Phase Averaged Deferred Correction for Multi-Timescale Systems

多时间尺度系统的相位平均延迟校正

• 批准号: EP/Y032624/1
• 负责人: Hossein Kafiabad
• 金额: $ 10.06万
• 依托单位: Durham University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY032624%2F1
• 关键词: Phase; Averaged; Deferred; Correction; Multi

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.

时间依赖性系统用微分方程（普通或部分微分方程）描述。这些系统无处不在，从海洋和大气流到金融市场或生物模型。要知道它们的行为，我们需要通过及时整合微分方程来求解它们。我们经常使用计算资源进行数字进行此操作。对于天气模型等复杂系统的这种计算非常苛刻，需要很长时间。为了降低计算时间，在现代科学计算中，我们将计算任务平行，并将其分配给同时计算它们的不同处理器。但是，在解决微分方程的时间集成的标准中，存在很大的障碍。时间积分是一个顺序过程，在任何时间步中计算解决方案都需要在先前的时间段上进行解决方案。因此，它不能轻易平行。非线性多时间计算系统的有效时间整合提出了一个额外的挑战。这些系统的快速模式与缓慢的模式结合在一起，找到它们的解决方案需要很小的时间段，以减慢整体计算。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（时间集成和快速振荡的平行化），该挑战有效地计算非线性多时间尺度系统的解决方案。我们计划开发的方法在快速振荡的阶段中平均考虑了平均的微分方程。平均值以系统的方式进行，因此可以轻松从平均解决方案中检索快速动态。这种平均值的最大优势是允许在不损害数字不稳定性的准确性或解决方案上爆炸的情况下允许更大的时间。但是，平均本身引入了一种新型的计算错误。为了减轻这种效果，我们通过降低平均窗口迭代校正平均解决方案。我们方法新颖性的一部分是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将其降低到整体计算时间。在开发了我们的方法并在简单示例上进行测试后，我们将其应用于结合快速波和缓慢涡旋的浅水模型。这可能是将拟议方法应用于海洋和天气预测模型中更复杂的地球物理流中的垫脚石。