Computational examples of reaction–convection–diffusion equations solution under the influence of fluid flow: First example

Garzón‐Alvarado, Diego Alexander; Galeano, Carlos; Mantilla, Juan Miguel

doi:10.1016/j.apm.2011.12.041

Cited by 16 publications

(6 citation statements)

References 23 publications

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“…In practical scenarios, the coupling of advection, difusion, and reaction processes results in intricate interactions, leading to temporal and spatial alterations within the ADR system. Te ADR equation is used in modeling contaminant transport [1,2], fuid fow [3], mass and heat transfer [4], chemical reaction process [5], difusion across a cell membrane [6], tumor cell expansion [7], population dynamics [8], and so on.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients

Tsega

2024

International Journal of Mathematics and Mathematical Sciences

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The advection-diffusion-reaction (ADR) equation is a fundamental mathematical model used to describe various processes in many different areas of science and engineering. Due to wide applicability of the ADR equation, finding accurate solution is very important to better understand a physical phenomenon represented by the equation. In this study, a numerical scheme for solving two-dimensional unsteady ADR equations with spatially varying velocity and diffusion coefficients is presented. The equations include nonlinear reaction terms. To discretize the ADR equations, the Crank–Nicolson finite difference method is employed with a uniform grid. The resulting nonlinear system of equations is solved using Newton’s method. At each iteration of Newton’s method, the Gauss–Seidel iterative method with sparse matrix computation is utilized to solve the block tridiagonal system and obtain the error correction vector. The consistency and stability of the numerical scheme are investigated. MATLAB codes are developed to implement this combined numerical approach. The validation of the scheme is verified by solving a two-dimensional advection-diffusion equation without reaction term. Numerical tests are provided to show the good performances of the proposed numerical scheme in simulation of ADR problems. The numerical scheme gives accurate results. The obtained numerical solutions are presented graphically. The result of this study may provide insights to apply numerical methods in solving comprehensive models of physical phenomena that capture the underlying situations.

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Section: Introductionmentioning

confidence: 99%

Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients

Tsega

2024

International Journal of Mathematics and Mathematical Sciences

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“…Formally, the two-dimensional version of the transport equation is a partial differential equation (PDE) defined in a rectangular domain Ω under some boundary conditions. Its mathematical formulation is described by equations (1), (2), and 3where u is the physical greatness to be evaluated, k is the diffusion coefficient, β x (x, y) and β y (x, y) are the velocities in x and y directions respectively, γ(x, y) is the function that defines the reaction process, f(x, y) is the source term, g is a function that defines the values of u in the border Γ, and h(x, y) is the function that defines the normal derivative value of u along the border Γ n [2]. 22…”

Section: Introductionmentioning

confidence: 99%

An Implementation of the Finite Differences Method for the Two-Dimensional Rectangular Cooling Fin Problem

Rodrigues¹

2019

IJITCS

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The transport or advection-diffusion-reaction equation is a well-known partial differential equation employed to model several types of flux problems. The cooling fin problem is a particular case of such an equation. This work presents a straightforward model for the rectangular cooling fin in a problem. The model was based on the finite differences numerical method and an efficient implementation was developed in a high-level mathematical programming language. The accuracy was evaluated with different granularity levels of meshes, and two distinct boundary conditions are compared. In the first one, only prescribed temperatures are assumed at the four tips of the domain. For the second scenario, it is assumed a heat flux at one tip of a fin with the same geometrical shape. The achieved solutions produced by the algorithm were able to depict the temperature along the whole fin surface accurately. Furthermore, the algorithm reaches relevant performance for meshes up to 4257 points where the CPU time was about 33 seconds.

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“…However, the numerical solution of those problems might become a challenge as it is well known that the discrete solutions generated by standard numerical methods is usually globally polluted by non-physical oscillations in the whole domain. Therefore, effective algorithms for the numerical solutions of those problems has captured the interest of a number of many researchers (Dimitrov and Kojouharov 2006, Garzon et al 2012, Stefano et al 2013.…”

Section: Introductionmentioning

confidence: 99%

Av-Avcı Problemleri için Kararlı Sonlu Eleman Yöntemleri Üzerine Bir Not

Sendur¹

2019

Afyon Kocatepe University Journal of Sciences and Engineering

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A numerical method that will improve and produce effective results for solving mathematical model for the system of predator-prey interactions which is defined by convection-diffusion-reaction problem is studied herein. We consider the Pseudo Residual-free Bubble (PRFB) method which is based on augmenting the finite element space by appropriate functions for the space discretization. The method is applied on different test problems and the numerical solutions are in good agreement with the result available in literature. The numerical results depict that the algorithm is efficient and feasible. Av-Avcı Problemleri için Kararlı Sonlu Eleman Yöntemleri Üzerine Bir Not Anahtar kelimeler Av-avcı denklem sistemleri; Konveksiyon-difüzyonreaksiyon; Kararlı Sonlu Eleman Yöntemi; Çok-ölçekli Yöntemler. Öz Bu çalışmada, konveksiyon-difüzyon-reaksiyon problemleri ile modellenebilen av-avcı denklem sistemlerinin simülasyonunda kullanılan sayısal çözüm tekniklerini iyileştirecek ve daha etkin sonuçlar üretecek sayısal bir yöntem önerilmiştir. Uzay ayrıklaştırması için, sonlu elemanlar metodunu uygularken seçilen polinom baz fonksiyonlarına ilaveten fonksiyon uzayının özel tip fonksiyonlarla (residual-free bubbles) zenginleştirilmesine dayanan Pseudo Residual-free Bubble (PRFB) yöntemi kullanılmıştır. Söz konusu yöntem, çeşitli test örneklerine uygulanmış olup elde edilen sayısal çözümlerin, literatürde mevcut olan sonuçlar ile iyi bir uyum içinde olduğu gözlemlenmiştir. Sayısal sonuçlar, önerilen yöntemin verimli ve uygulanabilir olduğunu göstermektedir.

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Computational examples of reaction–convection–diffusion equations solution under the influence of fluid flow: First example

Cited by 16 publications

References 23 publications

Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients

Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients

An Implementation of the Finite Differences Method for the Two-Dimensional Rectangular Cooling Fin Problem

Av-Avcı Problemleri için Kararlı Sonlu Eleman Yöntemleri Üzerine Bir Not

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