TAILIEUCHUNG - Practical Ship Hydrodynamics Episode 3

Tham khảo tài liệu 'practical ship hydrodynamics episode 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Introduction 31 LSOR line successive overrelaxation ADI alternating direction implicit line iterative ILU incomplete lower upper decomposition . SIP strong implicit procedure CG conjugate gradient method The various iterative methods differ in their prerequisite dominant main diagonal symmetry etc. convergence properties and numerical effort per iteration. Strongly implicit schemes such as SIP feature high convergence rates. The convergence is especially high for multigrid acceleration which today is almost a standard choice. Multigrid methods Multigrid methods use several grids of different grid size covering the same computational fluid domain. Iterative solvers determine in each iteration relaxation a better approximation to the exact solution. The difference between the exact solution and the approximation is called residual error . If the residuals are plotted versus the line number of the system of equations a more or less wavy curve appears for each iterative step. A Fourier analysis of this curve then yields high-frequency and low-frequency components. High-frequency components of the residual are quickly reduced in all solvers but the low-frequency components are reduced only slowly. As the frequency is defined relative to the number of unknowns respectively the grid fineness a given residual function is highly frequent on a coarse grid and low frequent on a fine grid. Multigrid methods use this to accelerate overall convergence by the following general procedure 1. Iteration relaxation of the initial system of equations until the residual is a smooth function . only low-frequent components are left. 2. Restriction transforming the residuals to a coarser grid . double the grid space . 3. Solution of the residual equation on the coarse grid. Since this grid contains for three-dimensional flow and grid space halving only 1 8 of the unknowns and the residual is relatively high frequent now only a fraction of the computational time is needed .

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