刚性微分方程高阶隐式离散解的快速迭代算法

项目名称： 刚性微分方程高阶隐式离散解的快速迭代算法

项目编号： No.11301575

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 陈浩

作者单位： 重庆师范大学

项目金额： 22万元

中文摘要： 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。

中文关键词： 刚性微分方程；隐式龙格-库塔方法；边值方法；迭代方法；预处理

英文摘要： 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

英文关键词： stiff differential equations；implicit Runge-Kutta methods；boundary value methods；iterative methods；preconditioning