刚性微分方程高阶隐式离散解的快速迭代算法

Advanced time discretization schemes for stiff systems of ordinary differential equations (ODEs), such as implicit Runge-Kutta and boundary value methods are praised for high order of accuracy and good stability. However, they give rise to linear algebraic systems which may be large and are difficult to solve, especially in the case when these methods are applied to time-dependent partial differential equations (PDEs). That is the main reason that these methods are rarely used for large, stiff systems of ODEs and time-dependent PDEs, despite the many appealing properties of such schemes. Therefore, the construction of efficient solution algorithms for the resulting linear systems is essential. The aim of this project is to study effective iterative methods for the linear systems from implicit Runge-Kutta and boundary value method discretizations of stiff systems. Since the coefficient matrices of the resulting systems of linear equations have the structure of a summation of two Kronecker products, we are interested in the design of a preconditioning strategy based on a Kronecker product approximate or equivalently an alternating direction splitting iteration, which is similar in spirit to the classical alternating direction implicit (ADI) method. We will study the convergence of the splitting iteration, the spectral properties of the preconditioned matrix and the guideline of choosing the optimal parameters for both ODEs and differential-algebraic systems. The potential of our approach will be illustrated by numerical experiments with a comparision made against some other strategies. The study of this project will provide a new way for the efficient implementation of implicit Runge-Kutta methods and boundary value methods, and promote more discussion on the efficient implementation of advanced time discretization schemes for stiff systems.

高阶隐式时间离散格式如隐式Runge-Kutta方法和边值方法因其具有高阶精度和优良的数值稳定性而非常适合刚性微分方程的求解。然而，当用其求解大规模刚性系统或空间半离散的偏微分方程时，求解其离散所得的大型线性代数系统的计算量非常大。正因如此，这些方法在偏微分方程数值计算领域应用很少。因而研究快速求解由这些方法所产生的代数系统是非常有意义的工作，本项目拟对这类线性代数系统的快速迭代算法进行探讨。 我们的想法是构造一类基于Kronecker积的交替分裂迭代格式，它的思想源自经典的交替方向隐式迭代方法(ADI)。我们将从此交替分裂迭代格式分别作为稳态迭代方法和作为Krylov子空间迭代方法的预处理子两个方面研究此方法的相关性质，及探讨此迭代法中出现参数的最优选取问题，并探索其在刚性微分方程和微分代数系统中的应用。 项目的研究将推进刚性系统高阶隐式方法实现途径的深入讨论，并提供新的思路和理论依据。

本项目以大型刚性微分方程高精度数值模拟为研究背景和需求牵引，开展预处理Krylov子空间方法等线性迭代方法研究。研究均以提高高阶隐式积分格式的计算效率为目的。针对高阶隐式龙格-库塔方法和边值方法的结构特点，设计了基于Kronecker积的分裂迭代方法并获得了相应的Kronecker积型预处理子。结合心电学Bidomain模型及刻画微尺度热传导的粘性波方程等实际应用问题的结构特点，在Kronecker积型分裂迭代方法的基础上，设计实现了分块Kronecker积型分裂迭代计算策略，获得了分块Kronecker积预处理子，有效加快了数值模拟的计算速度。所取得的研究成果丰富和发展了现有隐式龙格-库塔格式和边值方法的快速实现算法和理论，具有重要的理论和实际工程意义，对提高现有大型刚性问题的数值计算与数值仿真水平起到了重要作用。本项目组已在《Numerical Linear Algebra with Applications》、《BIT Numerical Mathematics 》、《Numerical Algorithm》及《Applied Mathematical Modelling》等国际SCI检索刊物上发表学术论文9篇，我们完成了项目的预定研究任务，并在部分研究内容上做了适当的延伸和扩展。