Numerical simulation for nonlinear space-fractional reaction convection-diffusion equation with its application

Anley, Eyaya Fekadie; Basha, Merfat; Hussain, Arafat; Dai, Binxiang

doi:10.1016/j.aej.2022.10.047

Cited by 5 publications

(1 citation statement)

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“…The fractional derivative, as a generalization of the integer derivative, is usually non-local and has the properties of memory and heredity, which can precisely describe the power-law dependence of the mean-square deviation on time in the process of anomalous diffusion [10,15]. Therefore, it has been extensively applied in physics [26,30], geology [2,3], financial analysis [1,23,27], cardiac science [5,6,19] and other fields. Based on the structure and electrophysiological activity character of heart tissue, the fractional FitzHugh-Nagumo monodomain model is used to capture the information transmission of anisotropic porous medium in the extra-cellular domain [19], which is a coupled nonlinear diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Banded Preconditioning with Shift Compensation for Solving Discrete Riesz Space-Fractional Diffusion Equations

Miao

2023

Preprint

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Based on the finite-difference method, the considered Riesz space-fractional diffusion equations result in a series of linear systems, whose coefficient matrices are composed of the identity matrix and the product of diagonal matrix and Toeplitz matrix. With the aid of the Toeplitz structure contained in the discrete linear system, a class of banded preconditioner with shift compensation (BSC preconditioner) is designed to improve the convergence rates of the Krylov subspace iteration methods. The BSC preconditioner can be viewed as a modification of the banded preconditioner with diagonal compensation (BDC preconditioner) proposed by F.-R. Lin, Y.-J. Wang and X.-Q. Jin (F.-R. Lin, S.-W. Yang and X.-Q. Jin, J. Comput. Phys., 256(2014), 109-117), but unlike the BDC preconditioner, it retains the same Toeplitz-like structure as the coefficient matrix of the discrete linear system. Although the theoretical result on the eigenvalues of the BDC-preconditioned matrix has not been given so far, the eigenvalue distributions of the BSC-preconditioned matrix and the BDC-preconditioned matrix can be demonstrated simultaneously by using the structure of the BSC preconditioner. In addition, theoretical analysis shows that the eigenvalues of the BSC-preconditioned matrix are real and located in a fixed positive interval, while the eigenvalues of the BDC-preconditioned matrix are clustered around those of the BSC-preconditioned matrix, which indicates that the BSC preconditioner is as effective as the BDC preconditioner. Numerical experiments reveal that both the BSC preconditioner and BDC preconditioner can significantly accelerate the convergence rates of the Krylov subspace iteration methods, and show h-independent convergence behavior.

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