Multiscale Numerical Strategies for Models with Quadratic Nonlinearity

拟议研究的主要目标是开发一种新的数学方法，以允许更快地对某些类型的偏微分方程组进行数值积分。这一建议中考虑的偏微分方程组在许多物理领域中非常常见，特别是在数值天气和气候研究中发挥着至关重要的作用。提出的方法背后的主要思想是，在许多应用中，主要感兴趣的量是大范围的平均量(例如，未来五到十年的平均温度变化，夏季的平均风速，平均海面温度)。因此，在这样的应用中，不需要精确地解决所有小尺度物理问题(例如，任何特定位置的局部风速)。另一方面，数值积分的时间步长往往受到这些小规模过程的限制(由于技术原因)。拟议的研究试图系统地修改基本的偏微分方程式，以减少小规模过程的整体影响，从而允许在数值模拟中有更大的时间步长。

The main goal of the proposed research is to develop a novel mathematical approach to allow faster numerical integration of certain types of Partial Differential Equations. The types of Partial Differential Equations considered in this proposal are very common in many areas of physics and, in particular, play a crucial role in numerical weather and climate studies. The main idea behind the proposed approach is that in many applications the main quantities of interest are large-scale averaged quantities (e.g., mean temperature changes over the next five to ten years, mean wind velocities during the summer, mean sea-surface temperature). Therefore, in such applications it is not necessary to resolve all small-scale physics (e.g., local wind speed at any particular location) accurately. On the other hand, the time-step of numerical integration is often limited (for technical reasons) by these small-scale processes. The proposed research seeks systematic modification of the underlying partial differential equations to reduce the overall influence of small-scale processes and, thus, to allow for a bigger time-step in numerical simulations.