二维Helmholtz方程的改进六阶紧致差分法

王肇君; 吴亭亭; 曾泰山

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

王肇君¹,
吴亭亭^1, ,,
曾泰山^{2, 3}

1.
山东师范大学数学与统计学院, 济南 250358
2.
华南师范大学数学科学学院, 广州 510631
3.
广东省大数据分析与处理重点实验室, 广州 510006

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
计量
- 文章访问数: 103
- HTML全文浏览量: 38
- PDF下载量: 44
- 被引次数: 0
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

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

1.
School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
2.
School of Mathematics, South China Normal University, Guangzhou 510631, China
3.
Guangdong Key Laboratory of Big Data Analysis and Processing, Guangzhou 510006, China

摘要

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。
- Helmholtz方程 /
- 紧致差分格式 /
- 数值频散
Abstract: To obtain a finite difference scheme with high accuracy for solving the 2D Helmholtz equation, an improved compact sixth-order difference scheme is constructed. Firstly, the truncation error of a compact sixth-order difference scheme with optimal parameters is presented. Then, some terms of the truncation error are approximated with second-order compact formulas to obtain an improved compact difference scheme. Next, the convergence analysis of the improved compact difference scheme is given, and it is proved that the proposed scheme enjoys sixth-order convergence. Based on minimizing the numerical dispersion, a refined choice strategy is proposed for choosing weight parameters. Compared with the compact sixth-order difference scheme with optimal parameters, numerical experiments show that, the numerical accuracy of the improved compact sixth-order difference scheme has been significantly improved, and the scheme's error is less dependent on the wavenumber k.
- Helmholtz equation /
- compact finite difference scheme /
- numerical dispersion

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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

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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

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表 1 加细的优化参数

Table 1. The refined optimal parameters

I_G	[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

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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

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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

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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

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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

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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

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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

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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

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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

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