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时间分数阶次扩散方程的多层扩充算法

陈剑 曾泰山

陈剑, 曾泰山. 时间分数阶次扩散方程的多层扩充算法[J]. 华南师范大学学报(自然科学版), 2020, 52(3): 106-110. doi: 10.6054/j.jscnun.2020051
引用本文: 陈剑, 曾泰山. 时间分数阶次扩散方程的多层扩充算法[J]. 华南师范大学学报(自然科学版), 2020, 52(3): 106-110. doi: 10.6054/j.jscnun.2020051
CHEN Jian, ZENG Taishan. The Multilevel Augmentation Method for Solving Time Fractional Subdiffusion Equation[J]. Journal of South China normal University (Natural Science Edition), 2020, 52(3): 106-110. doi: 10.6054/j.jscnun.2020051
Citation: CHEN Jian, ZENG Taishan. The Multilevel Augmentation Method for Solving Time Fractional Subdiffusion Equation[J]. Journal of South China normal University (Natural Science Edition), 2020, 52(3): 106-110. doi: 10.6054/j.jscnun.2020051

时间分数阶次扩散方程的多层扩充算法

doi: 10.6054/j.jscnun.2020051
基金项目: 

国家自然科学基金项目 11501106

国家自然科学基金项目 11671159

广东省自然科学基金项目 2018A0303130067

广东省自然科学基金项目 2016A030313835

广东省普通高校特色创新类项目 2016KTSCX024

详细信息
    通讯作者:

    曾泰山, 副教授, Email:zengtsh@m.scnu.edu.cn

  • 中图分类号: O241.82

The Multilevel Augmentation Method for Solving Time Fractional Subdiffusion Equation

  • 摘要: 基于L1公式和多尺度Galerkin方法, 对具有α阶Caputo导数的时间分数阶次扩散方程建立了全离散格式;证明了全离散格式存在唯一解和具有最优收敛阶O(hr+τ2-α), r为分片多项式的次数;在每个时间层,对全离散格式所得线性方程组, 设计了多层扩充算法进行高效求解, 并保持着最优收敛阶;最后, 给出数值算例来验证理论分析的正确性.
  • 图  1  u2, 5(x, t)在t=1时刻的时间收敛阶

    Figure  1.  The temporal convergence order of u2, 5(x, t) at t=1

    图  2  算法1与Gauss迭代法的计算时间对比

    Figure  2.  The comparison of computational time between algorithm 1 and the Gauss iteration method

    表  1  线性基底的数值结果

    Table  1.   The numerical results for linear basis

    m x(n) p=1 p=0 条件数
    u-u4, m1 Order u-u4, m0 Order
    0 15 2.009 5e+00 3.925 7e-02 1.216 4
    1 31 1.006 7e+00 0.997 2 9.823 9e-03 1.998 6 1.217 6
    2 63 5.035 8e-01 0.999 3 2.457 4e-03 1.999 2 1.217 9
    3 127 2.518 2e-01 0.999 8 6.143 7e-04 1.999 9 1.217 9
    4 255 1.259 1e-01 1.000 0 1.535 3e-04 2.000 6 1.217 9
    5 511 6.295 7e-02 1.000 0 3.831 5e-05 2.002 5 1.218 0
    6 1 023 3.147 8e-02 1.000 0 9.509 9e-06 2.010 4 1.218 0
    7 2 047 1.573 9e-02 1.000 0 2.316 5e-06 2.037 5 1.218 0
    下载: 导出CSV

    表  2  二次基底的数值结果

    Table  2.   The numerical results for quadratic basis

    m x(n) p=1 p=0 条件数
    u-u2, m1 Order u-u2, m0 Order
    0 7 1.577 5e+00 6.062 9e-02 1.215 1
    1 15 4.049 8e-01 1.961 7 7.804 1e-03 2.957 7 1.217 4
    2 31 1.019 1e-01 1.990 5 9.825 6e-04 2.989 6 1.217 8
    3 63 2.552 0e-02 1.997 6 1.230 5e-04 2.997 4 1.217 9
    4 127 6.382 6e-03 1.999 4 1.539 5e-05 2.998 7 1.217 9
    5 255 1.595 8e-03 1.999 9 1.990 7e-06 2.951 1 1.218 0
    下载: 导出CSV
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出版历程
  • 收稿日期:  2020-01-14
  • 刊出日期:  2020-06-25

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