A Nonlinear Monotone Finite Volume Scheme For Three-dimensional Diffusion Equations On Tetrahedral Meshes

In this thesis,we present,a new nonlinear monotone finite volume scheme for three-dimensional diffusion equations.Firstly,a nonlinear finite volume scheme for diffusion equations,which contains only cell-cent,ered unknowns,is given on the three-dimensional tetra-hedral meshes.It is a monotone scheme.In the construction process,cell-face unknowns and cell-vert.ex unknowns are introduced as auxiliary unknowns.In order to obt,ain a cell-centered finite volume scheme,the auxiliary unknowns need to be eliminated.In this thesis,a constraint treatment is used to e-liminate the cell-vertex unknowns.Meanwhile,according to the continuity of normal flux,the combination of cell-centered unknowns is used to eliminate the cell-face unknowns.The method ensures the positivity of the scheme.Fi-nally,in view of continuous diffusion coefficient and discontinuous diffusion coefficient,respectively,some numerical examples are given on distported tetra-hedral meshes.The numerical results indicate that our scheme almost obtains second-order accuracy for the solution and higher than first-order accuracy for the flux,and it's robust.For three-dimensional diffusion problems,the geometric structure is more complicated than the two-dimensional case.So the new difficulty of this thesis lies in the design of the scheme and the computational efficiency.The innova-tion lies in the fact,that while constructing the nonlinear two-point flux,this paper eliminates the non-negative assumption of auxiliary unknowns by the introduction of a parameter.It also preserves monotone.