A monotone nonlinear finite volume method for advection-diffusion equations on unstructured polyhedral meshes in 3D

We present a new monotone finite volume method for the advection-diffusion equation with a full anisotropic discontinuous diffusion tensor and a discontinuous advection field on 3D conformal polyhedral meshes. The proposed method is based on a nonlinear flux approximation both for diffusive and advective fluxes and guarantees solution non-negativity. The approximation of the diffusive flux uses the nonlinear two-point stencil described in [9]. Approximation of the advective flux is based on the second-order upwind method with a specially designed minimal nonlinear correction [26]. The second-order convergence rate and monotonicity are verified with numerical experiments.