Collaborative Research: Parallel Space-Time Solvers for Systems of Partial Differential Equations

Computer simulations and the mathematical methods supporting these are central to the modern study of engineering, biology, chemistry, physics, and other fields. Many simulations are computationally costly and require the large resources of modern supercomputers. New mathematical methods are urgently needed to efficiently utilize next generation supercomputers with millions to billions of processors. This project will develop new parallel-in-time algebraic multigrid methods for complex physical systems specifically designed for next generation computers. These new methods will add a new dimension of parallel scalability (time) and promise dramatically faster simulations in many important application areas, such as the gas and fluid dynamics problems considered (e.g., with relevance to wind turbines and viscoelastic flow). Graduate students will be involved and trained, and open source code will be developed. This project will develop fast, parallel, and flexible space-time solvers for systems of partial differential equations (PDEs). The project will focus on algebraic multigrid (AMG) within block preconditioning traditionally appropriate for large adaptively refined spatial systems. These techniques will be extended to general space-time systems with a flexible approach that allows for adaptive space-time refinement. This adaptivity helps to accurately resolve lower dimensional features such as shocks at a fraction of the cost and storage of uniform refinement. Furthermore, the project will produce new practical AMG theory for non-SPD (symmetric positive definite) problems as well as solvers for adaptively refined space-time discretizations for a variety of parabolic and hyperbolic PDEs including the Euler and Navier-Stokes equations and Cahn-Hilliard system. The project will design, analyze, and tune parallel AMG solvers that are robust, efficient, and fast over a wide range of PDEs and parameters and will contribute to the widely used packages MFEM and hypre. The solvers will be developed and tested for applications in wind turbines, as well the high Weissenberg number problem in viscoelastic flows. 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.