A higher‐order unconditionally stable scheme for the solution of fractional diffusion equation

In this paper, a higher‐order compact finite difference scheme with multigrid algorithm is applied for solving one‐dimensional time fractional diffusion equation. The second‐order derivative with respect to space is approximated by higher‐order compact difference scheme. Then, Grünwald–Letnikov appr...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences Vol. 44; no. 4; pp. 3004 - 3022
Main Authors: Ghaffar, Fazal, Ullah, Saif, Badshah, Noor, Khan, Najeeb Alam
Format: Journal Article
Language:English
Published: Freiburg Wiley Subscription Services, Inc 15.03.2021
Subjects:
Algorithms
Approximation
compact iterative scheme
convergence
Error analysis
Finite difference method
Fourier analysis
fractional diffusion equation
Mathematical analysis
Matrix methods
multigrid method
stability
Stability analysis
Truncation errors
ISSN:0170-4214, 1099-1476
Online Access:Get full text
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Summary:In this paper, a higher‐order compact finite difference scheme with multigrid algorithm is applied for solving one‐dimensional time fractional diffusion equation. The second‐order derivative with respect to space is approximated by higher‐order compact difference scheme. Then, Grünwald–Letnikov approximation is used for the Riemann–Liouville time derivative to get an implicit scheme. The scheme is based on a heptadiagonal matrix with eighth‐order accurate local truncation error. Fourier analysis is used to analyze the stability of higher‐order compact finite difference scheme. Matrix analysis is used to show that the scheme is convergent with the accuracy of eighth‐order in space. Numerical experiments confirm our theoretical analysis and demonstrate the performance and accuracy of our proposed scheme.
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ISSN:0170-4214
1099-1476
DOI:10.1002/mma.6406