本项目针对泥沙运移数值模拟过程中经常涉及到的几个重要数值代数问题，综合运用数值线性代数、矩阵分析、和计算流体力学的最新理论和方法进行系统深入的研究。主要研究内容是有限体积方法中压力修正类算法的收敛性、稳定性分析；求解泥沙运移数学模型离散后得到的大型稀疏线性代数方程组（一类鞍点问题）的高效迭代算法，和基于SIMPLE类算法的预条件子构造；TDMA算法的理论分析、改进和推广，以及预条件TDMA算法的构造及其应用。目的是通过严格而深入的理论分析，为工程界广泛使用的这些简洁高效的算法提供较充分完备的理论依据，并得到适用于河流泥沙运移数值模拟的高效数值计算方法，从而可以更好地应用于此类工程问题的数值模拟实践。取得的研究成果也可推广移植到相关研究领域，具有一定的理论意义和工程应用价值。

In this study, several important numerical algebra problems, which are motivated from the numerical simulations of the river sediment transportation, are chosen to be studied by using the latest theoretical achievements and methods of numerical linear algebra, matrix analysis, and Computational Fluid Dynamics. The main research contents are: the convergence and stability analysis of the pressure correction algorithms in the Finite Volume Method; the construction of some effective iterative algorithms and the preconditioners based on the SIMPLE-type algorithm for the large linear algebraic systems (a kind of saddle point problems) derived from the river sediment transportation; theoretical analysis and generalization of TDMA and the construction and applications of preconditioned TDMA. The aim of the project is to provide some strict and complete theoretical bases for these effective algorithms widely used in engineering, and develop some new effective numerical methods to simulate the river sediment transportation. It might also be used for some similar numerical simulation in engineering area. The project has clear engineering background and some theoretical significance.