Nonlinear wave equations in second-order form are fundamental to understanding phenomena in geophysics, plasma physics, quantum science, and beyond. However, accurately and efficiently simulating these equations remains a major challenge due to their complexity and sensitivity, which demand a careful balance of precision and speed, along with the use of stable numerical schemes to ensure reliable results. This project develops robust and efficient numerical algorithms for solving wave equations, optimized for high performance on both current and next-generation computing platforms. These computational tools will advance foundational research and have wide-ranging applications in areas where accurate wave prediction is critical. Beyond technical innovation, the project supports the development of a skilled scientific workforce by training graduate researchers and engaging students through reading groups and seminars. These educational initiatives promote participation in computational mathematics and contribute to the nation's continued leadership in science, engineering, and technological innovation. The main computational challenges associated with nonlinear second-order wave equations stem from their rich and intricate range of behaviors. These equations can exhibit solitary waves, solitons, finite-time blow-ups, singularities, and rapid oscillations. These equations, often derived from Euler–Lagrange equations, carry intrinsic geometric and energetic structures that c