Explore advanced numerical techniques in computational relativity through this comprehensive lecture, part of a summer school on gravitational-wave astronomy. Delve into modern methods used to solve Einstein's field equations and simulate high-energy astrophysical phenomena such as black hole mergers, neutron star collisions, and stellar collapses. Learn how these computational tools contribute to the analysis of gravitational-wave signals and multi-messenger observations. Gain insights from expert Geoffrey Lovelace on cutting-edge approaches in numerical relativity, essential for modeling complex spacetime dynamics and informing gravitational-wave astronomy research.