The modeling of atmospheric processes in the context of weather and climate simulations is an important and computationally expensive challenge. The temporal integration of the underlying PDEs requires a very large number of time steps, even when the terms accounting for the propagation of fast atmospheric waves are treated implicitly. Therefore, the use of parallel-in-time integration schemes to reduce the time-to-solution is of increasing interest, particularly in the numerical weather forecasting field.

We present a multi-level parallel-in-time integration method combining the Parallel Full Approximation Scheme in Space and Time (PFASST) with a spatial discretization based on Spherical Harmonics (SH). The iterative algorithm computes multiple time steps concurrently by interweaving parallel high-order fine corrections and serial corrections performed on a coarsened problem. To do that, we design a methodology relying on the spectral basis of the SH to coarsen and interpolate the problem in space.

The methods are evaluated on the shallow-water equations on the sphere using a set of tests commonly used in the atmospheric flow community. We assess the convergence of PFASST-SH upon refinement in time. We also investigate the impact of the coarsening strategy on the accuracy of the scheme, and specifically on its ability to capture the high-frequency modes accumulating in the solution. Finally, we study the computational cost of PFASST-SH to demonstrate that our scheme resolves the main features of the solution multiple times faster than the serial schemes.

在天气和气候模拟的背景下对大气过程进行建模是一个重要且计算量巨大的挑战。即使隐含处理了快速大气波传播的术语，底层PDE的时间积分也需要大量的时间步长。因此，使用实时并行积分方案来缩短求解时间越来越受到关注，尤其是在数值天气预报领域。

我们提出了一种多级并行时间积分方法，该方法将时空并行全近似方案（PFASST）与基于球谐函数（SH）的空间离散化相结合。迭代算法通过交织并行的高阶精细校正和对粗化问题执行的串行校正来同时计算多个时间步长。为此，我们设计了一种基于SH光谱基础的方法，可以对空间问题进行粗化和插值。

使用大气流动界中常用的一组测试，根据球体上的浅水方程对方法进行了评估。我们评估了PFASST-SH在细化后的收敛性。我们还研究了粗化策略对方案准确性的影响，尤其是对捕获解决方案中累积的高频模式的能力的影响。最后，我们研究了PFASST-SH的计算成本，以证明我们的方案比串行方案更快地解决了解决方案的主要特征。