Kinetic‐fluid derivation and mathematical analysis of the cross‐diffusion–Brinkman system

Bendahmane, Mostafa; Karami, Fahd; Zagour, Mohamed

doi:10.1002/mma.5139

Cited by 12 publications

(4 citation statements)

References 40 publications

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“…The above system (1.1)–(1.4) plays a key role in many applications, see [12] for enzyme reactions, [11] for a drug release model, and [10] for biological interactions. Theoretical analysis for the existence of weak solutions of the system was studied in [1] by Anaya et al and in [6] by Bendahmane et al, while the existence of classical solutions is unknown so far. Numerical solutions of the system (1.1)–(1.4) and related models have attracted much attention due to its wide applications, see [1–4, 9, 15, 20, 22–25] and references therein.…”

Section: Introductionmentioning

confidence: 99%

A new optimal error analysis of a mixed finite element method for advection–diffusion–reaction Brinkman flow

Gao,

Xie

2024

Numerical Methods Partial

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This article deals with the error analysis of a Galerkin‐mixed finite element methods for the advection–reaction–diffusion Brinkman flow in porous media. Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest‐order Raviart–Thomas element, the lowest‐order Nédélec edge element and piece‐wise constant discontinuous Galerkin element are used for the velocity, vorticity and pressure, respectively. The existing error estimate of this lowest‐order finite element method is only for all variables in spatial direction, which is not optimal for the concentration variable. This paper focuses on a new and optimal error estimate of a linearized backward Euler Galerkin‐mixed FEMs, where the second‐order accuracy for the concentration in spatial directions is established unconditionally. The key to our optimal error analysis is a new negative norm estimate for Nédélec edge element. Moreover, based on the computed numerical concentration, we propose a simple one‐step recovery technique to obtain a new numerical velocity, vorticity and pressure with second‐order accuracy. Numerical experiments are provided to confirm our theoretical analysis.

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Section: Introductionmentioning

confidence: 99%

A new optimal error analysis of a mixed finite element method for advection–diffusion–reaction Brinkman flow

Gao,

Xie

2024

Numerical Methods Partial

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“…The CDF system is nonlinear and coupled, and usually it is difficult to obtain its analytical solution. For this reason, some numerical methods have been developed for this type of system [11][12][13][14][15]. For CF system, Chertock et al [16] developed a high-resolution vorticity-based hybrid finite-volume finite-difference scheme to understand the interplay of gravity and chemotaxis in the formation of two-dimensional plumes.…”

Section: Introductionmentioning

confidence: 99%

A lattice Boltzmann model for the coupled cross-diffusion-fluid system

Zhan,

Chai,

Shi

2021

Preprint

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In this paper, we propose a lattice Boltzmann (LB) model for the generalized coupled crossdiffusion-fluid system. Through the direct Taylor expansion method, the proposed LB model can correctly recover the macroscopic equations. The cross diffusion terms in the coupled system are modeled by introducing additional collision operators, which can be used to avoid special treatments for the gradient terms. In addition, the auxiliary source terms are constructed properly such that the numerical diffusion caused by the convection can be eliminated. We adopt the developed LB model to study two important systems, i.e., the coupled chemotaxis-fluid system and the double-diffusive convection system with Soret and Dufour effects. We first test the present LB model through considering a steady-state case of coupled chemotaxis-fluid system, then we analyze the influences of some physical parameters on the formation of sinking plumes. Finally, the double-diffusive natural convection system with Soret and Dufour effects is also studied, and the numerical results agree well with some previous works.

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“…Many works used this method within different fields of application. For instance, chemotaxis phenomena in the basis of the famous Keller-Segel model [7], formation of patterns induced by cross-diffusion in a fluid [5,11]. In fact, this technique has been adopted to design a numerical scheme that preserves the asymptotic property introduced by [20,25].…”

Section: Introductionmentioning

confidence: 99%

Kinetic derivation of a time-dependent SEIRD reaction-diffusion system for COVID-19

Zagour

2021

Preprint

Self Cite

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In this paper, we propose a time-dependent Susceptible-Exposed-Infectious-Recovered-Died (SEIRD) reaction-diffusion system for the COVID-19 pandemic and we deal with its derivation from a kinetic model. The derivation is obtained by mathematical description delivered at the micro-scale of individuals. Our approach is based on the micro-macro decomposition which leads to an equivalent formulation of the kinetic model which couples the microscopic equations with the macroscopic equations. We develop a numerical asymptotic preservation scheme to solve the kinetic model. The proposed approach is validated by various numerical tests where particular attention is paid to the Moroccan situation against the actual pandemic.

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Kinetic‐fluid derivation and mathematical analysis of the cross‐diffusion–Brinkman system

Cited by 12 publications

References 40 publications

A new optimal error analysis of a mixed finite element method for advection–diffusion–reaction Brinkman flow

A new optimal error analysis of a mixed finite element method for advection–diffusion–reaction Brinkman flow

A lattice Boltzmann model for the coupled cross-diffusion-fluid system

Kinetic derivation of a time-dependent SEIRD reaction-diffusion system for COVID-19

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