波动方程的高精度数值解方法

doi:10.19596/j.cnki.1001-246x.8732

摘要/Abstract

摘要：

本文提出重心Lagrange插值配点法求解(2+1)维波动方程和(3+1)维波动方程。介绍了重心Lagrange插值法并且给出配点法的矩阵格式。波动方程的解函数和初边值条件均用Lagrange插值近似，利用配点法得到离散方程，获得波动方程的矩阵表达式。分别用附加法和置换法施加波动方程的初边值条件。数值算例表明：重心Lagrange插值配点法求解波动方程具有较高的计算精度和计算效率。

关键词: 重心Lagrange插值, 波动方程, 配点法, 矩阵格式

Abstract:

A barycentric Lagrange interpolation collocation method is proposed to solve the three-dimensional and four-dimensional wave equations. Firstly, the barycentric Lagrange interpolation method is introduced and the matrix format of the collocation method is given. Secondly, the solution function and initial boundary conditions of the wave equation are approximated by Lagrange interpolation. The discrete equation is obtained by collocation method, and the matrix expression of the wave equation is obtained. Finally, the initial and boundary conditions of the wave equation are imposed by the addition method and the replacement method respectively. Numerical examples show that the barycentric Lagrange interpolation collocation method has high computational accuracy and efficiency.

Key words: barycentric Lagrange interpolation, wave equation, collocation method, matrix scheme

中图分类号:

O469

袁洪旺, 王希胤, 李金. 波动方程的高精度数值解方法[J]. 计算物理, 2024, 41(4): 426-439.

Hongwang YUAN, Xiyin WANG, Jin LI. High Accuracy Numerical Solution of Wave Equation[J]. Chinese Journal of Computational Physics, 2024, 41(4): 426-439.

图/表 13

表1 不同节点数边界条件施加方法为附加法与置换法的绝对误差、相对误差和收敛阶

Table 1 Absolute error, relative error and the order of convergence of additional method and displacement method

节点数m=n=h	附加法			置换法
节点数m=n=h	E_a	rate	E_r	E_a	rate	E_r
9	9.704 1×10^-3		1.450 6×10^-3	9.801 2×10^-3		1.465 1×10^-3
11	1.021 9×10^-4	6.56	1.141 1×10^-5	2.373 0×10^-4	5.37	2.649 9×10^-5
13	9.277 3×10^-7	6.78	8.116 6×10^-8	3.526 0×10^-6	6.07	3.084 9×10^-7
15	8.489 6×10^-9	6.77	6.021 9×10^-10	3.623 9×10^-8	6.60	2.570 5×10^-9
17	5.010 3×10^-11	7.40	2.956 7×10^-12	1.781 3×10^-8	1.02	1.051 2×10^-9
19	3.873 4×10^-11	0.37	1.940 3×10^-12	2.787 4×10^-8	-0.6	1.396 3×10^-9

表2 三点中心差分法在t=1时刻的绝对误差、相对误差和收敛阶

Table 2 Absolute error, relative error and the order of convergence of three center difference method at t=1

节点数m=n=$\sqrt{h}$	三点中心差分法
节点数m=n=$\sqrt{h}$	E_a	rate	E_r
4	2.087 7×10^-2		1.443 9×10^-2
9	5.098 7×10^-3	2.03	1.322 4×10^-3
18	2.541 7×10^-3	1.00	3.102 2×10^-4
36	1.270 9×10^-3	1.00	7.534 0×10^-5
64	7.118 6×10^-4	0.84	2.344 5×10^-5

图1 当m = n = h = 19，边界条件施加方法为附加法，h = 10时的(a)数值解和(b)解析解

Fig.1 (a)Numerical solution and (b) analytical solution of additional method at h = 10 with m = n = h = 19

图2 当m = n = h = 19，边界条件施加方法为附加法，h = 10的误差

Fig.2 Domain error of additional method at h = 10 with m = n = h = 19

图3 当m = n = h = 19，边界条件施加方法为置换法，h = 10的(a)数值解和(b)解析解

Fig.3 (a)Numerical solution and (b) analytical solution of displacement method at h=10 with m = n = h = 19

