Computationally Efficient Hybrid Method for the Numerical Solution of the 2D Time Fractional Advection-Diffusion Equation

Salama, Fouad Mohammad; Ali, Norhashidah Hj. Mohd.

doi:10.33889/ijmems.2020.5.3.036

Cited by 6 publications

(3 citation statements)

References 15 publications

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“…They have been used for efficient computations of accurate numerical solutions to the two-dimensional fractional diffusion equation (Salama et al. 2022a ), two-dimensional fractional cable equation (Salama and Abd Hamid 2020 ; Khan et al. 2021 ), two-dimensional fractional reaction diffusion equation (Abdi et al.…”

Section: Introductionmentioning

confidence: 99%

Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

et al. 2023

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The time-fractional advection–diffusion reaction equation (TFADRE) is a fundamental mathematical model because of its key role in describing various processes such as oil reservoir simulations, COVID-19 transmission, mass and energy transport, and global weather production. One of the prominent issues with time fractional differential equations is the design of efficient and stable computational schemes for fast and accurate numerical simulations. We construct in this paper, a simple and yet efficient modified fractional explicit group method (MFEGM) for solving the two-dimensional TFADRE with suitable initial and boundary conditions. The proposed method is established using a difference scheme based on L1 discretization in temporal direction and central difference approximations with double spacing in spatial direction. For comparison purposes, the Crank–Nicolson finite difference method (CNFDM) is proposed. The stability and convergence of the presented methods are theoretically proved and numerically affirmed. We illustrate the computational efficiency of the MFEGM by comparing it to the CNFDM for four numerical examples including fractional diffusion and fractional advection–diffusion models. The numerical results show that the MFEGM is capable of reducing iteration count and CPU timing effectively compared to the CNFDM, making it well-suited to time fractional diffusion equations.

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Section: Introductionmentioning

confidence: 99%

Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

et al. 2023

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“…Jiang et al [ 11 ] used the sum of exponentials approximation to obtain the fast convolution algorithm. Salama et al [ 24 – 27 ] used hybrid Laplace transform-finite difference method to obtain the high efficient scheme.…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data

Bi-yun

Xiao

2023

Numer Algor

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In this paper, we focus on developing a high efficient algorithm for solving d -dimension time-fractional diffusion equation (TFDE). For TFDE, the initial function or source term is usually not smooth, which can lead to the low regularity of exact solution. And such low regularity has a marked impact on the convergence rate of numerical method. In order to improve the convergence rate of the algorithm, we introduce the space-time sparse grid (STSG) method to solve TFDE. In our study, we employ the sine basis and the linear element basis for spatial discretization and temporal discretization, respectively. The sine basis can be divided into several levels, and the linear element basis can lead to the hierarchical basis. Then, the STSG can be constructed through a special tensor product of the spatial multilevel basis and the temporal hierarchical basis. Under certain conditions, the function approximation on standard STSG can achieve the accuracy order with degrees of freedom (DOF) for and DOF for , where J denotes the maximal level of sine coefficients. However, if the solution changes very rapidly at the initial moment, the standard STSG method may reduce accuracy or even fail to converge. To overcome this, we integrate the full grid into the STSG, and obtain the modified STSG. Finally, we obtain the fully discrete scheme of STSG method for solving TFDE. The great advantage of the modified STSG method can be shown in the comparative numerical experiment.

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“…This motivates us to develop fast numerical schemes to solve them. This is useful for long time simulations, especially when attempting to solve multi-dimensional fractional problems [33][34][35][36]. It is well known that explicit group methods can diminish the computational complexity and reduce the computational time of numerical algorithms effectively [37][38][39][40][41][42][43][44].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Two-Dimensional Time Fractional Mobile/Immobile Equation Using Explicit Group Methods

Salama

Ali

2022

Int. J. Appl. Comput. Math

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In this paper, we shall present the development of two explicit group schemes, namely, fractional explicit group (FEG) and modified fractional explicit group (MFEG) methods for solving the time fractional mobile/immobile equation in two space dimensions. The presented methods are formulated based on two Crank-Nicolson (C-N) finite difference schemes established at two different grid spacings. The stability and convergence of order O(τ 2−α + h 2 ) are rigorously proven using Fourier analysis. Several numerical experiments are conducted to verify the efficiency of the proposed methods. Meanwhile, numerical results show that the FEG and MFEG algorithms are able to reduce the computational times and iterations effectively while preserving good accuracy in comparison to the C-N finite difference method.

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Computationally Efficient Hybrid Method for the Numerical Solution of the 2D Time Fractional Advection-Diffusion Equation

Cited by 6 publications

References 15 publications

Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data

Numerical Solution of Two-Dimensional Time Fractional Mobile/Immobile Equation Using Explicit Group Methods

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