Robust Numerical Methods for Nonlinear Wave Equations in Second-order Form

About This Grant

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 critically shape their dynamics. Standard numerical approaches typically reformulate them into first-order systems, which increase computational cost and potentially compromise key physical properties. Building on the investigator’s prior success with numerical methods for (semi-)linear second-order wave problems, this project aims to address these challenges. The goal is to design numerical schemes specifically tailored to the nonlinear second-order formulation, emphasizing stability, high accuracy, computational efficiency, and fidelity to the underlying physics. In particular, the research will focus on constructing fully discrete, structure-preserving energy discontinuous Galerkin methods and applying them to complex physical systems, such as geometric wave models and coupled first- and second-order hyperbolic systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.