A Cell-Centered Nonlinear Finite Volume Scheme Preserving Fully Positivity for Diffusion Equation

In the construction of existing nonlinear cell-centered finite volume schemes with monotonicity, it is required to assume that values of auxiliary unknowns are nonnegative. However, this assumption is not always satisfied, especially when accurate reconstruction of auxiliary unknowns is concerned on distorted meshes. In this paper we propose a new method to deal with this issue by introducing both edge unknowns and vertex unknowns as auxiliary unknowns. Edge unknowns are approximated by a convex combination of cell-centered unknowns and vertex unknowns by using the continuity of flux on cell edge. Vertex unknowns are approximated by a convex combination of cell-centered unknowns and edge unknowns. Our new method can assure that these weighted coefficients are nonnegative and the sum of these coefficients in each convex combination is one. The resulting scheme is a nonlinear monotone scheme with nonlinear coefficients depending on both edge unknowns and vertex unknowns, and a linear cell-centered finite volume scheme is formed at each nonlinear iteration by using the Picard linearized method. Numerical results show that our monotone scheme based on the new method of eliminating auxiliary unknowns is more accurate and robust than some existing monotone schemes.