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A Two-Grid Block-Centered Finite Difference Method for the Nonlinear Time-Fractional Parabolic Equation

Abstract

In this article, a two-grid block-centered finite difference scheme is introduced and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size H and a linear problem is solved on a fine grid of size h. Stability results are proven rigorously. Error estimates are established on non-uniform rectangular grid which show that the discrete \(L^{\infty }(L^2)\) and \(L^2(H^1)\) errors are \(O(\triangle t^{2-\alpha }+h^2+H^3)\). Finally, some numerical experiments are presented to show the efficiency of the two-grid method and verify that the convergence rates are in agreement with the theoretical analysis.

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Acknowledgements

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this paper. This work is supported by the National Natural Science Foundation of China Grant Nos. 11671233, 91330106.

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Correspondence to Hongxing Rui.

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Cite this article

Li, X., Rui, H. A Two-Grid Block-Centered Finite Difference Method for the Nonlinear Time-Fractional Parabolic Equation. J Sci Comput 72, 863–891 (2017). https://doi.org/10.1007/s10915-017-0380-4

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Keywords

  • Block-centered finite difference
  • Time-fractional parabolic equation
  • Nonlinear
  • Stability results
  • Error estimates
  • Numerical experiments

Mathematics Subject Classification

  • 26A33
  • 65M06
  • 65M12
  • 65M15
  • 65M55