A vertex‐centered and positivity‐preserving scheme for anisotropic diffusion equations on general polyhedral meshes

Abstract

We propose a new nonlinear positivity-preserving finite volume scheme for anisotropic diffusion problems on general polyhedral meshes with possibly nonplanar faces. The scheme is a vertex-centered one where the edgecentered, face-centered, and cell-centered unknowns are treated as auxiliary ones that can be computed by simple second-order and positivity-preserving interpolation algorithms. Different from most existing positivity-preserving schemes, the presented scheme is based on a special nonlinear two-point flux approximation that has a fixed stencil and does not require the convex decomposition of the co-normal. More interesting is that the flux discretization is actually performed on a fixed tetrahedral subcell of the primary cell, which makes the scheme very easy to be implemented on polyhedral meshes with star-shaped cells. Moreover, it is suitable for polyhedral meshes with nonplanar faces, and it does not suffer the so-called numerical heat-barrier issue. The truncation error is analyzed rigorously, while the Picard method and its Anderson acceleration are used for the solution of the resulting nonlinear system.Numerical experiments are also provided to demonstrate the second-order accuracy and well positivity of the numerical solution for heterogeneous and anisotropic diffusion problems on severely distorted grids. KEYWORDS diffusion equation, nonlinear two-point flux approximation, positivity-preserving, vertex-centered scheme 1 | INTRODUCTIONDiffusion equations are required to be solved efficiently and robustly on arbitrary distorted polygonal or polyhedral meshes in the numerical modelling of many physical problems, such as fluid flows in complex reservoir models, Lagrangian hydrodynamics with heat and radiative diffusion, and Lagrangian magnetohydrodynamics with magnetic diffusion. In some applications such as radiation hydrodynamics, the discrete solutions of diffusion equations should be nonnegative in order to avoid nonphysical quantities. The positivity-preserving or monotone property is one of the key requirements for diffusion schemes. In recent years, the study of positivity-preserving finite volume (FV) discretizations for diffusion problems has drawn great attention and many efforts have been devoted to this area.Since linear diffusion FV schemes do not unconditionally satisfy the positivity-preserving property and simultaneously provide a good approximation on distorted meshes or with high anisotropy ratio, 1,2 many researchers turn to

A novel vertex-centered finite volume method for solving Richards' equation and its adaptation to local mesh refinement

“…Our approach is based on a Vertex-Centered Finite Volume Method (hereinafter referred to as VCFVM) and sets all unknowns on grid vertices. It should be noted that the vertex-centered finite volume method has been widely studied for solving anisotropic diffusion/parabolic equations on general polygonal/polyhedral meshes in the past decade, see [41,40,45,36,5,29]. The numerical scheme proposed in this paper is partially inherited from [29], but substantial modifications in the spatial and temporal discretization have been made to consider the properties of Richards' equation, including the advection flux involved and the high nonlinearity between the soil hydraulic parameters and the soil water potential.…”