We describe a proof-of-concept development and application of a phase averaging technique to the nonlinear rotating shallow water equations on the sphere, discretised using compatible finite element methods. Phase averaging consists of averaging the nonlinearity over phase shifts in the exponential of the linear wave operator. Phase averaging aims to capture the slow dynamics in a solution that is smoother in time (in transformed variables) so that larger timesteps may be taken. We overcome the two key technical challenges that stand in the way of studying the phase averaging and advancing its implementation: 1) we have developed a stable matrix exponential specific to finite elements and 2) we have developed a parallel finite averaging proceedure. Following Peddle et al (2019), we consider finite width phase averaging windows, since the equations have a finite timescale separation. In our numerical implementation, the averaging integral is replaced by a Riemann sum, where each term can be evaluated in parallel. This creates an opportunity for parallelism in the timestepping method, which we use here to compute our solutions. Here, we focus on the stability and accuracy of the numerical solution. We confirm there is an optimal averaging window, in agreement with theory. Critically, we observe that the combined time discretisation and averaging error is much smaller than the time discretisation error in a semi-implicit method applied to the same spatial discretisation. An evaluation of the parallel aspects will follow in later work.

翻译：我们描述了相位平均技术的一种概念验证开发及其在球面非线性旋转浅水方程中的应用，该方程采用兼容有限元方法进行离散化。相位平均的本质是对线性波动算子指数中的非线性项进行相位偏移平均，旨在捕获解中在时间上更平滑（经变换变量后）的慢动态过程，从而允许采用更大的时间步长。我们克服了研究相位平均及其实现推广的两项关键技术挑战：1) 开发了专属于有限元的稳定矩阵指数计算方案；2) 提出了并行有限平均处理程序。遵循Peddle等人（2019）的方法，由于方程存在有限时间尺度分离，我们采用有限宽度相位平均窗口。在数值实现中，平均积分被替换为黎曼和，其中每一项均可并行计算。这为时间步进方法创造了并行化机会，并在此用于求解。本文重点研究数值解的稳定性与精度。我们证实了与理论一致的最优平均窗口存在性。关键发现是：在相同空间离散化条件下，时间离散化与平均的联合误差远小于半隐式方法的时间离散化误差。并行性能评估将在后续工作中展开。