Parallel-in-Time Multigrid Methods for Hyperbolic Problems, with a Focus on the Shallow Water Equations
The coming massive parallelism of exascale computing presents a pressing challenge for the many DOE simulations of time-dependent partial differential equations, which typically use traditional sequential time stepping methods. Since this traditional approach is inherently serial, it presents a sequential bottleneck when moving to exascale computing, because future performance gains will come through greater concurrency, not faster clock speeds. Thus, the goal of this work is to research parallelism in time, i.e., methods that compute multiple time values simultaneously, not sequentially. The focus will be on hyperbolic problems of programmatic interest to DOE, with the goal of enabling scalable simulations of time-dependent hyperbolic problems on future architectures. The difficulty lies in the fact that hyperbolic problems are well-known to be difficult for parallel-in-time methods, with the most common parallel-in-time method, Parareal, diverging in many cases. Here, the chosen methodology for scalably solving these hyperbolic space-time equations is multigrid-reductionin-time (MGRIT), because multigrid (when it works) is a powerful, optimal, and scalable solver for discretized PDEs. To further research in this area, this project will explore a model hyperbolic problem (the shallow water equations) in the context of recent advances (e.g., by Wingate and Haut) in constructing improved coarse time-grid time-propagators by using the slow asymptotic structure of the equations. In particular, we will investigate if such coarse time-propagators offer significant advantages in an MGRIT setting, and explore any broader insights gained into parallel-in-time for hyperbolic problems.