Nonlinear finite volume discretization for transient diffusion problems on general meshes

Positive Scharfetter-Gummel finite volume method for convection-diffusion equations on polygonal meshes

“…Their extensions to complex flows in porous media have been investigated in [7,19]. Contrary to upwind baseddiscretizations, central positive nonlinear schemes have been proposed in [9,25,27]. These strategies make use of some singular potential functionals to reinforce the solution positivity in the case of nonlinear diffusion or linear diffusion with a drift.…”

Weighted positive nonlinear finite volume method for dominated anisotropic diffusive equations

“…The convergence proof of the proposed numerical scheme is an adaptation of the ones elaborated in [10,25,27]. The following Lemma is essential, it allows to apply the compactness criterion in time [4].…”

“…An alternative formulation was conceived in [25] to limit the impact of the artificial diffusion, but the accuracy remains of first order. Alternatively, accurate schemes based on logarithmic formulations of the potential function were suggested in [9,27]. The advantage is to tackle quite general meshes and fully tensors.…”

Node-Diamond approximation of heterogeneous and anisotropic diffusion systems on arbitrary two-dimensional grids

“…This can lead to possible negative eigenvalues of the stiffness matrix entailing spurious oscillation on the solution and loss of convergence. Among schemes enabling consistency and coercivity, we first cite the DDFV (Discrete Duality Finite Volume) method [6,14,25,26,29]. The construction of approximate fluxes, and also the gradient on diamond subsets, involve cell and vertex unknowns which may increase the computational cost.…”