scholarly journals A Convergent Entropy Diminishing Finite Volume Scheme for a Cross-Diffusion System

2020 ◽  
Vol 58 (5) ◽  
pp. 2684-2710
Author(s):  
Clément Cancès ◽  
Benoît Gaudeul
Keyword(s):  
Finite Volume ◽  
Diffusion System ◽  
Finite Volume Scheme ◽  
Volume Scheme ◽  
Cross Diffusion

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.

scholarly journals Finite-volume scheme for a degenerate cross-diffusion model motivated from ion transport

2018 ◽  
Vol 35 (2) ◽  
pp. 545-575 ◽  
Author(s):  
Clément Cancès ◽  
Claire Chainais-Hillairet ◽  
Anita Gerstenmayer ◽  
Ansgar Jüngel
Keyword(s):  
Ion Transport ◽  
Diffusion Model ◽  
Finite Volume ◽  
Finite Volume Scheme ◽  
Volume Scheme ◽  
Cross Diffusion

scholarly journals Convergence of a finite volume scheme for a system of interacting species with cross-diffusion

2020 ◽  
Vol 145 (3) ◽  
pp. 473-511 ◽  
Author(s):  
José A. Carrillo ◽  
Francis Filbet ◽  
Markus Schmidtchen
Keyword(s):  
Finite Volume ◽  
Coupled System ◽  
Convergence Result ◽  
The Self ◽  
Finite Volume Scheme ◽  
Discrete Energy ◽  
Volume Scheme ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Interacting Species

Abstract In this work we present the convergence of a positivity preserving semi-discrete finite volume scheme for a coupled system of two non-local partial differential equations with cross-diffusion. The key to proving the convergence result is to establish positivity in order to obtain a discrete energy estimate to obtain compactness. We numerically observe the convergence to reference solutions with a first order accuracy in space. Moreover we recover segregated stationary states in spite of the regularising effect of the self-diffusion. However, if the self-diffusion or the cross-diffusion is strong enough, mixing occurs while both densities remain continuous.

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.

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