Monotone Finite Volume Method on General Meshes

Abstract

Cell-centered finite volume (FV) discretizations are appealing for the approximate solution of boundary value problems since they are locally conservative and applicable to general meshes, i.e., to meshes with general polyhedral cells. In this chapter, we introduce nonlinear flux discretizations which result in monotone FV schemes at the cost of scheme nonlinearity, even if it is applied to a linear partial differential equation (PDE) such as diffusion and convection-diffusion equations. Also, we give two examples of linear two-point flux vector discretization of the diffusion equation in the mixed formulation and the Navier-Stokes equations. Such flux vector discretizations are stable in spite of degrees of freedom collocated at cell centers, are applicable to systems of PDEs, and demonstrate monotone numerical solutions.