A Nonlinear Positive Preserving Finite Volume Scheme For Three-dimensional Convection-diffusion Equations

In this paper,a positive finite volume scheme for three-dimensional convection-diffusion equations on tetrahedral meshes is presented.Firstly,a cell-centered nonlinear positive finite volume scheme for convection-diffusion equations,which contains only cell-centered unknowns,is construct-ed.Local conservation is used to construct the diffusive flux and obtained a nonlinear two-point flux method.A corrected upwind scheme is applied to construct the convective flux,and all information used comes from the upstream side of the cell-face.Secondly,in order to obtain a cell-centered scheme,a constraint treatment is used to eliminate the cell-vertex unknowns,and the cell-vertex unknowns obtained by this method is positive.Finally,for scalar and tensor diffusion coefficients,diffusion-dominated and convection-dominated problems,numerical examples on different meshes(including dis-torted and TetGen dissecting tetrahedral meshes)are given respectively.The numerical results shows that our scheme appears to be approximate second-order accuracy for the solution and higher than first-order accuracy for the flux,and it's robust.The difficulties of this paper is that the slope restriction in the upwind scheme of the convective flux is quite complex,and a large computation is produced.The challenge of this paper is not dependent on the slope restriction in the upwind scheme.A new high-precision method is obtained based on the geometric structure of tetrahedron,Taylor expansion and direction gradient transformation.Comparison with the existed schemes,our method solves the convection terms more conveniently,improves the computation more efficiency.