The Study Of Vertex-centered And Positivity-preserving Finite Volume Schemes For Diffusion Problems

Vertex-centered and positivity-preserving finite volume schemes of anisotropic diffusion problems on polygonal/polyhedral meshes is mainly studied in this paper.Firstly,a vertex-centered nonlinear positivity-preserving scheme for diffusion problms on arbitrary polygons is constructed.The vertex-centered unknowns are primary while the edge-midpoint and cell-centered unknowns are treated as auxiliary ones.It is the first pure vertex-centered nonlinear positivity-prcsci*ving scheme in the world.Unlike most existing pcositivity-preserving schemes,the construction of the scheme is based con a special nonlinear two-point flux approximation that has a fixed stencil and does not require the convex decomposition of the co-normal and hence avoids the tedious search process.The auxiliary unknowns arc intcipolated by very simple algo?rithms.Moreover,the present scheme does not suffer from the so-called numerical heat-barrier issue without any special treatment.The positivity-preserving property and the truncation er-ror are analyzed rigorously.To improve the computational efficiency,Picard method and its Anderson acceleration are discussed.Numerical experiments demonstrate the second-order ac-curacy and well positivity of the solution for anisotropic or discontinuous problems on severely distorted grids.The high efficiency of the Anderson acceleration is also verified.Next we extend the vertex-centered scheme to three-dimensional diffusion problems and present the nonlinear vertex-centered and positivity-preserving scheme on general polyhedrons.The three-dimensional vertex-centered scheme inherits the advantages of the two-dimensional ones and can be applied to the meshes with non-planar faces.Since the one-sided flux discretiza-tion on the dual face is done on a fixed tetrahedral subcell of the primary cell,the algorithm is very simple to implement on the polyhedral meshes.Then,the vertex-centered positivity-preserving scheme is applied to simulate the two-dimensional two/three-temperature radiation diffusion equations.Numerical results show that the scheme can commendably simulate radiation diffusion system with strong discontinuity,strong nonlinearity and tightly coupling.Finally,a decoupled and positivity-preserving discrete duality finite volume scheme for diffusion problems on general polygons is constructed.As with the existing discrete duality finite volume schemes,two sets of finite volume equations on the primary and dual meshes are built,respectively.The transpose of the coefficient matrix for the equations of the cell-centered unknowns is an M-matrix while that for the vertex unknowns is symmetric and positive definite.By employing a certain truncation technique for the vertex unknowns,the positivity-preserving property for both cateaories of unknowns is guaranteed.Local conservation is strictly nain-tained for the cell-centered unknowns and conditionally maintained for the vertex unknown-s.In contrast to existing rionlincar positivity-preserving schemes,the new scheme requires no nonlinear iterations for linear problems.For nonlinear problems,the positivity-preserving mechanism of the new scheme is decoupled from its nonlinear iteration so that any nonlinear solver can be adopted.Moreover the positivity of the discrete solution is proved and the well-posedness is analyzed rigorously for linear problems.For comparison,a nonlinear cell-centered positivity-preserning scheme based on interpolation truncation is proposed.Numerical exper-iments show that the decoupled scheme has high computational efficiency and can effectively avoid the numerical heat-barrier issue while maintaining the positivity-preservins property.