Local Fourier Analysis Of Multigrid Method For Solving Incompressible Flow

Multigrid methods are generally considered as one of the fastest numerical methods which have an optimally computational complexity for solving partial differential equations (PDEs). They are mainly applied to solve algebraic system of equations from the discretized differential equations. Especially, they are regarded as being fastest numerical algorithm for the solution of elliptic differential equations, i.e. the convergence is independent on the mesh size, and the number of arithmetic operations is proportional to the number N of discretized unknowns. It is because of the superiority that multigrid methods are the indispensable numerical algorithm in computational fluid dynamics. The dissertation originates the following two scientific research projects which are sponsored by the National Natural Science Foundation of China (Grant No.51279071), the Doctoral Foundation of Ministry of Education of China (Grant No.2013531413002). The multigrid method for solving incompressible flow is researched by means of local Fourier anlysis. The main Contents are summarized as follows.1. In the first, the multigrid methods and its iterative scheme are introdued. Secondly, the element of local Fourier analysis (LFA) is presented, and the smoothing factor and h-ellipicity of discretizing operators are defined, respectively. With the help of relation between different Fourier components and the definition of invariant subspace, the Fourier representations of intergrid operators are investigated with two different coarsening methods. Meanwhile, the calculation of two-grid asymptotic convergence factor is given. In the forth chaper, the smoothing analysis process of multicolor point relaxation in multigrid method is investigated by local Fourier analysis. As a key starting point of the problems under consideration, the mathematical constitutions among Fourier modes with various frequencies was constructed as a base to expand two-color to multicolor smoothing analyses. Two different invariant subspaces based on the 2h-harmonics for the two-color relaxation with two and four Fourier modes were constructed, and successfully used in smoothing analysis process of the Poisson equation for the two-color point Jacobi relaxation. The two-color smoothing analysis was generalized to the multicolor smoothing analysis problems by multigrid method based on the invariant subspaces constructed. The relaxation of the process is general. Therefore, the result possesses universality.2. Firstly, the partial differential equations of Stokes system are discretized on the staggered and nonstaggered grid, respectively. The two different multigrid relaxations, distributive and collective relaxation, are implemented into the discrete Stokes system. In the staggered discrete system, the distributive relaxation is applied. The result is obtained that the smoothing property is determined by Laplace operator, and the related smoothing factor is computed. Then, in the nonstaggered discrete system, the distributive and collective relaxations are considered in the Fourier harmonic space of two-color relaxation. By means of LFA, the smoothing analyses of two relaxations are investigated, and the analytical expressions on the parameter of artificial pressure term are obtained. The analysis results show that the relaxation is convergence and independent on mesh size, but depends on the parameter in the artificial elliptic pressure term.3. With the help of optimal red-black Jacobi point relaxation in multigrid method, the convergence of two-level grid algorithms are researched in the form of theoretical analysis by LFA. Then, the multigrid relaxation is applied to the first-order up-wind discretization form of the convection diffusion equation, which is developed by LFA. The impact of convection-dominant parameters on the smoothing properties and convergences of muligrid methods are discussed. Meanwhile,the first-order up-wind discretization form of Oseen flow is obtained by applying Godunov-type flux difference splitting approach based on Riemann solvers. The convergence analysis of two kinds of the cycle algorithms, V-cycle and W-cycle, on multigrid method for solving the discretizing equations is given. Furthermore, smoothing analysis of collective symmetrical alternating line Gauss-Seidel relaxation is investigated by means of local Fourier analysis. The numerical results show that the collective symmetrical alternating line Gauss-Seidel relaxation has good smooth properties, and the convergences of W-cycle algorithm is better than V-cycle algorithm in multigrid method solving Oseen flow with different Reynolds numbers.4. The discretized incompressible flow system is given on the staggered grid, i.e. only the space variables are discretized. The distributive relaxation is carried on the discrete system. Then the smoothing properties of the multigrid relaxation is relied on the time-dependent convection diffusion operator, which is dealt with two multigrid methods, space-time and waveform multigrid methods, respectively, whose smoothing analysis are developed by LFA. In the space-multigrid method, space and time variables are discretized simultaneously, where the partial derivative of time is discretized by the first-order backward Euler scheme, and the first-order up-wlnd discretization form is suited to the space variables. In the same time, the coarsened scheme is chosen to the space coarsening only. Thus, the smoothing properties of various space-time multigrid relaxations are analyzed by LFA. In the waveform multigrid method, the time-dependent problem is converted into the constant problem with complex coefficient by Laplace transformation. The smoothing analysis of various relaxations applied to the waveform method is discussed by LFA. The relation between the smoothing properties and both convection-dominant parameters and Reynolds number is investigated, and the optimal smoothing factor and the corresponding choice of relaxation parameters are given.All the above theorem and methods are partially applied to the scientific research projects which are managed by my supervisor. For example, the algorithm of National Natural Science Foundation of China, called full Eulerian parallism multigrid modeling of rotating turbulent flow in hydro turbine, is investigated by means of them.