| Citation: |
Huanyan Jian, Tingzhu Huang, Xile Zhao, Yongliang Zhao. FAST SECOND-ORDER ACCURATE DIFFERENCE SCHEMES FOR TIME DISTRIBUTED-ORDER AND RIESZ SPACE FRACTIONAL DIFFUSION EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1359-1392. doi: 10.11948/2156-907X.20180247
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FAST SECOND-ORDER ACCURATE DIFFERENCE SCHEMES FOR TIME DISTRIBUTED-ORDER AND RIESZ SPACE FRACTIONAL DIFFUSION EQUATIONS
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Abstract
The aim of this paper is to develop fast second-order accurate difference schemes for solving one- and two-dimensional time distributed-order and Riesz space fractional diffusion equations. We adopt the same measures for one- and two-dimensional problems as follows: we first transform the time distributed-order fractional diffusion problem into the multi-term time-space fractional diffusion problem with the composite trapezoid formula. Then, we propose a second-order accurate difference scheme based on the interpolation approximation on a special point to solve the resultant problem. Meanwhile, the unconditional stability and convergence of the new difference scheme in L2-norm are proved. Furthermore, we find that the discretizations lead to a series of Toeplitz systems which can be efficiently solved by Krylov subspace methods with suitable circulant preconditioners. Finally, numerical results are presented to show the effectiveness of the proposed difference methods and demonstrate the fast convergence of our preconditioned Krylov subspace methods. -
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References
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Huanyan Jian, Tingzhu Huang, Xile Zhao, Yongliang Zhao. FAST SECOND-ORDER ACCURATE DIFFERENCE SCHEMES FOR TIME DISTRIBUTED-ORDER AND RIESZ SPACE FRACTIONAL DIFFUSION EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1359-1392. doi: 10.11948/2156-907X.20180247
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Figure 1. Exact solutions (lines) and numerical solutions (symbols) of Example 4.1: (a)
$ \beta $ = 1.3 at$ T $ = 1.5 (stars), 1.2 (rhombus), 0.8 (triangles); (b)$ \beta $ = 1.8 at$ T $ = 1.5 (stars), 1.2 (rhombus), 0.8 (triangles) -
Figure 2. Spectrum of original matrice (red) and R. Chan's-based preconditioned matrice (blue) for Example 4.1 at time level (a)
$ n = 0 $ and (b)$ n = 1 $ , respectively, when$ M $ =$ N $ = 128,$ J $ = 50,$ \beta $ = 1.8, and$ T $ = 1.5. -
Figure 3. Spectrum of original matrice (red) and R. Chan's-based preconditioned matrice (blue) for Example 4.1 at time level (a)
$ n = 0 $ and (b)$ n = 1 $ , respectively, when$ M $ =$ N $ = 256,$ J $ = 50,$ \beta $ = 1.8, and$ T $ = 1.5 -
Figure 4. The solution surfaces obtained from Example 4.3 at
$ J $ = 50,$ \widetilde{M} $ = 40 and$ N $ = 10: (a) the exact solution with$ T = 1 $ ,$ \beta $ =$ \gamma $ = 1.8; (b) the numerical solution with$ T = 1 $ ,$ \beta $ =$ \gamma $ = 1.8 by the scheme (3.12)-(3.14); (c) the exact solution with$ T = 0.5 $ ,$ \beta $ =$ \gamma $ = 1.3; (d) the numerical solution with$ T = 0.5 $ ,$ \beta $ =$ \gamma $ = 1.3 by the scheme (3.12)-(3.14); -
Figure 5. Spectrum of original matrice (red) and R. Chan's-based preconditioned matrice (blue) for Example 4.3 at time level (a)
$ n = 0 $ and (b)$ n = 1 $ , respectively, when$ \widetilde{M} $ =$ N $ = 64,$ J $ = 50,$ \beta $ =$ \gamma $ = 1.5, and$ T $ = 1.5 -
Figure 6. Spectrum of original matrice (red) and R. Chan's-based preconditioned matrice (blue) for Example 4.3 at time level (a)
$ n = 0 $ and (b)$ n = 1 $ , respectively, when$ \widetilde{M} $ =$ N $ = 128,$ J $ = 50,$ \beta $ =$ \gamma $ = 1.5, and$ T $ = 1.5