Positivity‐preserving scheme without separated method for 2D convection–diffusion equations on polygonal meshes

Abstract

AbstractWe present a finite volume scheme for 2D steady convection–diffusion equations on arbitrary convex polygonal meshes. The method uses auxiliary unknowns on the cell‐edge and cell‐node of the mesh, and the continuous flux is computed to be a two‐point nonlinear flux. Since the transpose of the coefficient matrix is an M‐matrix, it is guaranteed that the scheme is positive. We propose a new strategy, which aims that providing a scheme without separated method to compute the value on the cell‐edge without previous reconstruct. Numerical results show that our scheme is positive on polygonal meshes for both diffusion‐dominated and convection‐dominated problems and has second‐order accuracy for solutions.