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 [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.The discrete maximum principle (DMP) and local mass conservation are important properties of a numerical scheme for the approximate solution of the steady state advection-diffusion equation. An accurate discretization method satisfying DMP is hard to develop. We address the monotonicity condition as the simplified version of the DMP, which guarantees only solution non-negativity. A number of physical quantities (concentration, temperature, etc.) are non-negative by their nature and their approximations should be non-negative as well. We present a nonlinear finite volume (FV) method on conformal polyhedral meshes that satisfies the monotonicity condition for a wide range of problem coefficients. We admit a jumping diffusion coefficient represented by full anisotropic tensors, a jumping advection coefficient, which may be produced by the Darcy equation in multimaterial media, and both diffusion-dominated and advection-dominated regimes. The presented method is the extension of numerical schemes [9, 26] developed for the 3D diffusion equation [9] and the 2D advection-diffusion equation with continuous coefficients [26].The major difficulty encountered in the design of a monotone numerical scheme is suppressing unwanted spurious (non-physical) oscillations in the numerical solution. These oscillations may appear in advection-dominated problems due to internal shocks and boundary layers, and in diffusion-dominated problems in highly anisotropic media due to inappropriate approximations of the diffusive flux.In the finite element (FE) context, efficient damping of spurious oscillations in advection-dominated regimes has been developed within the streamline upwind

A vertex‐centered and positivity‐preserving scheme for anisotropic diffusion equations on general polyhedral meshes

“…By (21) which verifies (40). (41) is a natural consequence of (42) and the definition of B ϵ K;e;σ in (22).…”

“…Following the idea of Potier, 3 Kapyrin 19 proposed a family of positivity-preserving schemes on unstructured tetrahedral meshes. Under the framework of Lipnikov et al, 9 positivity-preserving FV schemes for diffusion equations were developed in Danilov and Vassilevski 20,21 on conformal polyhedral meshes with planar faces, and this method was further extended to advection-diffusion equations 22 and multiphase flow model 23 on unstructured polyhedral meshes. In addition, piecewise linear transformation was proposed in Vidović et al 18 to obtain a complicated 3D positivity-preserving interpolation method, and it was extended in another study 24 to discretize the diffusive flux under the assumption that the structure of the domain is locally layered.…”

Two-phase water flooding simulations on dynamic adaptive octree grids with two-point nonlinear fluxes

“…In the general case, the linear scheme may not provide approximation at all. The detailed description of the nonlinear TPFA in the 3D case can be found in [7,18]. Here we sketch the method presentation for interior faces and diffusive fluxes.…”

“…Following [7,10,[12][13][14]18,23,26,28] we use the nonlinear FV method based on nonlinear two-point flux approximation (nonlinear TPFA) pioneered by Le Potier [23]. Its compact discretization is secondorder accurate even on K-non-orthogonal grids and preserves the positivity of the differential solution.…”

Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions

“…To guarantee solution monotonicity for arbitrary meshes, a number of nonlinear methods have been proposed in both FE [12,31] and finite volume [4,16,21,24,25,27,28,29,33,38,39] frameworks. We present here a procedure for postprocessing non-monotone FE solution which produces a corrected solution satisfying both monotonicity and DMP requirements, and also preserving the order of accuracy.…”