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Efficient time integration schemes are necessary to capture the complexprocesses involved in atmospheric flows over long periods of time. In thiswork, we propose a high-order, implicit-explicit numerical scheme that combinesMulti-Level Spectral Deferred Corrections (MLSDC) and the Spherical Harmonics(SH) transform to solve the wave-propagation problems arising from theshallow-water equations on the rotating sphere.  The iterative temporal integration is based on a sequence of correctionsdistributed on coupled space-time levels to perform a significant portion ofthe calculations on a coarse representation of the problem and hence to reducethe time-to-solution while preserving accuracy. In our scheme, referred to asMLSDC-SH, the spatial discretization plays a key role in the efficiency ofMLSDC, since the SH basis allows for consistent transfer functions betweenspace-time levels that preserve important physical properties of the solution.  We study the performance of the MLSDC-SH scheme with shallow-water test casescommonly used in numerical atmospheric modeling. We use this suite of testcases, which gradually adds more complexity to the nonlinear system ofgoverning partial differential equations, to perform a detailed analysis of theconvergence rate of MLSDC-SH upon refinement in time. We illustrate the goodstability properties of MLSDC-SH and show that the proposed scheme achieves upto eighth-order accuracy in time. Finally, we study the conditions in whichMLSDC-SH achieves its theoretical speedup, and we show that it cansignificantly reduce the computational cost compared to single-level SpectralDeferred Corrections (SDC).

Multi-Level Spectral Deferred Corrections Scheme for the Shallow Water Equations on the Rotating Sphere