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Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation

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Abstract

We propose a locally one dimensional (LOD) finite difference method for multidimensional Riesz fractional diffusion equation with variable coefficients on a finite domain. The numerical method is second-order convergent in both space and time directions, and its unconditional stability is strictly proved. The matrix algebraic equations of the proposed second-order schemes are almost the same as the ones of the popular first-order finite difference method for fractional operators. And the matrices involved in the schemes of different convergence orders have completely same structure and the computational count for matrix vector multiplication is \(\fancyscript{O}(N \text{ log } N)\); and the computational costs for solving the matrix algebraic equations of the second-order and first-order schemes are almost the same. The LOD-multigrid method is used to solve the resulting matrix algebraic equation, and the computational count is \(\fancyscript{O}(N \text{ log } N)\) and the required storage is \(\fancyscript{O}(N)\), where \(N\) is the number of grid points. Finally, extensive numerical experiments are performed to show the powerfulness of the second-order scheme and the LOD-multigrid method.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Minghua Chen, Yantao Wang, Xiao Cheng & Weihua Deng

Authors
  1. Minghua Chen
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  2. Yantao Wang
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  3. Xiao Cheng
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  4. Weihua Deng
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Corresponding author

Correspondence to Weihua Deng.

Additional information

Communicated by Jan Hesthaven.

This work was supported by the Program for New Century Excellent Talents in University under Grant No. NCET-09-0438, the National Natural Science Foundation of China under Grant No. 10801067 and No. 11271173, and the Fundamental Research Funds for the Central Universities under Grant No. lzujbky-2010-63 and No. lzujbky-2012-k26.

Appendix

Appendix

For a general linear system

$$\begin{aligned} A_hu_h=f_h, \end{aligned}$$

we employ the following V-cycle MGM (Algorithm 1–2) to solve one dimensional (2.8) and V-cycle LOD-MGM (Algorithm 1–3) solve two dimensional (2.13). Solve the three dimensional system (1.1) by Algorithm 1, 2 and 4.

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Chen, M., Wang, Y., Cheng, X. et al. Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation. Bit Numer Math 54, 623–647 (2014). https://doi.org/10.1007/s10543-014-0477-1

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Keywords

Mathematics Subject Classfication (2010)

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