Collaborative Research: Euler-Based Time-Stepping with Optimal Stability and Accuracy for Partial Differential Equations

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Principled simulations of many real-world problems (such as fluid flow, geophysical phenomena, and quantum mechanics) require an evolution in time with high accuracy, yet in a structurally simple and robust fashion. This project develops novel time integration methods for complex multi-physics problems, while not incurring fundamental problems that reduce the accuracy or stability of many existing methods. The developed methods are founded in new mathematical theories, and are used to devise more accurate and robust simulations of shallow water flows with dispersive effects, which are important in the understanding of tsunamis, storm surge, and coastal flooding. This project will support one graduate student for two years at NJIT and one graduate student per year at the second institution, Temple.This project develops methods for the time integration of differential equations that are implemented as sequences of generalized Euler steps, including: multistage diagonally implicit Runge-Kutta (DIRK) and multistep implicit-explicit (IMEX) methods. Such methods are significant as they reduce the implementation burden on a practitioner to the solution of a fully- or semi-implicit Euler step for their initial-boundary-value problem. The key research contributions are: (A) a full algebraic theory of weak stage order, and its use to design optimized high-order DIRK methods devoid of order reduction; (B) a stability theory for IMEX methods applied to differential algebraic equations, and the co-design of IMEX splittings and scheme coefficients to overcome stability limitations prevalent in existing methods. Applications include new efficient time-stepping for the dispersive shallow water equations and related differential algebraic equations. The collaborative mentoring of graduate students at two campuses is an important component of this project.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.