Formulation and numerical computation of the low frequency mean flow of fluids

高振动的偏微分方程组(PDE)在许多科学应用中占据主导地位。我们现在正处在一场计算革命的门口，拥有新的、不同种类的计算机体系结构，可以实现显著更高的空间分辨率。阻碍利用这些新计算机的主要问题之一是，随着空间分辨率的提高，时间步长必须减小。例如，对于网格分辨率为300公里的大气模式，通常使用20分钟的时间步长。将空间分辨率提高到1公里需要将时间步长减少到4秒，这使得高分辨率模拟不适用于科学探索。然而，在理论流体动力学中，低频是通过共振波和近谐振波的非线性耦合而产生的。这些频率主导着天气和气候应用的动态。揭示所述共振行为的振荡偏微分方程组的新公式可用于推导描述问题的非线性共振和“近共振”部分的平均流的方程，这反过来又导致构造能够采用比以前可能的更大时间步长的数值方法。在这个项目中，我建议以理论流体力学和数值分析的结果为基础，1)发展考虑共振和“接近共振”频率的平均流的新公式，以及2)开发和分析新的数值格式，不忽略共振和“接近共振”频率，目的是超越目前的限制，采取大的时间步长。这些方案将通过数值分析的方式进行研究，并在理想化的领域中实施，主要是在天气和气候模拟方面的应用。将探索的数值方案包括时间并行方案，因为这种方案可以大大加快现代计算机体系结构上的计算速度。发展的流体力学理论和数值算法有可能显著缩短天气和气候模型在新的计算机体系结构上的求解时间。

Highly oscillatory systems of partial differential equations (PDEs) dominate many scientific applications. We are now on the doorstep of a computing revolution with new, heterogeneous computer architectures which could enable significantly higher spatial resolution. One of the main issues standing in the way of taking advantage of these new computers is that as the spatial resolution increases, the time step must decrease. As an example, for an atmosphere model with a grid resolution of 300 kilometers, a 20 minute time step is generally used. Increasing the spatial resolution to 1 kilometer requires a reduction of the time step to 4 seconds, making high resolution simulations impractical for scientific exploration. However, it is known in theoretical fluid dynamics that low frequencies are created through nonlinear coupling of resonant and “near-resonant” waves. These frequencies dominate the dynamics of weather and climate applications. New formulations of the oscillatory PDEs that expose the described resonance behaviour can be used to derive equations for the mean flow which describes the nonlinear resonant and “near-resonant” part of the problem, and this, in turn, leads to constructing numerical methods able to take larger time steps than previously possible. In this project, I propose building on results from theoretical fluid dynamics and numerical analysis to 1) develop new formulations of the mean flow which take both the resonant and the “near-resonant” frequencies into account and 2) develop and analyse new numerical schemes, that do not neglect the resonant and also the “near-resonant” frequencies, with the goal of taking large time steps beyond current limitations. These schemes will be investigated by means of numerical analysis and implemented in idealized domains, mainly within the application of weather and climate simulation. The numerical schemes that will be explored include parallel-in-time schemes, because such schemes can contribute to substantially speeding up computations on modern computer architectures. The developed fluid dynamics theory and numerical algorithms have the potential to significantly decrease the time-to-solution for weather and climate models on new computer architectures.