Multiscale Computations of Stiff Oscillatory Ordinary Differential Equations

刚性振动常微分方程是许多重要科学领域中常见的数学模型，从经典的天体力学到现代物理学和化学中的原子动力学。随着计算机能力的提高，这些模型变得越来越重要。基于这类系统的大量模拟正在进行，以预测例如金属中裂纹和位错扩展的性质，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。挑战是在物理上相关的时间内模拟这些高度振荡的相互作用，而不使输出质量受到数值误差的影响。这一方案的研究旨在发展一种新的多尺度计算方法，以便能够准确而高效地计算由振荡之间的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.