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时间分数阶扩散波方程的无单元Galerkin法分析

吴迪 李小林

吴迪,李小林. 时间分数阶扩散波方程的无单元Galerkin法分析 [J]. 应用数学和力学,2022,43(2):1-9 doi: 10.21656/1000-0887.420172
引用本文: 吴迪,李小林. 时间分数阶扩散波方程的无单元Galerkin法分析 [J]. 应用数学和力学,2022,43(2):1-9 doi: 10.21656/1000-0887.420172
WU Di, LI Xiaolin. Element-Free Galerkin Method for Time Fractional Diffusion-Wave Equations[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420172
Citation: WU Di, LI Xiaolin. Element-Free Galerkin Method for Time Fractional Diffusion-Wave Equations[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420172

时间分数阶扩散波方程的无单元Galerkin法分析

doi: 10.21656/1000-0887.420172
基金项目: 国家自然科学基金(面上项目)(11971085);重庆市高校创新研究群体项目(CXQT19018);重庆市教委科学技术研究项目(重大项目)(KJZD-M201800501);重庆市研究生教育教学改革研究项目(yjg203063)
详细信息
    作者简介:

    吴迪(1997—),女,硕士(E-mail:1284526709@qq.com

    李小林(1983—),男,教授(通讯作者。 E-mail:lxlmath@163.com

  • 中图分类号: O241.82

Element-Free Galerkin Method for Time Fractional Diffusion-Wave Equations

  • 摘要: 利用无单元Galerkin法,对Caputo意义下的时间分数阶扩散波方程进行了数值求解和相应误差理论分析。首先用L1逼近公式离散该方程中的时间变量,将时间分数阶扩散波方程转化成与时间无关的整数阶微分方程;然后采用罚函数方法处理Dirichlet边界条件,并利用无单元Galerkin法离散整数阶微分方程;最后推导该方程无单元Galerkin法的误差估计公式。数值算例证明了该方法的精度和效果。
  • 图  1  算例1在$ \alpha = 1.65 $$ h = {1 \mathord{\left/ {\vphantom {1 {50}}} \right. } {50}} $$ \tau {\text{ = }}{1 \mathord{\left/ {\vphantom {1 {40}}} \right. } {40}} $时的近似解和误差:(a) 近似解;(b) 误差

    Figure  1.  Graphs of approximate solution andresulting error with $ \alpha = 1.65 $, $ h = {1 \mathord{\left/ {\vphantom {1 {50}}} \right. } {50}} $ and $ \tau {\text{ = }}{1 \mathord{\left/ {\vphantom {1 {40}}} \right. } {40}} $ for example 1: (a) the approximate solution; (b) the resulting error

    图  2  (a)常数罚因子$\beta $和(b)变化罚因子$ \beta = {C_\beta }{h^{ - 2}} $对误差和收敛性的影响:(a) 常数罚因子$\beta $;(b) 变化罚因子$ \beta = {C_\beta }{h^{ - 2}} $

    Figure  2.  Error and convergence for fixed penalty factor $\beta $ and variable penalty factors $ \beta = {C_\beta }{h^{ - 2}} $: (a) the fixed penalty factor $\beta $; (b) the variable penalty factors $ \beta = {C_\beta }{h^{ - 2}} $

    图  3  算例1关于时间步长$\tau $和空间步长$h$的收敛性:(a) 时间步长$\tau $;(b) 空间步长$h$

    Figure  3.  Convergence with respect to the time step $\tau $and the space step $h$ for example 1: (a) the time step $\tau $; (b) the space step $h$

    图  4  影响域参数Sscale对误差的影响

    Figure  4.  Influence of the parameter Sscale of the influence domain on error

    图  5  算例2关于时间步长$\tau $和空间步长$h$的收敛性:(a) 时间步长$\tau $;(b) 空间步长$h$

    Figure  5.  Convergence with respect to the time step $\tau $and the space step $h$ for example 2: (a) the time step $\tau $; (b) the space step $h$

    表  1  比较有限元法和无单元Galerkin法的$ {L_\infty } $误差

    Table  1.   Comparison of $\; {L_\infty } \;$ error between the finite element and element-free Galerkin methods

    $\alpha {\text{ = }}1.25$$\alpha {\text{ = }}1.75$
    FEMEFGFEMEFG
    $h = \tau = {1 / 4}$$5.168{\text{ }}8 \times {10^{ - 3}}$$1.872{\text{ }}1 \times {10^{ - 3}}$$1.088{\text{ }}5 \times {10^{ - 2}}$$1.123{\text{ }}6 \times {10^{ - 2}}$
    $h = {1 / 8}$,$\tau {\text{ = }}{1 / {64}}$$2.261{\text{ }}6 \times {10^{ - 3}}$$2.010{\text{ }}6 \times {10^{ - 4}}$$2.390{\text{ }}4 \times {10^{ - 3}}$$3.343{\text{ }}6 \times {10^{ - 4}}$
    $h = {1 /{16}}$,$\tau {\text{ = }}{1 / {1{\text{ }}024}}$$6.744{\text{ }}8 \times {10^{ - 4}}$$4.078{\text{ }}5 \times {10^{ - 5}}$$6.769{\text{ }}3 \times {10^{ - 4}}$$4.078{\text{ }}5 \times {10^{ - 5}}$
    $h = \tau = {1 \mathord{\left/ {\vphantom {1 8}} \right. } 8}$$2.479{\text{ }}5 \times {10^{ - 3}}$$3.926{\text{ }}4 \times {10^{ - 4}}$$6.368{\text{ }}5 \times {10^{ - 3}}$$5.180{\text{ }}5 \times {10^{ - 3}}$
    $h = {1 /{16}}$,$\tau {\text{ = }}{1 / {128}}$$6.752{\text{ }}0 \times {10^{ - 4}}$$4.078{\text{ }}5 \times {10^{ - 5}}$$7.099{\text{ }}6 \times {10^{ - 4}}$$1.507{\text{ }}3 \times {10^{ - 4}}$
    下载: 导出CSV

    表  2  比较$ h = {{\text{π}} / {40}} $时三种方法的${L_2}$误差

    Table  2.   Compare of ${L_2}$ errors obtained from three numerical methods with $ h = {{\text{π}} / {40}} $

    $ \tau $Galerkin FEMADI FEMEFG
    ${1 / 4}$$2.519{\text{ }}0 \times {10^{ - 1}}$$5.451{\text{ }}2 \times {10^{ - 2}}$$1.898{\text{ }}5 \times {10^{ - 2}}$
    ${1 / 8}$$ 2.614{\text{ }}5 \times {10^{ - 2}} $$1.691{\text{ }}1 \times {10^{ - 2}}$$1.179{\text{ }}1 \times {10^{ - 3}}$
    ${1 / {16}}$$8.581{\text{ }}7 \times {10^{ - 3}}$$5.536{\text{ }}5 \times {10^{ - 3}}$$7.670{\text{ }}7 \times {10^{ - 4}}$
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-06-23
  • 录用日期:  2021-06-23
  • 修回日期:  2021-08-27
  • 网络出版日期:  2022-01-07

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