High Accuracy Numerical Solution of Wave Equation

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

Abstract

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

CLC Number:

O469

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

Figures/Tables 13

References 26

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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

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

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

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

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