A Cascadic Multigrid Method Based On The Linear-Preserving Finite Volume Scheme On Distorted Meshes

In this thesis,we study a cascadic multigrid method based on the linearly preserved finite volume scheme on the distorted quadrilateral grid.The finite volume method is widely used in the calculation of fluid mechanics,and the multigrid method is a fast iterative method for solving algebraic equations.First,consider the finite volume discretization of the general diffusion equation.The discretization process needs to satisfy the linearity preservation constraint,that is,when the accuracy of the diffusion equation is required When the solution is a linear function of the independent variables and the diffusion coefficient is constant,each discrete step is exactly true.In the process of constructing the finite volume scheme,the unknowns of the element center and the unknowns of the element nodes will appear at the same time,and a method of eliminating the unknowns of the nodes that satisfies the linearity preservation constraint is given,so that the finite volume scheme containing only the unknowns of the element center is obtained.The scheme can also be called the nine-point scheme.Then,the stability analysis and convergence analysis of the scheme are carried out,and it is proved that the scheme has stability and is first-order convergent.Then,an interpolation operator that can be used for non-nested grids is constructed,which can obtain the cell node values of fine meshes from the cell center values of the coarse meshes,and use this interpolation operator to design a cascadic multigrid method.The new cascadic multigrid method can not only provide a better initial iterative value,but also degenerate the nine-point scheme into a five-point scheme,thereby improving the efficiency of the solution.Finally,the numerical simulation of the diffusion equation with different diffusion coefficients is carried out.For equations whose diffusion coefficient is constant and the analytical solution is a linear function,the linearity-preserving finite-volume discrete scheme is directly solved,and the numerical results show that the scheme is linearly accurate.For the equations whose diffusion coefficients are discontinuous constant coefficient,tensor coefficient and discontinuous tensor coefficient,the new cascadic multigrid method is used to solve it,and the numerical results obtained are compared with the Biconjugate Stable Gradient Method(BICGSTAB)directly.The numerical results show that,the results of the new cascadic multigrid method and the BICGSTAB method can approximate the second-order accuracy,while the new cascadic multigrid method is far in solution time and iteration times much less than the BICGSTAB method,and when the degree of distortion of the mesh is different,it can maintain nearly second-order accuracy,which confirms the high efficiency and stability of the new cascadic multigrid method.20 pictures,4 tables,63 references...