Research On The Application Of Extrapolation Cascadic Multigrid Methods For Solving Diffusion Equations

Extrapolation cascadic multigrid(EXCMG)method is an efficient algorithm for solving partial differential equations,where the extrapolation interpolation operator and completed extrapolation operator can construct the accurate initial values and numerical solutions,respectively.In this thesis,some generalized EXCMG methods with low computational cost and high accuracy are investigated for several types of discretization linear or nonlinear systems of some diffusion type equations.The main results are as follows.Firstly,a second-order linearly implicit difference scheme is proposed for the three-dimensional nonlinear Fitzhugh-Nagumo equation,the existence and uniqueness of the numerical solution is proved,and the stability and con-vergence of the difference scheme in~∞-norm are also established.This scheme is second-order convergent both in time and space variables,and un-conditionally stable.Based on the obtained discretization linear system,the EXCMG method is considered at each time level,an approximate solution with third-order accuracy can be obtained by extrapolation and quadratic in-terpolation,which can greatly reduce the number of iterations on next finer grid.Some numerical results verify that the EXCMG method combined with linearly discretization scheme can improve the efficiency of solving such non-linear parabolic equations.Secondly,we develop an EXCMG-Newton method to solve nonlinear system arising from fourth-order compact finite difference scheme for the semilinear Poisson equation.With the help of extrapolation and quartic in-terpolation techniques,a quite good initial guess of Newton iteration on the neighboring finer grid can be acquired,and the linear Jacobi system of New-ton method is solved by SSOR-Bi CGStab solver.In addition,the completed extrapolation operator can provide an extrapolated solution with sixth-order accuracy cheaply.Three numerical examples are given to demonstrate that the EXCMG-Newton method can achieve high accuracy and keep less cost.Thirdly,a fourth-order compact difference scheme on a nonuniform grid is derived for the convection-diffusion equation with boundary layer or lo-cal singularity.From the perspective of constructing accurate iterative initial value and high precision extrapolated solution,we propose an extrapolation cascade multigrid(EXCMG-NG)method for non-uniform grids,the extrapo-lation interpolation operator and complete extrapolation operator on the non-uniform grid are designed by using average strategy.Numerical experiments show that the EXCMG-NG method can obtain the initial guess with fifth-order accuracy and the extrapolated solution with sixth-order accuracy,which verify that this method has the advantages of high accuracy and less computational cost.Finally,based on the trisected grids,we present an new cell-centered ex-trapolation cascadic multigrid(CEXCMG)method to solve large linear sys-tems arising from finite volume discretization of the diffusion equation with discontinuous and anisotropic coefficients.The constructed cell-centered ex-trapolation interpolation operator can provide a third-order initial value ap-proximation to the numerical solution of the discretization system on the next finer grid,and the cell-centered completed extrapolation operator is designed to produce a third-order extrapolated solution on the whole fine grid direct-ly.Five numerical examples are tested to confirm that the CEXCMG method can acquire the third-order extrapolated solution,and the efficiency is higher than the commonly used the aggregation-based algebraic multigrid(AGMG)method.