Multiscale Computations of Stiff Oscillatory Ordinary Differential Equations

刚性振荡常微分方程是从经典天体力学到现代物理化学中的原子动力学等许多重要科学领域中常见的数学模型。 随着计算机能力的提高，这些模型越来越重要。 正在进行基于这种系统的许多模拟，以预测例如金属中裂纹和位错的传播的性质，以及某些蛋白质在什么条件下改变形状并具有可用于药物设计的特定功能。 许多基本现象是通过快速振荡之间的微妙相互作用产生的。挑战是在物理相关的时间内模拟这些高度振荡的相互作用，而不会因数值误差而损坏输出质量。 该方案的研究旨在开发一种新的多尺度计算方法，以便能够长时间准确有效地计算振荡之间相互作用引起的有趣现象。 还将对算法进行相应的数学分析。

Stiff oscillatory ordinary differential equations are common mathematical models in many important scientific fields, ranging from classical celestial mechanics to atomistic dynamics in modern physics and chemistry. These models are gaining ever more importance as the computer capability increases. Numerous simulations based on such systems are being performed in order to predict the nature, for example, of propagation of cracks and dislocations in metals and under what condition certain proteins change shape and take on specific functions that can be used in drug design. Many fundamental phenomena are generated through delicate interactions between rapid oscillations.The challenge is to simulate these highly oscillatory interactions over physically relevant time without having the quality of the output being damaged by numerical errors. The research of this proposal aims at developing a new multi-scale computational approach such that the interesting phenomena resulting from interaction between oscillations can be computed accurately and efficiently over long time. The corresponding mathematical analysis of the algorithms will also be performed.