An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains

Feng, Libo; Liu, Fawang; Turner, Ian

doi:10.1016/j.cam.2019.06.035

Cited by 18 publications

(5 citation statements)

References 53 publications

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“…where (x r , y r ) is the mid-point of the control face and m i is the number of sub-control volumes associated with the node i. For more details, the readers can refer to [29]. Then from (4), we can derive the following ODE system…”

Section: The Implementation Of the Control Volume Methodsmentioning

confidence: 99%

The use of a time-fractional transport model for performing computational homogenisation of 2D heterogeneous media exhibiting memory effects

Feng,

Turner,

Perre

et al. 2021

Preprint

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In this work, a two-dimensional time-fractional subdiffusion model is developed to investigate the underlying transport phenomena evolving in a binary medium comprised of two sub-domains occupied by homogeneous material. We utilise an unstructured mesh control volume method to validate the model against a derived semi-analytical solution for a class of two-layered problems. This generalised transport model is then used to perform computational homogenisation on various two-dimensional heterogenous porous media. A key contribution of our work is to extend the classical homogenisation theory to accommodate the new framework and show that the effective diffusivity tensor can be computed once the cell problems reach steady state at the microscopic scale. We verify the theory for binary media via a series of well-known test problems and then investigate media having inclusions that exhibit a molecular relaxation (memory) effect. Finally, we apply the generalised transport model to estimate the bound water diffusivity tensor on cellular structures obtained from environmental scanning electron microscope (ESEM) images for Spruce wood and Australian hardwood. A highlight of our work is that the computed diffusivity for the heterogeneous media with molecular relaxation is quite different from the classical diffusion cases, being dominated at steady-state by the material with memory effects.

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Section: The Implementation Of the Control Volume Methodsmentioning

confidence: 99%

The use of a time-fractional transport model for performing computational homogenisation of 2D heterogeneous media exhibiting memory effects

Feng,

Turner,

Perre

et al. 2021

Preprint

Self Cite

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show abstract

“…Green's formula is used to deal with the diffusion term while a lumped mass approach [19] with the other terms in equation (2.7). Hence, we obtain:…”

Section: Finite-volume Schemementioning

confidence: 99%

Solving a generalized fractional diffusion equation with variable fractional order and moving boundary by two numerical methods: FDM vs FVM

Li¹,

Li²,

Hu³

2023

Phys. Scr.

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In this paper, both the finite difference method (FDM) and the finite volume method (FVM) are employed to solve the fractional partial differential diffusion equation with temporal dimension and one spatial dimension. In this case, the boundary on the right of the domain is moving with time, while the variable fractional order is depicted as a function of both time and space. Special technique has been proposed to deal with the moving boundary which not only involves the computational difficulty and also accumulates the error. The accuracy and computational resource consumption of the two methods are compared in four designed cases with different functions of moving boundaries and fractional orders. The results show that the computation cost of FDM and FVM is almost the same in problems with one-dimensional space, but the accuracy of the FDM is higher than that of the FVM. Besides, compared with linear cases, the computational accuracy of both methods decreases significantly with nonlinear functions of fractional derivative and moving boundary.

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“…For example, nonpolynomial quintic splines have been recently employed for one‐ and two‐dimensional time fractional diffusion problems, 24,25 homotopy perturbation method for system of FDEs was introduced in Abdulaziz et al, 26 Hu and Zhang investigated fractional cable equation via implicit compact difference method, 27 and finite element method was implemented to space‐time fractional Fokker–Planck equation in Deng 28 . Feng et al 29 proposed an unstructured mesh control volume method for the investigation of two‐dimensional space fractional diffusion equations with variable coefficients defined on a convex domains. Vong and Wang 30 deployed compact difference scheme to deal with the solution of 2D fractional Klein–Gordon equation, while Cui 31 utilized fourth‐order compact scheme for the sine–Gordon equation.…”

Section: Introductionmentioning

confidence: 99%

A collocation method for time‐fractional diffusion equation on a metric star graph with η$$ \eta $$ edges

Faheem

Khan

2023

Math Methods in App Sciences

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A wavelet collocation method based on Haar wavelet is proposed to investigate the numerical solution of time fractional diffusion equation (TFDE) on a metric star graph. We have utilized the Riemann-Liouville definition of fractional integral operator together with Haar wavelet to obtain the Riemann-Liouville fractional integral operator for Haar wavelet (RLFIO-H). The application of RLFIO-H and Haar wavelet to TFDE on metric star graph returns a system of linear algebraic equations. Solving this system yields wavelet coefficients and subsequently the solution. The convergence analysis of the proposed method is discussed to establish the theoretical authenticity of the method. The proposed method is tested on four benchmark problems to validate the theoretical findings. Moreover, the comparison of the developed method with the existing method shows the superiority and accuracy of the proposed method. To the best of the author's knowledge, the proposed work is one of the first collocation method in the literature that deals with the solution of TFDE on metric star graph using wavelet numerically.

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An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains

Cited by 18 publications

References 53 publications

The use of a time-fractional transport model for performing computational homogenisation of 2D heterogeneous media exhibiting memory effects

The use of a time-fractional transport model for performing computational homogenisation of 2D heterogeneous media exhibiting memory effects

Solving a generalized fractional diffusion equation with variable fractional order and moving boundary by two numerical methods: FDM vs FVM

A collocation method for time‐fractional diffusion equation on a metric star graph with η$$ \eta $$ edges

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