Volterra泛函微分方程多步Runge-Kutta方法的数值分析及应用

A series of results of multi-step Runge-Kutta methods for nonlinear stiff Volterra functional differential equations (VFDEs), which are numerical contractivity, asymptotic stability, numerical dissipativity and B-convergence on the infinite interval, will be established. These results can be regarded as improvements and extension of the existing general theory of VFDEs and their numerical methods (mainly for Runge-Kutta method) established in recent years, including the B-stability, B-convergence, contractivity and asymptotic stability theory. These works will make the numerical theory of functional differential equations more systematic and perfect. It is as a guide to provide a theoretical basis and effective numerical methods for several special types of stiff functional differential equation, such as ordinary differential equation, delay differential equation，delay integro-differential equations and so on. Moreover, it provides new methods and technologies in the large-scale scientific computing of modern science technology and practical engineering problems. In addition, it will construct high order accuracy numerical schemes based on mutil-step Runge-Kutta method for fractional differential equation, enriching and developing numerical theories for fractional differential equation. Due to various functional differential equations are often confronted in modern science technology and practical engineering problems, this project not only has significance in science but also owns great potential applications.

本项目将为非线性刚性Volterra泛函微分方程多步Runge-Kutta方法建立具有更为普遍指导意义的收缩性、渐近稳定性、散逸性及无限区间上的B-收敛理论，这是近年来已建立的刚性泛函微分方程数值方法（主要是Runge-Kutta方法）的一般数值理论，包括B-稳定、B-收敛、收缩性及渐近稳定性理论的进一步发展，从而使得泛函微分方程数值理论更加系统和完善；并以此为指南，为几类经常遇到的特殊类型的刚性泛函微分方程如常微分方程、延迟微分方程、延迟积分微分方程等的数值求解提供统一的理论基础及高效数值方法，为现代科学技术及工程实际问题中大规模科学计算提供新的方法和技术。同时构造基于多步Runge-Kutta方法的分数阶微分方程的高精度数值计算格式，丰富和发展分数阶微分方程的数值理论。由于现代科学技术及工程实际问题中经常遇到上述各类泛函微分方程，本项目研究不仅具有重要科学意义，而且具有广阔应用前景。

本项目研究了几类非线性泛函微分方程的Runge-Kutta方法、单支方法和多步Runge-Kutta方法的稳定性、散逸性和收敛性，这些方程包括中立型延迟微分方程、中立型延迟积分微分方程、中立型泛函积分微分方程等。本项目建立了应用于各类方程的数值方法的稳定性、散逸性和收敛性理论。由于非线性泛函微分方程广泛应用于自动控制、生态学、环境科学、电力系统等许多工程和科学领域且种类繁多，而很多泛函微分方程的理论解难以获得，因此对不同类型的方程的理论和数值方法的研究尤为重要. 该项目所获得的成果，是已有的泛函微分方程数值理论的进一步发展，从而使得泛函微分方程数值理论更加系统和完善；并以此为指南，为几类经常遇到的特殊类型的刚性泛函微分方程如常微分方程、延迟微分方程、延迟积分微分方程等的数值求解提供统一的理论基础及高效数值方法，为现代科学技术及工程实际问题中大规模科学计算提供了新的方法和技术。所以该项目成果不仅具有较大的理论意义，也有较大的实际应用价值。