Research parallel-in-time computational methods

With the stagnation of processor core performance, further reductions in the time-to-solution for geophysical fluids problems are becoming increasingly difficult with standard time integrators. “Parallel-in-time” methods expose and exploit additional parallelism in the time dimension. Inherently sequential in traditional methods, parallel-in-time methods enable the potential acceleration of geophysical fluid simulations on massively parallel machines.

CISL scientists have been collaborating with Dr. Martin Schreiber from the University of Exeter, a visitor to NCAR over the past two summers. Our collaboration is focused on the rational approximation of exponential integrators (REXI) method. This research is in an early stage, but it has shown that linear oscillatory operators allow taking arbitrarily long time steps. The method is based on a sum of inverse problems that can be solved independently in parallel. Hence REXI is well suited for modern massively parallel computers in the post-Dennard scaling era.

In the past, study and development of the REXI approach has been limited to linearized problems on the periodic 2D plane. The work performed in this CISL collaboration extends the REXI time-stepping method to the linear shallow-water equations (SWE) on the rotating sphere, thus moving the method one step closer to solving fully nonlinear fluid problems of geophysical interest on the sphere. The rotating sphere poses particular challenges for finding an efficient solver due to the zonal dependence of the Coriolis term. Our most recent work has extended the method to nonlinear SWE problems, albeit with time-step limitations, and has explored deep mathematical connections between REXI and other implicit time-integration schemes.

This exploratory work resulted in an efficient REXI solver based on spherical harmonics. Further, an Eigenvalue analysis of the methods dispersion relations indicated superior properties of the REXI time-stepping method compared to other standard time-integration methods. Our results indicate that REXI is not only able to take larger time steps, but that REXI can also be used to gain higher accuracy compared to other methods.

This work is supported by NSF Core funding through the CISL Visitor Program.