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由微分方程组(DES)来模拟的。一般情况下，DES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，分别是有限差分或有限元方法来完成的。然而，最近发现了新的所谓的几何积分方法，它们准确地保留了许多DES的定性特征(与传统方法相反)，导致了关键的稳定性改进。该项目将动力系统理论和传统的数值离散事件系统相结合，将改进、扩展和系统化这一新的几何积分领域。

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.