  Efficient time integration schemes are necessary to capture the complex
processes involved in atmospheric flows over long periods of time. In this
work, we propose a high-order, implicit-explicit numerical scheme that combines
Multi-Level Spectral Deferred Corrections (MLSDC) and the Spherical Harmonics
(SH) transform to solve the wave-propagation problems arising from the
shallow-water equations on the rotating sphere.
  The iterative temporal integration is based on a sequence of corrections
distributed on coupled space-time levels to perform a significant portion of
the calculations on a coarse representation of the problem and hence to reduce
the time-to-solution while preserving accuracy. In our scheme, referred to as
MLSDC-SH, the spatial discretization plays a key role in the efficiency of
MLSDC, since the SH basis allows for consistent transfer functions between
space-time levels that preserve important physical properties of the solution.
  We study the performance of the MLSDC-SH scheme with shallow-water test cases
commonly used in numerical atmospheric modeling. We use this suite of test
cases, which gradually adds more complexity to the nonlinear system of
governing partial differential equations, to perform a detailed analysis of the
convergence rate of MLSDC-SH upon refinement in time. We illustrate the good
stability properties of MLSDC-SH and show that the proposed scheme achieves up
to eighth-order accuracy in time. Finally, we study the conditions in which
MLSDC-SH achieves its theoretical speedup, and we show that it can
significantly reduce the computational cost compared to single-level Spectral
Deferred Corrections (SDC).
