New structures in geometric numerical integration

物理学、天文学、化学和地球科学中的许多科学现象都是用微分方程来模拟的。一般来说，微分方程没有封闭形式的解，必须依靠数值积分。传统上，这是使用例如龙格-库塔法或线性多步法，分别是有限差分法或有限元法来完成的。然而，最近，新的所谓的“几何”集成方法，保留定性功能的许多DE完全（而不是传统的方法）已被发现，导致至关重要的稳定性的改善。结合动力系统理论和传统的数值DE方面，这个项目将改进，扩展和系统化几何积分这一新领域。

Many scientific phenomena in physics, astronomy, chemistry, and geoscience, are modelled by differential equations (DEs). Generally DEs have no closed form solutions, and one must rely on numerical integration. Traditionally this is done using, for example, Runge-Kutta methods or linear multistep methods, respectively finite difference or finite element methods. Recently, however, novel so-called ‘geometric’ integration methods that preserve qualitative features of many DEs exactly (as opposed to traditional methods) have been discovered, leading to crucial stability improvements. Combining aspects of dynamical systems theory and traditional numerical DEs, this project will improve, extend, and systematise this new field of geometric integration.