical methods.