非线性复合刚性偏泛函和分数阶微分代数系统的正则分裂算法理 论及其应用研究

本课题组已在延迟和分数阶微分代数系统的高效数值算法及其理论方面获得了较好成果，并在非线性复合刚性发展方程的正则Euler分裂算法及其理论研究方面获得了重要进展。本申请项目拟将上述两方面成果相结合并研究：.(1)非线性复合刚性偏泛函微分代数系统的并行正则分裂算法及其理论与应用；.(2)非线性复合刚性偏泛函微分代数系统的区间分裂算法及其理论与应用；.(3)非线性复合刚性分数阶偏泛函微分代数系统的并行正则分裂算法及其理论与应用；.(4)非线性复合刚性对流扩散反应偏泛函微分方程和辐射扩散方程的并行正则分裂算法及其理论与应用。.在大规模科学计算中，使用并行正则分裂方法，可在确保计算质量和精度的基础上，大幅度提高计算速度。由此可见，本申请项目的研究不仅可为上述各相关研究领域带来新的重要进展，而且对于促进我国高新技术数值模拟研究的快速发展同样具有重要意义。

The efficient algorithms and their theories for delay and fractional-order differential-algebraic systems have been obtained, and researches on the canonical Euler splitting algorithms and their theories for nonlinear composite stiff evolution equations have made great progress in our group. Based on the research results of the above-mentioned two aspects, this project further studies the following contents:.(1) the construction and theories of parallel canonical splitting algorithms for nonlinear composite stiff partial functional differential-algebraic systems (PFDAEs);.(2)the construction and theories of interval splitting algorithms for nonlinear composite stiff PFDAEs;.(3) the construction and theories of parallel canonical splitting algorithms for nonlinear composite stiff fractional-order PFDAEs;.(4) the construction and theories of parallel canonical splitting algorithms for convection diffusion reaction partial functional differential equations and radiation diffusion equations.. In the large-scale scientific computing, the computing speed can be significantly improved on basis of ensuring the computing quality and accuracy by using canonical parallel splitting algorithms. Therefore, The expected results of this project not only can bring new important progress in the above research fields, but also can promote the rapid development of high-tech numerical simulation research in China.