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 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 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.

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