几类多时间尺度微分方程系统的组合方法

针对几类重要的多时间尺度微分方程初值问题系统(含时滞、随机刚性问题，多刚性、多参数、时滞、随机奇异摄动问题及相应的微分代数系统，刚性高振荡问题)及奇异摄动分数阶微分方程等其它系统，重点研究基于系统分离或算子分裂的组合方法及其它方法在定、变步长情形下的线性与非线性稳定性、保真性、定量误差和长时间误差性态，克服刚性、时滞、高振荡、奇异摄动、随机、记忆性等复杂因素部分并存所带来的实质困难，解决求解过程中所遇到的一些重要和困难的理论、算法、应用问题，丰富和发展多时间尺度系统和分数阶微分方程的算法理论。在此基础上，改进已有的求解这些系统的算法，构造新的适用于这些问题类的能有效实现与应用的高效保真组合算法及其它算法；进一步将所获的部分理论结果、部分高效算法推广、应用到（时滞、随机）刚性偏微分初边值问题等相关问题的计算；为科学工程中相关应用领域的研究和发展提供有益的帮助。

In this project, we mainly discuss the linear and nonlinear stability, structure-preserving properties, quantitative error behaviours and long time error behaviours of combined methods and other methods based on the system partitions and the operator splits for the initial value problem systems of differential equations (such as delay or stochastic stiff problems, multiply-stiff，multiple-parameter, delay or stochastic singular perturbation problems and the corresponding differential-algebraic systems, and stiff oscillatory problems) and singular perturbation fractional differential equations in fixed or variable step-sizes. Some essential difficulties from the coexistence of a part of the complex factors including stiffness, delay, high oscillation, sinular perturbation, stochasticity and nonlocality etc. are overcome. Some important and difficult problems in theory, algorithm and application are solved which appear in the process of the numerical solution to enrich and develop the algorithm theories of the system with multiple time scales and fractional differential equations. Based on the above work, for these systems, some known algorithms are improved, and some new efficient truth-preserving combined algorithms and other algorithms are given which can be implemented and applied efficiently. The obtained theoretical results and numerical methods will be extended and applied to the initial-boundary value problems of delay or stochastic partial differential equations and the other problems. These will be beneficial to the researches and developments in the related application fields of sciences and engineering.