Exponential Integrators for Weather and Climate Simulations using Global Spectral Methods

This work studies exponential integrator (EI) methods in the context of climate and weather simulations. We put our focus on a rational approximation of exponential integrators (REXI) which can be expressed via U (∆t) ≈ n β n (∆tL + α n ) −1 U (0) where α, β ∈ C, L a linear operator (with oscillatory or diffusive stiffness) and the initial conditions, U (0) (see e.g. [Clancy and Lynch, 2011; Haut et al., 2015]). The additional degrees of freedom for parallelization are made available by the independent terms in the sum which can be solved in parallel. This formulation enables us to overcome the time step limitation due to the linear stiffness. A special focus is put on PDEs relevant for climate and weather simulations.<br>Such EI methods have a rich history (see e.g. [Moler and Van Loan, 2003; Hochbruck and Ostermann, 2010]) and are of increasing interest in the parallel-in-time (PinT) community. In the context of PinT methods, these EI formulations were first mentioned by [Gander and Guettel, 2013] as the ParaEXP method. In the context of the development of dynamical cores for climate and weather simulations, recent research [Clancy and Lynch, 2011; Garcia et al., 2014; Clancy and Pudykiewicz, 2013] exploited EI formulations and also showed promising results regarding higher accuracy. Additionally, EI are a crucial component when applying the asymptotic PinT approach [Haut and Wingate, 2014] to a more realistic problem. Parallel performance and numerical accuracy studies of the REXI method were conducted for the linear shallow-water equations (SWE) on the plane [Schreiber et al., 2016] using spectral solvers for Fourier space and finite-difference formulations. Up to two orders of magnitude of speedup for oscillatory linear stiff problems were presented in this work. The successful application of spectral solvers to solve for each REXI term motivated an investigation of EI to the linearized SWE including the effect of a rotating sphere. Spherical Harmonics are used which exploit the low-bandwidth structure of the linear rotational SWE in spectral space[Schreiber and Loft, 2017].<br>In this talk, we will present our current research on extending our previous work to the full non-linear PDEs. First, we will give a brief overview of the results to solve the linear SWE on the plane and rotating sphere with EI. Second, we will discuss time splitting approaches and study the direct application of EI formulations which can be expressed via the aforementioned sum and their extension to non-linearities (see e.g. [Cox and Matthews, 2002; Luan and Ostermann, 2016]). Finally, we will compare EI formulations with state-of-the art time stepping methods which are currently used in operational weather forecasts.<br>ERRATA:page 21: \leq =&gt; \geqpage 21: exp(i) =&gt; exp(i0)