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 fully discrete and energy-dissipating finite-volume scheme for aggregation-diffusion equations

2020 ◽  
Vol 30 (13) ◽  
pp. 2487-2522
Author(s):  
Rafael Bailo ◽  
José A. Carrillo ◽  
Hideki Murakawa ◽  
Markus Schmidtchen
Keyword(s):  
Finite Volume ◽  
Flow Structure ◽  
Gradient Flow ◽  
Diffusion Equations ◽  
Finite Volume Scheme ◽  
Time Step ◽  
Discrete Energy ◽  
Volume Scheme ◽  
Fully Discrete ◽  
Local Aggregation

We study an implicit finite-volume scheme for nonlinear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced in [R. Bailo, J. A. Carrillo and J. Hu, Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient flow structure, arXiv:1811.11502 ]. Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved.

A finite-volume scheme for gradient-flow equations with non-homogeneous diffusion

2021 ◽  
Vol 89 ◽  
pp. 150-162
Author(s):  
Julien Mendes ◽  
Antonio Russo ◽  
Sergio P. Perez ◽  
Serafim Kalliadasis
Keyword(s):  
Finite Volume ◽  
Gradient Flow ◽  
Finite Volume Scheme ◽  
Flow Equations ◽  
Volume Scheme

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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