Analysis and construction of cell-centered finite volume scheme for diffusion equations on distorted meshes

A finite volume scheme solving diffusion equation on non-rectangular meshes is introduced by Li [Deyuan Li, Hongshou Shui, Minjun Tang, On the finite difference scheme of two-dimensional parabolic equation in a non-rectangular mesh, J. Numer. Meth. Comput. Appl. 4 (1980) 217 (in Chinese), D.Y. Li, G.N. Chen, An Introduction to the Difference Methods for Parabolic Equation, Science Press, Beijing, 1995 (in Chinese)], which is the so-called nine-point scheme on arbitrary quadrangles. The vertex unknowns can be represented as some weighted combination of the cell-centered unknowns, but it is difficult to choose the suitable combination coefficients for the multimaterial computation on highly distorted meshes. We present a nine-point scheme for discretizing diffusion operators on distorted quadrilateral meshes, and derive a new expression for vertex unknowns. The stability and convergence of the resulting scheme are proved. We give numerical results for various test cases which exhibit the good behavior of our scheme.