Time integration methods are one of the main building blocks for solving PDEs. In his talk, Dr. Martin Schreiber will give an overview of challenges on efficiently solving PDEs on upcoming high-performance (Exascale) architectures and possible solutions by recently developed time integration methods and their relevance to climate and weather simulations.
Abstract:
Weather and climate simulations face new challenges due to changes in computer architectures caused by physical limitations. From a pure computing perspective, algorithms are required to cope with stagnating or even decreasing per-core speed and increasing on-chip parallelism. Although this leads to an increase in the overall on-chip compute performance, data movement is increasingly becoming the most critical limiting factor. All in all, these trends will continue and already led to research on partly disruptive mathematical and algorithmic reformulations of dynamic cores, e.g. using (additional) parallelism in the time dimension.
This presentation provides an overview and introduction to the variety of newly developed and evaluated time integration methods for dynamical cores, all aimed at improving the ratio of wall clock time to error:
First, I will begin with rational approximations of exponential integrator methods in their various forms: Terry Haut's rational approach of exponential integrators (T-REXI), Cauchy contour integral methods (CI-REXI) on the complex plane and their relationship to Laplace transformations, and diagonalized Butcher's Tableau (B-REXI).
Second, Semi-Lagrangian (SL) methods are often used to overcome limitations on stable time step sizes induced by nonlinear advection. These methods show superior properties in terms of dispersion accuracy, and we have used this property with the Parareal parallel-in-time algorithm. In addition, a combination of SL with REXI is discussed, including the challenges of such a formulation due to Lagrangian formulation.
Third, the multi-level time integration of spectral deferred correction (ML-SDC) will be discussed, focusing on the multi-level induced truncation of nonlinear interactions and the importance of viscosity in this context. Based on this, the "Parallel Full Approximation Scheme in Space and Time" (PFASST) adds a time parallelism that allows even higher accelerations on the time-to-solution compared to ML-SDC and traditional time integration methods.
All studies were mainly conducted based on the shallow water equations (SWE) on the f-plane and the rotating sphere to investigate horizontal aspects of dynamical cores for weather and climate simulation. Overall, our results motivate further investigation and combination of these methods for operational weather/climate systems.
(With contributions and more from Jed Brown, Francois Hamon, Richard Loft, Michael Minion, Matthew Normile, Nathanaël Schaeffer, Andreas Schmitt, Pedro S Peixoto).