Explore the foundations of modern numerical methods in computational relativity through this comprehensive lecture, the first in a series by Geoffrey Lovelace. Delve into advanced techniques used to solve Einstein's field equations and model complex astrophysical phenomena such as black hole mergers, neutron star collisions, and gravitational wave events. Gain insights into the mathematical formulations, numerical algorithms, and computational strategies employed in cutting-edge gravitational wave research. Learn how these methods contribute to our understanding of high-energy astrophysics and support observations from gravitational wave detectors, electromagnetic telescopes, and neutrino observatories. Suitable for graduate students and researchers in gravitational wave astronomy, this lecture provides essential knowledge for those seeking to advance their skills in numerical relativity and computational astrophysics.