时滞发展型微分方程的高效隐显方法

Implicit-explicit(IMEX) methods are a class of important time splitting methods, it not only satisfy the stability requirement and improve the computation efficiency, but also obviously reduce the computation cost. It included IMEX linear multistep methods, IMEX Runge-Kutta methods and IMEX general linear methods, it have made rapid development and applied widely. Meanwhile, the research of numerical and exact solutions for delay (partial) differential equations and Volterra (partial) functional differential equations are still a very active branch in the field of differential equations. This project aims to research innovative and efficient IMEX methods for some evolutionary differential equations with delay (including variable delay and distributed delay), and establish their mathematical theories. In particular, we consider three problems: (1)For the stiff delay differential equations arise from spatial discretization of delay parabolic differential equations, based on the existing IMEX methods, we will construct the new effective algorithms, and consider the stability and convergence properties; (2) We will develop the finite element methods with IMEX time-stepping for evolutionary differential equations with delay, reaction-diffusion equations with distributed delay and Volterra partial functional differential equations; (3)We will develop the local discontinuous Galerkin methods with IMEX time-marching for the delayed wave equations. This project also aims to enrich and develop the algorithm theory and efficient algorithm for (partial) functional differential equations, and provide algorithm support for the related applications.

隐显方法是一类重要的时间离散分裂方法，能兼顾稳定性和计算效率的要求，且明显降低计算量。主要包括隐显线性多步方法、隐显Runge-Kutta方法和隐显一般线性方法等，其研究发展迅速，已获广泛应用。同时，时滞（偏）微分方程和Volterra 型（偏）泛函微分方程数值解与精确解的研究仍然是微分方程领域中的一个非常活跃分支。本项目拟研究几类时滞（包括变时滞和分布时滞）发展型微分方程的高效隐显方法及其理论，主要研究内容为：(1)对时滞抛物型微分方程空间离散后得到的刚性时滞常微分方程组，在已有数值算法基础上，构造新型有效的隐显方法，并分析所得格式的稳定性与收敛性；(2)时滞发展型微分方程、带分布时滞的反应扩散方程和Volterra型偏泛函微分方程的有限元隐显方法；(3)时滞波动方程的局部间断有限元隐显方法。本项目旨在丰富和发展（偏）泛函微分方程算法理论及高效算法，为相关应用提供算法支撑。

本课题组通过项目的研究，探讨了提高计算精度与减少计算量、存储量的高效算法。针对非线性刚性初值问题，我们提出了一类新的隐显Rosenbrock-RK方法，给出了此方法的阶条件，并获得了相应的误差分析结果。发展了求解时滞发展型微分方程的隐显多步方法，首先考虑了隐显多步有限元方法求解一类时滞反应对流扩散问题，克服解的时间偏导数具有间断性所导致的困难，得到了方法的收敛性结果；其次研究了隐显单支方法求解一类刚性Volterra时滞积分微分方程初值问题时的稳定性与误差分析；还研究了一维和二维非线性抛物型偏积分微分方程的隐显BDF方法及其误差分析。应用间断有限元方法求解刚性泛函微分方程和中立型时滞微分方程，克服解的时间导数具有间断性所导致的困难，分别获得了方法的稳定性结果和超收敛结果。另外，针对一类带弱奇异核的非线性分数阶积分微分方程和非线性分数阶时滞微分方程初值问题，利用强A-稳定Runge-Kutta方法，构造了高阶离散离散格式，并对数值方法进行了误差分析和稳定性分析。