A parallel time integrator for solving the linearized shallow water equations on the rotating sphere

Abstract

Summary With the stagnation of processor core performance, further reductions in the time to solution for geophysical fluid problems are becoming increasingly difficult with standard time integrators. Parallel‐in‐time exposes and exploits additional parallelism in the time dimension, which is inherently sequential in traditional methods. The rational approximation of exponential integrators (REXI) method allows taking arbitrarily long time steps based on a sum over a number of decoupled complex PDEs that can be solved independently massively parallel. Hence, REXI is assumed to be well suited for modern massively parallel super computers, which are currently trending. To date, the study and development of the REXI approach have been limited to linearized problems on the periodic two‐dimensional plane. This work extends the REXI time stepping method to the linear shallow‐water equations 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. Here, we present an efficient REXI solver based on spherical harmonics, showing the results of a geostrophic balance test, a comparison with alternative time stepping methods, an analysis of dispersion relations indicating superior properties of REXI, and finally, a performance comparison on the Cheyenne supercomputer. Our results indicate that REXI not only can take larger time steps but also can be used to gain higher accuracy and significantly reduced time to solution compared with currently existing time stepping methods.

Exponential integrators with parallel-in-time rational approximations for the shallow-water equations on the rotating sphere

“…Exponential integrators allow for solving the linearized SWE (L g ) on the rotating sphere (including L c ) with very high accuracy [9,38]. Furthermore, exponential integrators have been shown to support time steps as long as 1.5 days [38].…”

“…Exponential integrators allow for solving the linearized SWE (L g ) on the rotating sphere (including L c ) with very high accuracy [9,38]. Furthermore, exponential integrators have been shown to support time steps as long as 1.5 days [38]. It has also been shown that SH provides an ideal basis for REXI to time integrate the linear parts of the SWE [9], even on the rotating sphere [38].…”

“…Furthermore, exponential integrators have been shown to support time steps as long as 1.5 days [38]. It has also been shown that SH provides an ideal basis for REXI to time integrate the linear parts of the SWE [9], even on the rotating sphere [38]. However, the non-linearities N(U) in the SWE pose additional restrictions regarding the stable time integration size, the resulting accuracy and finally the wallclock time which will be studied in the present work.…”

Semi-Lagrangian Exponential Integration with Application to the Rotating Shallow Water Equations

“…Krylov solvers, such as those presented in [28], are used in [15] and [25] for the matrix exponentiation of a dynamic linearization of the shallow water system. Furthermore, [48] adopts a rational approximation based on [27] for the rotating SWE on the plane, which is also used for the sphere in [47,49] with a global spectral spherical harmonics representation. This rational approximation approach calculates the matrix exponentials with a very high degree of parallelism, so the additional computational costs of the calculating such exponential may be absorbed by extra compute nodes to still reduce the time-to-solution.…”

“…Small scale horizontal gravity waves play an important role in the large structure of the middle atmosphere, particularly for climate simulations [38]. Exponential integrators provide a way to obtain large time-steps without affecting these small-scale waves, preserving superior linear dispersion properties (see [47,14]).…”