图4 当m = n = h = 19，边界条件施加方法为置换法，h = 10的误差

Fig.4 Domain error of displacement method at h = 10 with m = n = h = 19

表3 不同节点数边界条件施加方法为附加法与置换法的绝对误差、相对误差和收敛阶

Table 3 Absolute error, relative error and the order of convergence of additional method and displacement method

节点数m=n=h	附加法			置换法
节点数m=n=h	E_a	rate	E_r	E_a	rate	E_r
9	6.718 7×10^-4		2.157 3×10^-5	7.267 6×10^-4		2.333 6×10^-5
11	3.321 9×10^-6	7.66	8.174 0×10^-8	8.275 7×10^-6	6.46	2.036 4×10^-7
13	1.856 5×10^-8	7.48	3.644 2×10^-10	6.241 0×10^-8	7.05	1.225 1×10^-9
15	1.077 7×10^-10	7.43	1.738 3×10^-12	4.258 6×10^-8	0.55	6.868 9×10^-10
17	1.227 3×10^-10	-0.2	1.664 1×10^-12	1.977 4×10^-7	-2.2	2.681 2×10^-9
19	3.116 7×10^-10	-1.3	3.616 9×10^-12	1.413 5×10^-7	0.50	1.640 4×10^-9

图5 当m = n = h = 19，边界条件施加方法为附加法，h = 1的(a)数值解和(b)解析解

Fig.5 (a)Numerical solution and (b) analytical solution of additional method at h = 1 with m = n = h = 19

图6 当m = n = h = 19，边界条件施加方法为附加法，h = 1时的误差

Fig.6 Domain error at h = 1 with additional method and m = n = h = 19

图7 当m = n = h = 15，边界条件施加方法为置换法，h = 1的(a)数值解和(b)解析解

Fig.7 (a) Numerical solution and (b) analytical solution of displacement method at h = 1 with m = n = h = 15

图8 当m = n = h = 15，边界条件施加方法为置换法，h = 1时刻的误差

Fig.8 Domain error of displacement method at h = 1 with m = n = h = 15

表4 不同节点数边界条件施加方法为附加法与置换法的绝对误差、相对误差和收敛阶

Table 4 Absolute error, relative error and the order of convergence with additional method and displacement method

节点数m=n=s=h	附加法			置换法
节点数m=n=s=h	E_a	rate	E_r	E_a	rate	E_r
7	5.578 9×10^-1		7.490 9×10^-2	5.069 6×10^-2		6.807 1×10^-3
9	5.774 6×10^-2	3.27	4.728 8×10^-3	2.031 4×10^-2	1.32	1.663 5×10^-3
11	1.259 3×10^-3	5.52	6.939 6×10^-5	1.143 6×10^-3	4.15	6.302 0×10^-5
13	2.137 9×10^-5	5.88	8.465 9×10^-7	3.270 5×10^-5	5.13	1.295 1×10^-6
15	2.961 0×10^-7	6.17	8.830 6×10^-9	5.837 7×10^-7	5.81	1.741 0×10^-8
17	2.488 4×10^-9	6.89	5.789 9×10^-11	7.225 8×10^-9	6.34	1.681 2×10^-10

表5 三点中心差分法在t=1时刻的绝对误差、相对误差和收敛阶

Table 5 Time absolute error, relative error and the order of convergence with three center difference method at t=1

节点数m=n=s=$\sqrt{h}$	三点中心差分法
节点数m=n=s=$\sqrt{h}$	E_a	rate	E_r
4	3.349 9×10^-1		2.441 2×10^-1
9	2.202 1×10^-1	0.61	3.690 7×10^-2
17	1.579 5×10^-1	0.48	9.359 0×10^-3
31	1.158 1×10^-1	0.45	2.672 8×10^-3
62	8.132 4×10^-2	0.51	6.473 3×10^-4

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