A decoupled and positivity-preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes

Abstract

We suggest a new positivity-preserving discrete duality finite volume (DDFV) scheme for anisotropic diffusion problems on general polygonal meshes. Like existing DDFV schemes, this scheme is built on the primary and dual meshes and has finite volume (FV) equations for both vertex and cell-centered unknowns. The transpose of the coefficient matrix for the FV equations of the cell-centered unknowns is an M-matrix while that for the vertex unknowns is not an M-matrix but a symmetric and positive definite matrix. By employing a certain truncation technique for the vertex unknowns, the positivity-preserving property for both categories of unknowns is guaranteed. Local conservation is strictly maintained for the cell-centered unknowns and conditionally maintained for the vertex unknowns. Since the FV equations of the vertex unknowns can be solved independently, the two sets of FV equations are decoupled. In contrast to existing nonlinear 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. In numerical experiments, the new scheme is examined extensively and compared with two positivity-preserving cell-centered schemes and a nonlinear DDFV scheme. Numerical results show that the present DDFV scheme achieves second-order accuracy and preserves the positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids. The high efficiency of the scheme is also demonstrated by the comparison of computation time and number of nonlinear iterations.