Phase Averaged Deferred Correction for Multi-Timescale Systems

时变系统用微分方程（常微分方程或偏微分方程）来描述。这些系统无处不在，从海洋和大气流动到金融市场或生物模型。为了了解它们的行为，我们需要通过对它们进行时间积分来解微分方程。我们经常用我们的计算资源来做这个。对于像天气模型这样复杂的系统，这样的计算要求很高，而且需要很长时间。为了减少计算时间，在现代科学计算中，我们将计算任务并行化，并将它们分配给不同的处理器同时进行计算。然而，求解微分方程的时间积分的并行化存在很大的障碍。时间积分是一个连续的过程，其中在任何时间步长的解计算都需要在前一个时间步长的解。因此，它不容易并行化。非线性多时间尺度系统的有效时间集成提出了新的挑战。这些系统的快模态与慢模态耦合在一起，找到它们的解需要非常小的时间步长，从而降低了整体计算速度。该项目通过开发一种新型的并行时间积分器来解决这两个挑战（时间积分的并行化和快速振荡），该积分器可以有效地计算非线性多时间尺度系统的解。我们计划开发的方法考虑快速振荡相位上的微分方程平均值。以一种系统的方式进行平均，这样就很容易从平均解中检索快速动力学。这种平均的最大优点是可以在不影响精度或由于数值不稳定而导致的解决方案爆炸的情况下使用更长的时间。然而，平均本身在计算中引入了一种新的误差。为了减轻这种影响，我们通过降低平均窗口来迭代地修正平均解。我们的方法的一个新颖之处在于，以一种可以并行计算的方式设计这些校正层，因此使用多个处理器来降低总体计算时间。在开发了我们的方法并在简单的例子上进行了测试之后，我们将其应用于一个包含快波和慢涡的浅水模型。这可以为将所提出的方法应用于更复杂的海洋地球物理流动和天气预报模式奠定基础。

The time-dependent systems are described with differential equations (either ordinary or partial differential equations). These systems are everywhere from ocean and atmospheric flows to financial markets or biological models. To know their behaviour, we need to solve the differential equations by integrating them in time. We often do this numerically using our computational resources. Such computation for complicated systems like weather models is very demanding and takes a long time. To lower the computational time, in modern scientific computing we parallelise the computational tasks and assign them to different processors that compute them simultaneously. However, there is a big obstacle in the parallisation of time integration for solving differential equations. Time integration is a sequential process, in which computing the solution at any timestep requires the solution at previous timesteps. Hence, it cannot be parallelised easily. The efficient time integration of nonlinear multi-timescale systems poses an additional challenge. The fast modes of these systems are coupled with the slow modes and finding their solution requires very small timesteps that slow down the overall computation. This project addresses these two challenges (parallelisation of time integration and fast oscillations) by developing a novel parallel time integrator that efficiently computes the solution of nonlinear multi-timescale systems.The method that we plan to develop considers the differential equations averaged over the phase of fast oscillations. The averaging is done in a systematic way such that it will be easy to retrieve the fast dynamics from the averaged solution. The biggest advantage of this averaging is to allow taking larger times without compromising too much on the accuracy or the solution blowing up due to numerical instabilities. The averaging itself, however, introduces a new type of error in computation. To mitigate this effect, we iteratively correct the averaged solution by lowering the averaging window. A part of our method's novelty is designing these correction layers in a way that can be computed in parallel and hence using several processors to lower to the overall computation time. After developing our method and testing it on simple examples, we apply it to a model of shallow waters that incorporates fast waves and slow vortices. This can be a stepping-stone for the application of the proposed method in more complicated geophysical flows in the ocean and weather prediction models.