Compact Finite Difference Methods For Highly Anisotropic Elliptic Equations

In this thesis,we mainly study the solution of the highly anisotropic elliptic model(SP-model)used to describe the autonomous electric field in the annular Tokamak fusion device.When the highly anisotropy parameter is 0,the elliptic equation becomes an ill-posed problem.By using the traditional numerical method to discretize the SP-model directly,an ill-posed linear system will be obtained.The condition number of coefficient matrix can be very big,the result will be used in linear system analytical standard numerical methods have unacceptable numerical error.In order to obtain accurate results,the mesh must be refined to compensate for the influence of the anisotropy parameters.However,for large anisotropy ratios,high numerical costs will be incurred.Therefore,in order to solve the numerical problems of highly anisotropic elliptic equations,in order to develop a high-order numerical scheme that can give the parameters independent of anisotropy,the following work is done in this thesis:Firstly,in order to avoid the numerical discretization of highly anisotropic elliptic equation directly,this thesis introduces auxiliary variables through the asymptotic method based on micro-macro decomposition,and eliminates the rigid term(containing term)about the variables in the model,thus obtaining two asymptotic equations equivalent to the original highly anisotropic elliptic equation.Secondly,in order to obtain a higher order numerical solution without refining the mesh,the highly anisotropic elliptic equation and its equivalent asymptotic equation are solved in a fourth-order compact finite difference framework.Finally,by analyzing the global error and solving a specific numerical example,we prove and verify experimentally that the convergence of the numerical scheme of the highly anisotropic elliptic equation under the fourth-order compact finite difference framework is affected by the value of the anisotropy parameters.When the anisotropy parameter is equal to a large value,the compact finite difference scheme of highly anisotropic model converges in the fourth order.When the anisotropy parameter is equal to a smaller value tending to 0,the solution error is large and there is no convergence.For asymptotic equations in under the framework of the fourth-order compact finite difference numerical format,we demonstrated and verified for arbitrary,the format always maintain fourth-order convergence,namely the asymptotic equations in fourth-order compact finite difference under the framework of the convergence of numerical format is independent of the anisotropy parameters,which shows the stability of the solution proposed in this thesis.