The finite volume element method for the two-dimensional space-fractional convection–diffusion equation

Bi, Yanan; Jiang, Ziwen

doi:10.1186/s13662-021-03524-4

Cited by 4 publications

(11 citation statements)

References 38 publications

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“…Table 1 shows the effect of applying uniform and non-uniform PDQM on computation of solute concentration υ(x, t) at β = 1.85, C x = 2, L = 8. It is noticed that the L ∞ error is decreasing when the grid points are increasing, that is, the PDQM is stability in the x-direction, as well as the computed results via non-uniform grids are higher agree with earlier numerical solutions [13,42,43,59] than uniform ones. Also, L ∞ error (1.0885 × 10 −6 ) and performance time (0.17 s) for non-uniform PDQM achieved the least.…”

Section: Numerical Resultsmentioning

confidence: 53%

“…The primary objective of our paper is to evaluate the performance, validity, efficiency, and accuracy of the developed techniques. We confirm this by comparing the computed results with existing numerical solutions [13,42,43,59] and analytical solutions [40,41,44]. To assess the convergence and accuracy of the developed methods, we use the error computation method outlined in [13,42,43,59]:…”

Section: Numerical Resultsmentioning

confidence: 68%

“…We confirm this by comparing the computed results with existing numerical solutions [13,42,43,59] and analytical solutions [40,41,44]. To assess the convergence and accuracy of the developed methods, we use the error computation method outlined in [13,42,43,59]:…”

Section: Numerical Resultsmentioning

confidence: 68%

“…Additionally, we design a MATLAB code for each approach to obtain a numerical solution for three problems to be examined. When compared to earlier analytical [40,41] and numerical [13,42,43] methodologies, the derived numerical results attain excellent efficiency and accuracy. Furthermore, we present some parametric studies to highlight the reliability of our methods with the influence of fractional order derivative, the velocity, and positive diffusion parameters on the results.…”

Section: Introductionmentioning

confidence: 87%

“…Solving the fractional convection-diffusion equation numerically is a difficult task, especially for high-dimensional cases, due to the fractional derivative characteristic of the space fractional derivative differential operator. Various numerical methods have been proposed for approximating one-dimensional fractional convection-diffusion equations, including the finite difference method [9], Galerkin method [10], collocation method [11], homotopy analysis transform method [12], and finite volume element method [13]. Ding et al [14] suggested an improved matrix transform numerical approach to solve the onedimensional space fractional advection-dispersion model, and an analytical solution was obtained using the Padé approximation.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

Mustafa,

Ragb,

Salah

et al. 2023

Fractal Fract

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The aim of this study is to utilize a differential quadrature method with various kernels, such as Lagrange interpolation and discrete singular convolution, to tackle problems related to the Riesz fractional diffusion equation and the Riesz fractional advection–dispersion equation. The governing equation for convection and diffusion depends on both spatial and transient factors. By using the block marching technique, we transform these equations into an algebraic system using differential quadrature methods and the Caputo-type fractional operator. Next, we develop a MATLAB program that generates code capable of solving the fractional convection–diffusion equation in (1+2) dimensions for each shape function. Our goal is to ensure that our methods are reliable, accurate, efficient, and capable of convergence. To achieve this, we conduct two experiments, comparing the numerical and graphical results with both analytical and numerical solutions. Additionally, we evaluate the accuracy of our findings using the L∞ error. Our tests show that the differential quadrature method, which relies mainly on the discrete singular convolution shape function, is a highly effective numerical approach for fractional convective diffusion problems. It offers superior accuracy, faster convergence, and greater reliability than other techniques. Furthermore, we study the impact of fractional order derivatives, velocity, and positive diffusion parameters on the results.

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Section: Numerical Resultsmentioning

confidence: 53%

Section: Numerical Resultsmentioning

confidence: 68%

Section: Numerical Resultsmentioning

confidence: 68%

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

Mustafa,

Ragb,

Salah

et al. 2023

Fractal Fract

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On solutions of convection-diffusion equation in fluid

Nyaupane,

Uprety,

Sapkota

2024

Sci World

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Convection and diffusion are fundamental transport mechanisms that govern the flow of mass, heat, or other quantities in various physical, biological and engineering systems. The work overviews theoretical underpinnings, mathematical formulations and numerical methods to analyze the convection-diffusion process in one and two dimensions, both in steady and transient states. We have compared the analytical and numerical results using finite difference and finite volume methods for fluid flow and temperature distribution. The numerical solutions are in agreement with the analytical solution in one dimension. The work compares convection-diffusion with convection-dominating for larger velocities of fluid flow. The solution obtained from FVM is in line with an analytical solution than the solution obtained from FDM. The numerical simulation for the transient flow is also discussed.

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Linearized Crank–Nicolson Scheme for the Two-Dimensional Nonlinear Riesz Space-Fractional Convection–Diffusion Equation

2023

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In this paper, we study the nonlinear Riesz space-fractional convection–diffusion equation over a finite domain in two dimensions with a reaction term. The Crank–Nicolson difference method for the temporal and the weighted–shifted Grünwald–Letnikov difference method for the spatial discretization are proposed to achieve a second-order convergence in time and space. The D’Yakonov alternating–direction implicit technique, which is effective in two–dimensional problems, is applied to find the solution alternatively and reduce the computational cost. The unconditional stability and convergence analyses are proved theoretically. Numerical experiments with their known exact solutions are conducted to illustrate our theoretical investigation. The numerical results perfectly confirm the effectiveness and computational accuracy of the proposed method.

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The finite volume element method for the two-dimensional space-fractional convection–diffusion equation

Cited by 4 publications

References 38 publications

Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

On solutions of convection-diffusion equation in fluid

Linearized Crank–Nicolson Scheme for the Two-Dimensional Nonlinear Riesz Space-Fractional Convection–Diffusion Equation

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