半线性微分方程的数值理论及其应用

This project is mainly concerned with the numerical theory of semilinear differential equations and its applications. The semilinear differential equations consist of a linear part (usually a stiff part) and a nonlinear part, and these equations mainly come from spatial discretizations of some evolution equations. It has great practical value to analyze the numerical methods for semilinear differential equations. In this project, exponential Runge-Kutta methods with continuous extension will be presented for the semilinear differential equations with integer-order on the basis of exponential integrators, and the convergence and stability will be analyzed. Furthermore, the nonlinear numerical stability of exponential integrators for the semilinear differential equations with integer-order will be defined and the algebra conditions of the nonlinear stability of exponential integrators will be given out for various number fields and spatial dimensions. In addition, a series of Mittag-Leffler (ML) integrators, such as ML-Runge-Kutta methods, ML-Rosenbrock methods, ML-linear multistep methods, ML-general linear methods and multistep ML-collocation methods will be constructed for the semilinear fractional differential equations. Moreover, the corresponding fast computation, convergence and stability of these methods will also be studied. Meanwhile, some numerical examples will be given to demonstrate the above conclusions. This project can not only enrich the numerical theory of differential equations, but also give some effective algorithms for the development of the corresponding applied science.

本项目主要研究半线性微分方程的数值理论及其应用。半线性微分方程由一个线性项（通常为刚性）和一个非线性项组成，主要来源于空间离散化的演化方程，其数值方法的分析具有重要的应用价值。本项目拟在半线性整数阶微分方程指数方法的基础上构造带有连续扩张的指数Runge-Kutta方法并分析其收敛性及稳定性，进一步拟给出此类方程指数方法的非线性数值稳定概念并根据数域及维数的不同分别给出指数方法非线性稳定的代数条件。此外，本项目拟针对半线性分数阶微分方程构造一系列Mittag-Leffler（ML）积分方法，例如ML-Runge-Kutta方法、ML-Rosenbrock方法、ML-线性多步方法、ML-一般线性方法及多步ML-配置方法，进一步分析上述方法的快速计算、收敛性和稳定性。同时，通过数值算例验证所得的结论。本项目的工作不但能够丰富微分方程的数值理论，而且也可以为相关应用科学的发展提供有效的算法支持。

半线性微分方程主要来源于演化方程的空间离散化之后的常微分方程，因此其数值方法的研究具有重要的理论意义和应用价值。本项目主要研究半线性微分方程的数值理论及其应用。具体地，本项目分别研究了半线性微分方程叠加型Runge-Kutta方法、Magnus指数积分方法的格式构造、阶条件和稳定性分析，延迟微分方程指数型数值方法、对称Runge-Kutta方法、谱方法、边值方法的收敛性和稳定性，分数阶微分方程高阶向后差分公式、配置方法、有限元方法的收敛性和数值稳定性，分数阶微分方程简化Tikhonov正则化方法、最优滤波方法的收敛性等问题。共发表SCI论文33篇，培养博士4名，硕士16名。本项目的研究成果不但能够丰富微分方程的数值理论，而且也可以为相关应用科学的发展提供有效的算法支持。