Delve into the fifth lecture of a comprehensive series on the theory and numerical methods for partial differential equations, presented by Harald Pfeiffer at the International Centre for Theoretical Sciences. Explore advanced concepts in numerical relativity as part of a two-week summer school on gravitational-wave astronomy. Gain insights into solving field equations of General Relativity and magnetohydrodynamics using supercomputers, essential for modeling mergers of black holes and neutron stars, massive star collapses, and accreting black holes. Learn how these numerical simulations contribute to understanding gravitational-wave signals and multi-messenger observations in astrophysics. Benefit from graduate-level instruction in mathematical formulation, relativistic hydrodynamics, and modern computational methods in the context of gravitational-wave research and high-energy astrophysical phenomena.