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二维Helmholtz方程的改进六阶紧致差分法

王肇君 吴亭亭 曾泰山

王肇君, 吴亭亭, 曾泰山. 二维Helmholtz方程的改进六阶紧致差分法[J]. 华南师范大学学报(自然科学版), 2022, 54(2): 90-100. doi: 10.6054/j.jscnun.2022031
引用本文: 王肇君, 吴亭亭, 曾泰山. 二维Helmholtz方程的改进六阶紧致差分法[J]. 华南师范大学学报(自然科学版), 2022, 54(2): 90-100. doi: 10.6054/j.jscnun.2022031
WANG Zhaojun, WU Tingting, ZENG Taishan. An Improved Compact Sixth-order Finite Difference Scheme for the 2D Helmholtz Equation[J]. Journal of South China normal University (Natural Science Edition), 2022, 54(2): 90-100. doi: 10.6054/j.jscnun.2022031
Citation: WANG Zhaojun, WU Tingting, ZENG Taishan. An Improved Compact Sixth-order Finite Difference Scheme for the 2D Helmholtz Equation[J]. Journal of South China normal University (Natural Science Edition), 2022, 54(2): 90-100. doi: 10.6054/j.jscnun.2022031

二维Helmholtz方程的改进六阶紧致差分法

doi: 10.6054/j.jscnun.2022031
基金项目: 

山东省自然科学基金项目 ZR2021MA049

山东省自然科学基金项目 ZR2020MA031

广东省自然科学基金项目 2018A0303130067

中山大学广东省计算科学重点实验室开放课题 2021022

广东省大数据分析与处理重点实验室开放基金项目 202101

详细信息
    通讯作者:

    吴亭亭, Email: tingtingwu@sdnu.edu.cn

  • 中图分类号: O241.82

An Improved Compact Sixth-order Finite Difference Scheme for the 2D Helmholtz Equation

  • 摘要: 为得到求解二维Helmholtz方程的高精度差分法, 构造了一种改进六阶紧致差分格式: 首先, 给出一种带优化参数的六阶紧致差分格式的截断误差; 然后, 对此截断误差的部分项进行二阶紧致逼近, 得到一种改进紧致差分格式; 其次, 对该格式进行了收敛性分析, 证明其为六阶收敛的; 最后, 基于极小化数值频散的思想, 给出该格式优化参数的加细选取策略。与带优化参数的六阶紧致差分格式相比, 数值实验说明改进六阶紧致差分格式的数值精度有了显著提高, 且其误差对波数k的依赖性更低。
  • 图  1  问题1的离散L2- 范数误差曲线

    Figure  1.  The curves of error in the discrete L2- norm for Problem 1

    图  2  问题2的离散L2- 范数误差曲线

    Figure  2.  The curves of error in the discrete L2- norm for Problem 2

    表  1  加细的优化参数

    Table  1.   The refined optimal parameters

    IG [2.5, 3] [3, 4] [4, 5] [5, 6] [6, 8] [8, 10] [10, 400]
    d 0 0 0 0 0 0 0
    e 0.007 2 0.006 0 0.005 1 0.004 7 0.004 4 0.004 0 0.003 6
    下载: 导出CSV

    表  2  a=1, k=24时的离散L2- 范数误差

    Table  2.   The error in the discrete L2- norm for a=1, k=24

    差分格式 N=32 N=64 N=128 N=256 N=512
    OC6 6.296 6 0.007 5 8.868 6e-05 1.420 9e-06 2.228 6e-08
    IC6 1.748 9 8.063 2e-04 6.458 7e-07 3.265 9e-09 3.380 7e-11
    下载: 导出CSV

    表  3  a=3, k=24时的离散L2- 范数误差

    Table  3.   The error in the discrete L2- norm for a=3, k=24

    差分格式 N=32 N=64 N=128 N=256 N=512
    OC6 0.179 7 9.924 7e-04 1.187 0e-05 1.882 4e-07 2.946 1e-09
    IC6 0.047 2 3.915 6e-05 1.777 8e-07 4.731 6e-09 8.216 3e-11
    下载: 导出CSV

    表  4  a=3, k=48时的离散L2- 范数误差

    Table  4.   The error in the discrete L2- norm for a=3, k=48

    差分格式 N=64 N=128 N=256 N=512 N=1 024
    OC6 0.892 1 0.003 4 4.037 8e-05 6.460 6e-07 1.013 0e-08
    IC6 0.430 1 3.161 0e-04 2.062 9e-07 1.952 6e-09 2.944 4e-11
    下载: 导出CSV

    表  5  a=1, N=256时的离散L2- 范数误差

    Table  5.   The error in the discrete L2- norm for a=1, N=256

    差分格式 N=30 N=40 N=50 N=60 N=70
    OC6 8.384 7e-06 8.254 1e-05 4.833 3e-04 0.002 4 0.009 2
    IC6 2.930 9e-08 5.287 1e-07 4.874 2e-06 6.508 4e-05 4.717 0e-04
    下载: 导出CSV

    表  6  β=100, k=100.005 0时的离散L2- 范数误差

    Table  6.   The error in the discrete L2- norm for β=100, k=100.005 0

    差分格式 N=128 N=256 N=512 N=700 N=1 024
    OC6 0.287 1 0.002 5 3.058 3e-05 4.647 6e-06 4.747 8e-07
    IC6 0.451 0 0.001 4 1.784 5e-05 2.634 9e-06 2.638 7e-07
    下载: 导出CSV

    表  7  β=199.997 5, k=200时的离散L2- 范数误差

    Table  7.   The error in the discrete L2- norm for β=199.997 5, k=200

    差分格式 N=512 N=700 N=800 N=900 N=1 024
    OC6 0.004 3 6.465 8e-04 2.947 2e-04 1.294 6e-04 5.252 6e-05
    IC6 0.001 5 2.168 9e-04 8.918 0e-05 4.167 2e-05 1.842 6e-05
    下载: 导出CSV

    表  8  β=499.999 0, k=500时的离散L2- 范数误差

    Table  8.   The error in the discrete L2- norm for β=499.999 0, k=500

    差分格式 N=700 N=800 N=900 N=1 000 N=1 100
    OC6 1.486 5 0.307 5 0.199 1 0.113 1 0.030 0
    IC6 1.050 7 0.268 7 0.086 5 0.033 1 0.014 3
    下载: 导出CSV

    表  9  N=700时的离散L2- 范数误差

    Table  9.   The error in the discrete L2- norm for N=700

    差分格式 N=100 N=200 N=300 N=400 N=450
    OC6 4.643 9e-06 6.465 8e-04 0.013 6 0.182 7 0.281 2
    IC6 2.648 7e-06 2.168 9e-04 0.005 6 0.091 1 0.163 3
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-09-02
  • 网络出版日期:  2022-05-12
  • 刊出日期:  2022-04-25

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