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

Abstract

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 [Danilov and Vassilevski, Russ. Numer. Anal. Math. Modelling 24: 207–227, 2009]. Approximation of the advective flux is based on the second-order upwind method with a specially designed minimal nonlinear correction [Lipnikov, Svyatskiy, and Vassilevski, J. Comp. Phys. 229: 4017–4032, 2010]. The second-order convergence rate and monotonicity are verified with numerical experiments.