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）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求解微分方程。我们经常使用我们的计算资源在数字上做这件事。对于像天气模型这样的复杂系统，这样的计算是非常苛刻的，并且需要很长时间。为了降低计算时间，在现代科学计算中，我们将计算任务并行化，并将它们分配给同时计算它们的不同处理器。然而，时间积分法求解微分方程的并行化存在着很大的障碍。时间积分是一个连续的过程，其中计算任何时间步的解都需要以前时间步的解。因此，不能轻易地将其平行化。非线性多时间尺度系统的有效时间积分提出了额外的挑战。这些系统的快模式与慢模式相结合，找到它们的解需要非常小的时间步长，这会降低整体计算的速度。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（时间积分和快速振荡的并行化），该并行时间积分器可以有效地计算非线性多时标系统的解。我们计划开发的方法考虑了在快速振荡相位上平均的微分方程。以系统的方式进行平均，使得容易从平均解中检索快速动态。这种求平均值的最大优点是允许花费更长的时间，而不会因为数值不稳定性而对精度或解的爆破造成太大的影响。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校正平均解。我们的方法的新奇性的一部分是设计这些校正层的方式，可以并行计算，因此使用几个处理器，以降低整体计算时间。在开发我们的方法和测试它的简单的例子，我们将其应用到一个模型的浅水沃茨，包括快波和慢涡。这可以是一个踏脚石，所提出的方法在更复杂的地球物理流动的海洋和天气预报模型的应用。

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.