线性化离散不可压Navier-Stokes方程的预处理算法研究

Numerical solution of the incompressible Navier-Stokes equations is an active research field. Based on the artificial compressibility method, the high-order upwind compact finite difference schemes and the linearization method can be employed to discretize two-dimensional steady-state incompressible Navier-Stokes equations effectively. After discretization, an approximate solution of the system of linear equations can be computed by the approximate factorization alternating direction implicit (AF-ADI) method, the LU-SGS method, and so on. These approaches are essentially based on approximate factorizations of the discretized incompressible Navier-Stokes equations, they may, however, cause factorization errors, which could affect the overall computational accuracy of the discretized solution and the global convergent rate of the iteration process. In order to overcome the deficiencies of these methods, the project will carry out a discussion about solving the system of linear equations, and further analyze the structure and properties of the coefficient matrix, and construct several effective preconditioners, and estimate accurate eigenvalue bound of the preconditioned matrix, and design a class of low computational complexity, stable and effective precondition algorithm, and extend the precondition algorithm to the unsteady incompressible Navier-Stokes equations.

不可压Navier-Stokes方程的数值求解是一个活跃的研究领域。结合人工压缩性方法的高阶迎风紧致差分格式和线性化方法可以有效地数值离散二维定常不可压Navier-Stokes方程。离散后所得到的线性代数方程组通常利用逼近分解交替方向隐式(AF-ADI)，LU-SGS等方法求解。这类方法是基于离散方程的近似分解，从而会引入分解误差，该误差会影响数值解的计算精度和全局收敛速度。为了克服这类方法的不足，本项目将围绕这类线性代数方程组的求解问题展开，深入分析系数矩阵的结构和性质，构造高效预处理子，在理论上给出预处理矩阵特征值精确界的估计，设计出一类计算量小，稳定性好，收敛率高的预处理算法，并将这类预处理算法推广到非定常不可压Navier-Stokes方程。

基于人工压缩方法，利用隐式有限差分格式和线性化方法离散非定常不可压Navier-Stokes方程，在每一个伪时间层，得到一个有结构的线性代数方程组并需要数值解。在本项目中，我们提出了几类预处理方法。数值结果表明了这几类方法的有效性。进一步地，我们还发现结合的预处理方法和线性化方法可以有效地求解离散化不可压Navier-Stokes方程。项目的研究基本上实现了预期目标。