scholarly journals Study of a Finite Volume Scheme for the Drift-Diffusion System. Asymptotic Behavior in the Quasi-Neutral Limit

2014 ◽  
Vol 52 (4) ◽  
pp. 1666-1691 ◽  
Author(s):  
M. Bessemoulin-Chatard ◽  
C. Chainais-Hillairet ◽  
M.-H. Vignal
Keyword(s):  
Asymptotic Behavior ◽  
Finite Volume ◽  
Diffusion System ◽  
Finite Volume Scheme ◽  
Drift Diffusion ◽  
Volume Scheme

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.

FINITE VOLUME APPROXIMATION FOR DEGENERATE DRIFT-DIFFUSION SYSTEM IN SEVERAL SPACE DIMENSIONS

2004 ◽  
Vol 14 (03) ◽  
pp. 461-481 ◽  
Author(s):  
CLAIRE CHAINAIS-HILLAIRET ◽  
YUE-JUN PENG
Keyword(s):  
Finite Volume ◽  
Parabolic Equations ◽  
Diffusion System ◽  
Degenerate Parabolic Equations ◽  
Finite Volume Scheme ◽  
Drift Diffusion ◽  
Finite Volume Discretization ◽  
Degenerate Parabolic ◽  
Volume Approximation ◽  
Finite Volume Approximation

This paper is devoted to a finite volume discretization for multidimensional nonlinear drift-diffusion system. Such system arises in semiconductors modelling and is composed of two degenerate parabolic equations and an elliptic one. We prove the convergence of the finite volume scheme and then the existence of solutions to the problem. Several numerical tests show the efficiency of the method.

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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