A Finite Volume Scheme for Diffusion Problems on General Meshes Applying Monotony Constraints

Abstract

International audienceIn order to increase the accuracy and the stability of a scheme dedicated to the approximation of diffusion operators on any type of grids, we propose a method which locally reduces the curvature of the discrete solution where the loss of monotony is observed. The discrete solution is shown to fulfill a variational formulation, thanks to the use of Lagrange multipliers. We can then show its convergence to the solution of the continuous problem, and an error estimate is derived. A numerical method, based on Uzawa's algorithm, is shown to provide accurate and stable approximate solutions to various problems. Numerical results show the increase of precision due to the application of the method

Finite volume approximation of a diffusion–dissolution model and application to nuclear waste storage

“…At all times, the saturation presents a discontinuity in x = 0.5, due to the fact that, since both phases flow at this location, the gas and the liquid pressures must remain continuous. Therefore, the saturation must respect the equation P (1) c (S (1) l (0.5, t)) = P (2) c (S (2) l (0.5, t)), where upper indices (1) and (2) denote the values and functions respectively available in regions M1 and M2, at all times t > 0, which leads to different left and right limits of the saturation in x = 0.5. At large times, the liquid pressure becomes constant, which is expected since the water phase is mobile in the whole domain, and therefore the liquid saturation, resulting from the capillary curves and the difference between the gas and the liquid pressures, becomes constant in M1 and M2.…”

“…In case of highly distorted meshes, this parameter can be adjusted in order to prevent from oscillations [2]. We assume that the permeability k takes the constant value k K in K.…”

“…l (0) P (1) c (S (1) l (∞)) = P (2) c (S (2) l (∞)), P l (∞) = P g (∞) − P (1) c (S (1) l (∞)) = P g (∞) − P (2) c (S (2) l (∞)).…”

The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes

“…In [21], based on repair technique and constrained optimization, two approaches have been suggested to enforce discrete extremum principle for linear finite element solutions of general elliptic equations with self-adjoint operator on triangular meshes. A finite volume scheme for diffusion problems on general meshes applying monotony constraints is presented and analyzed in [3].…”

A monotone nonlinear finite volume method for approximating diffusion operators on general meshes

“…However, it is known that none of these methods does satisfy the discrete maximum principle except in the particular case of isotropic diffusion equations discretized on some particular meshes like rectangle meshes or, more generally, Delaunay–Voronoi meshes and admissible meshes . When the approximated solution must be positive (for example, temperature, energy, concentration, and so on) this drawback can become embarrassing and can require repair techniques or monotony constraints , in order to enforce the approximated solution to be positive.…”