The Methods To Construct The Discrete Conserved Flux In The Finite Volume Scheme Preserving Extremum Principle For Three-Dimensional Diffusion Equations

In this paper,the cell-centered finite volume,which satisfies the discrete extremum principle scheme for the three-dimensional diffusion equation on tetrahedral meshes is studied.And the method of obtaining discrete conserved flux is focused on.A discrete flux expression containing only the cell-center unknown can be got when constructing the scheme.In order to satisfy the discrete extremum principle,a difference term which can be obtained by rewriting the discrete flux is needed in the discrete flux.Conserved discrete flux is obtained by the sum of difference term and weighted combination of nonconservative discrete flux.In this paper,the general form of the nonlinear coefficient is given.Three groups of coefficients are selected for preliminary analysis,and the flux errors obtained by using different coefficients are compared.Subsequent numerical examples verify the accuracy and performance of the scheme.For the smoothing coefficient problem,numerical examples on orthogonal and distorted grids with scalar and tensor diffusion coefficients are given respectively.For the discontinuous coefficient problem,numerical examples on orthogonal and distorted grids with scalar diffusion coefficient are given.The results show that the scheme guarantees the approximate second-order accuracy of the numerical solution and about 1.5-order accuracy of the flux,and the flux error obtained by using the coefficients 3 is the smallest.This is consistent with the analysis conclusion and meets the discrete extremum principle.It is robust and effective.The innovation and difficulty of this paper are:The computational com-plexity of three-dimensional diffusion problem is large.How to make the coefficient matrix as sparse as possible and reduce the computational time is a difficult problem.Conserved discrete flux is obtained in a new way.The general scheme of the sum of difference terms and nonlinear weighted noncon-servative discrete fluxes is used to avoid a series of more detailed case-by-case discussions.