scholarly journals Finite Volume approximation of a two-phase two fluxes degenerate Cahn-Hilliard model

2021 ◽  
Author(s):  
Clément Cancès ◽  
Flore Nabet
Keyword(s):  
Finite Volume ◽  
Mass Conservation ◽  
Continuous Model ◽  
Finite Volume Scheme ◽  
Two Phase ◽  
Existence Of A Solution ◽  
Entropy Dissipation ◽  
Continuous Problems ◽  
Volume Approximation ◽  
Compactness Arguments

We study a time implicit Finite Volume scheme for degenerate Cahn-Hilliard model proposed in [W. E and P. Palffy-Muhoray. Phys. Rev. E , 55:R3844–R3846, 1997] and studied mathematically by the authors in [C. Canc\`es, D. Matthes, and F. Nabet. Arch. Ration. Mech. Anal. , 233(2):837-866, 2019]. The scheme is shown to preserve the key properties of the continuous model, namely mass conservation, positivity of the concentrations, the decay of the energy and the control of the entropy dissipation rate. This allows to establish the existence of a solution to the nonlinear algebraic system corresponding to the scheme. Further, we show thanks to compactness arguments that the approximate solution converges towards a weak solution of the continuous problems as the discretization parameters tend to 0. Numerical results illustrate the behavior of the numerical model.

scholarly journals A Fully Implicit Finite Volume Scheme for a Seawater Intrusion Problem in Coastal Aquifers

Water ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1639
Author(s):  
Abdelkrim Aharmouch ◽  
Brahim Amaziane ◽  
Mustapha El Ossmani ◽  
Khadija Talali
Keyword(s):  
Finite Volume ◽  
Coastal Aquifers ◽  
Nonlinear Partial Differential Equations ◽  
Time Integration ◽  
Mass Conservation ◽  
Coupled System ◽  
Finite Volume Scheme ◽  
Time Step ◽  
Volume Scheme ◽  
Fully Implicit

We present a numerical framework for efficiently simulating seawater flow in coastal aquifers using a finite volume method. The mathematical model consists of coupled and nonlinear partial differential equations. Difficulties arise from the nonlinear structure of the system and the complexity of natural fields, which results in complex aquifer geometries and heterogeneity in the hydraulic parameters. When numerically solving such a model, due to the mentioned feature, attempts to explicitly perform the time integration result in an excessively restricted stability condition on time step. An implicit method, which calculates the flow dynamics at each time step, is needed to overcome the stability problem of the time integration and mass conservation. A fully implicit finite volume scheme is developed to discretize the coupled system that allows the use of much longer time steps than explicit schemes. We have developed and implemented this scheme in a new module in the context of the open source platform DuMu X . The accuracy and effectiveness of this new module are demonstrated through numerical investigation for simulating the displacement of the sharp interface between saltwater and freshwater in groundwater flow. Lastly, numerical results of a realistic test case are presented to prove the efficiency and the performance of the method.

scholarly journals Numerical Analysis of a Nonlinear Free-Energy Diminishing Discrete Duality Finite Volume Scheme for Convection Diffusion Equations

2018 ◽  
Vol 18 (3) ◽  
pp. 407-432 ◽  
Author(s):  
Clément Cancès ◽  
Claire Chainais-Hillairet ◽  
Stella Krell
Keyword(s):  
Finite Volume ◽  
Diffusion Equations ◽  
Finite Volume Scheme ◽  
Time Behavior ◽  
Convection Diffusion ◽  
Volume Scheme ◽  
Existence Of Positive Solutions ◽  
Long Time ◽  
Nonlinear Form ◽  
Compactness Arguments

AbstractWe propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy/energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. We establish the existence of positive solutions to the scheme. Based on compactness arguments, the convergence of the approximate solution towards a weak solution is established. Finally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters tend to 0 and when time goes to infinity.

scholarly journals Convergence of a finite-volume scheme for a degenerate-singular cross-diffusion system for biofilms

2020 ◽  
Author(s):  
Esther S Daus ◽  
Ansgar Jüngel ◽  
Antoine Zurek
Keyword(s):  
Finite Volume ◽  
System Modeling ◽  
Gradient Flow ◽  
Continuous Model ◽  
Diffusion System ◽  
Finite Volume Scheme ◽  
Biofilm Growth ◽  
Volume Scheme ◽  
Cross Diffusion ◽  
Flux Approximation

Abstract An implicit Euler finite-volume scheme for a cross-diffusion system modeling biofilm growth is analyzed by exploiting its formal gradient-flow structure. The numerical scheme is based on a two-point flux approximation that preserves the entropy structure of the continuous model. Assuming equal diffusivities the existence of non-negative and bounded solutions to the scheme and its convergence are proved. Finally, we supplement the study by numerical experiments in one and two space dimensions.

Sign in / Sign up

Export Citation Format

Share Document