Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method

Hussain, Arafat; Zheng, Zhoushun; Anley, Eyaya Fekadie

doi:10.3390/math8111869

Cited by 15 publications

(12 citation statements)

References 49 publications

(52 reference statements)

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“…where N B is the number of boundary points and N = N I + N B . Therefore, applying the collocation points to (4) and ( 5), we can obtain fc j g, j = 1, 2, 3, ⋯, N through (4) and (5). Then, we can obtain the approximate solution to (3) at all given points.…”

Section: The Radial Basis Function Collocationmentioning

confidence: 99%

See 1 more Smart Citation

A Modified RBF Collocation Method for Solving the Convection-Diffusion Problems

Chuathong

2023

Abstract and Applied Analysis

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The main purposes of this study are to propose the modified radial basis function (RBF) collocation method using a hybrid radial basis function to solve the convection-diffusion problems numerically and to choose the optimal shape parameter of radial basis functions. The modified numerical scheme is tested on a benchmark problem with varying shape parameters. The root mean square error and maximum error are used to validate the accuracy and efficiency of the method. The proposed method can be a good alternative to the radial basis function collocation method to improve accuracy and results.

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Section: The Radial Basis Function Collocationmentioning

confidence: 99%

“…The convection-diffusion problem is important in many branches of science and engineering governed by the convection-diffusion equation [1][2][3][4][5]. The convectiondiffusion equation is a fundamental equation that combines convection and diffusion processes to represent the problem process.…”

Section: Introductionmentioning

confidence: 99%

A Modified RBF Collocation Method for Solving the Convection-Diffusion Problems

Chuathong

2023

Abstract and Applied Analysis

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“…Numerical solution of CDEs has become an important means of analysis. Robust numerical methods for solving CDEs are the Boundary Element Method (ΒΕΜ) [3], [4], the Finite Volume Method (FVM) [5], [6], the Finite Difference Method (FDM) [7], [8], the Finite Element Method (FEM) [9]- [11], and the Mesh Free Method (MFM) [12], [13]. Although these methods can be used to solve a wide range of engineering problems in solid mechanics, fluid mechanics and heat transfer, they also have some drawbacks.…”

Section: Introductionmentioning

confidence: 99%

Finite Line Method for Solving Convection–diffusion Equations

Gao

LIU

DING

2022

WIT Transactions on Engineering Sciences

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In this paper, a creative collocation-type numerical method, the Finite Line Method (FLM), is proposed for solving general convection-diffusion equations. The method is based on the use of a finite number of lines crossing each collocation point, and the Lagrange polynomial interpolation formulation to construct the shape functions over each line. The directional derivative technique is proposed to derive the first-order partial derivatives of any physical variables with respect to the global coordinates for the high-dimensional problems from the lines' ones and the high-order derivatives are evaluated from a recurrence formulation. The derived spatial partial derivatives are directly substituted into the governing partial differential equations and related boundary conditions of the convection-diffusion equations to set up the system of equations. The finite number of lines crossing each collocation point is called the line set. To evaluate the convection and diffusion terms accurately, two different line sets are used for these two terms, which are called the convection line set and central line set, respectively. The former is formed according to the velocity direction and is used for performing the upwind scheme in the computation of the convection term, and the latter is formed by the crossed lines including the collocation point at the center. A numerical example will be given to verify the correctness and stability of the proposed method.

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“…Momani and Yıldırım in [24] have obtained the approximate analytical solution of CDE via He's homotopy perturbation method. Since the analytical solution of fractional-order problems in most of the cases is difficult to obtain, the researchers have developed numerous numerical methods for the investigation of the solutions of fractional-order problems such as the finite difference method [25], finite element method [26], finite volume method [27], and meshless methods [28][29][30].…”

Section: Introductionmentioning

confidence: 99%

“…Gao and Sun [38] obtained the numerical approximation of fractional sub-diffusion equation via finite difference scheme. Hussain et al [27] analyzed the solution of CDE numerically using the finite element method. In computational modeling, the well-known classical mesh-based methods such as finite difference, finite elements, and finite volume have received much interest in recent years.…”

Section: Introductionmentioning

confidence: 99%

On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations

Kamran

Kamal

Rahmat

et al. 2021

Mathematical Problems in Engineering

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In this paper, we present an efficient method for the numerical investigation of three-dimensional non-integer-order convection-diffusion equation (CDE) based on radial basis functions (RBFs) in localized form and Laplace transform (LT). In our numerical scheme, first we transform the given problem into Laplace space using Laplace transform. Then, the local radial basis function (LRBF) method is employed to approximate the solution of the transformed problem. Finally, we represent the solution as an integral along a smooth curve in the complex left half plane. The integral is then evaluated to high accuracy by a quadrature rule. The Laplace transform is used to avoid the classical time marching procedure. The radial basis functions are important tools for scattered data interpolation and for solving partial differential equations (PDEs) of integer and non-integer order. The LRBF and global radial basis function (GRBF) are used to produce sparse collocation matrices which resolve the issue of the sensitivity of shape parameter and ill conditioning of system matrices and reduce the computational cost. The application of Laplace transformation often leads to the solution in complex plane which cannot be generally inverted. In this work, improved Talbot’s method is utilized which is an efficient method for the numerical inversion of Laplace transform. The stability and convergence of the method are discussed. Two test problems are considered to validate the numerical scheme. The numerical results highlight the efficiency and accuracy of the proposed method.

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Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method

Cited by 15 publications

References 49 publications

A Modified RBF Collocation Method for Solving the Convection-Diffusion Problems

A Modified RBF Collocation Method for Solving the Convection-Diffusion Problems

Finite Line Method for Solving Convection–diffusion Equations

On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations

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