Construction and Implementation of Efficient Numerical Methods for Ordinary Differential Equations

科学和工程中的许多问题都是用大型微分方程组来模拟的。设计并实现准确、高效、可靠、稳健的数值模拟方法是本课题研究的目的。这些方法应该能够利用这种系统中经常存在的特殊结构来提高计算效率。这类具有特殊结构的系统的例子包括计算流体动力学、分子生物学、量子力学和热传导中的许多问题，它们的数值求解需要比经典方法更强大的新数值技术。本提案将研究的新技术是基于最近的一般线性方法理论和指数积分器，它们处理微分系统的不同特定部分。提出的一些方案适合在并行计算环境中实现。

Many problems in science and engineering are modelled by large systems of differential equations. It is the purpose of the research described in this proposal to design and implement accurate, efficient, reliable, and robust methods for the numerical simulations of such systems. These methods should be capable of exploiting special structure often present in such systems to increase the computational efficiency. Examples of such systems with special structure include many problems in computational fluid dynamics, molecular biology, quantum mechanics, and heat transfer, and their numerical solution requires new more powerful numerical techniques than the classical methods. The novel techniques which will be investigated in this proposal are based on the recent theory of general linear methods and on exponential integrators which treat differently specific parts of the differential systems. Some of the proposed schemes are appropriate for implementation in a parallel computing environment.