非线性波动方程的高效保能量数值算法

谢建强; 汪灿

doi:10.21656/1000-0887.460090

非线性波动方程的高效保能量数值算法

doi: 10.21656/1000-0887.460090

谢建强^,,
汪灿

安徽大学数学科学学院，合肥 230601

基金项目:

国家自然科学基金 12201005

详细信息

通讯作者:
谢建强(1988—)，男，副教授，博士，硕士生导师(通信作者. E-mail: xiejq1025@163.com)

中图分类号: O241.82
计量
- 文章访问数: 90
- HTML全文浏览量: 32
- PDF下载量: 21
- 被引次数: 0
出版历程
- 收稿日期: 2025-05-06
- 修回日期: 2025-06-17
- 刊出日期: 2026-01-01

An Efficient Energy-Preserving Numerical Algorithm for Nonlinear Wave Equations

XIE Jianqiang^,,
WANG Can

School of Mathematical Sciences, Anhui University, Hefei 230601, P.R.China

摘要

摘要: 将降阶法、Lagrange乘子方法和紧致差分法相结合，对非线性波动方程建立一个时间二阶和空间四阶收敛精度的保能量数值算法，证明所提算法保持原始能量守恒性质，并给出相应算法的计算步骤.数值算例验证所提算法的正确性和有效性.
- 非线性波动方程 /
- 降阶法 /
- Lagrange乘子方法 /
- 紧致差分法 /
- 保能量数值算法
Abstract: An energy-preserving numerical algorithm, which is 2nd-order in time and 4th-order in space, for nonlinear wave equations was developed based on the order reduction method, the Lagrange multiplier method and the compact difference method. The discrete original energy conservation property of the suggested algorithm was proven. The computational procedure of the associated algorithm was exhibited. Numerical results validate the exactness and effectiveness of the proposed algorithm.
- nonlinear wave equation /
- order reduction method /
- Lagrange multiplier method /
- compact finite difference method /
- energy-preserving numerical algorithm

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图 1 不同步长数值解的误差曲面(μ=0.5)

Figure 1. The error surfaces of numerical solution with different steps (μ=0.5)

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图 2 不同步长数值解的误差曲线(μ=0.5)

Figure 2. The error curves of numerical solution with different steps (μ=0.5)

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图 3 η^k的误差曲线

Figure 3. The curve of errorsη^k

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图 4 t=100时能量的相对误差曲线

Figure 4. The curve of relative errors of energy at t=100

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图 5 t=500时能量的相对误差曲线

Figure 5. The curve of relative errors of energy at t=500

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图 6 t=1 000时能量的相对误差曲线

Figure 6. The curve of relative errors of energy at t=1 000

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图 7 均匀介质中的波传播

Figure 7. The propagation of wave in the homogenous medium

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图 8 双层介质中的波传播

Figure 8. The propagation of wave in 2-layer media

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图 9 三层介质中的波传播

Figure 9. The propagation of wave in the 3-layer media

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算法差分格式(16)—(20)
输入 u⁰, v⁰, β, a, b, m, n, T
输出 u, v, η
1 for k=0: n－1 do
2 通过式(27)—(28)计算p^k+1, q^k+1
3 使用Newton迭代法求解式(29)，得到η^k+1
4 将η^k+1代入式(26)，计算得到u^k+1
5 由式(25)计算得v^k+1
6 将u^k+1, v^k+1, η^k+1分别保存到u, v, η
7 end for
8 return u, v, η

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表 1 取不同步长时数值解的误差和空间收敛阶(τ=h²)

Table 1. Errors and spatial convergence orders of the solution with different steps (τ=h²)

		h=1/4	h=1/8	h=1/16	h=1/32
μ=0.3	M_u	1.38E-3	1.32E-4	8.73E-6	5.51E-7
	R_{l^∞, h}	-	3.39	3.91	3.99
	L_u	2.98E-3	2.89E-4	1.92E-5	1.21E-6
	R_{l², h}	-	3.37	3.91	3.99
	H_u	4.12E-3	4.00E-4	2.66E-5	1.68E-6
	R_{H¹, h}	-	3.36	3.91	3.99
μ=0.5	M_u	1.12E-3	7.60E-5	4.83E-6	3.02E-7
	R_{l^∞, h}	-	3.89	3.98	4.00
	L_u	2.51E-3	1.68E-4	1.06E-5	6.67E-7
	R_{l², h}	-	3.90	3.98	4.00
	H_u	3.54E-3	2.37E-4	1.50E-5	9.43E-7
	R_{H¹, h}	-	3.90	3.98	4.00
μ=0.7	M_u	6.77E-4	4.48E-5	2.84E-6	1.78E-7
	R_{l^∞, h}	-	3.92	3.98	3.99
	L_u	1.54E-3	1.02E-4	6.47E-6	4.06E-7
	R_{l², h}	-	3.92	3.98	3.99
	H_u	2.08E-3	1.38E-4	8.76E-6	5.50E-7
	R_{H¹, h}	-	3.91	3.98	3.99

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表 2 取不同步长时数值解的误差和时间收敛阶(τ=h)

Table 2. Errors and temporal convergence orders of the solution with different steps (τ=h)

		h=1/20	h=1/40	h=1/80	h=1/160
μ=0.3	M_u	9.71E-4	3.08E-4	8.50E-5	2.19E-5
	R_{l^∞, τ}	-	1.66	1.86	1.95
	L_u	2.16E-3	6.89E-4	1.90E-4	4.91E-5
	R_{l², τ}	-	1.65	1.86	1.95
	H_u	3.03E-3	9.64E-4	2.66E-4	6.87E-5
	R_{H¹, τ}	-	1.65	1.86	1.95
μ=0.5	M_u	7.48E-4	1.95E-4	4.96E-5	1.25E-5
	R_{l^∞, τ}	-	1.94	1.97	1.99
	L_u	1.62E-3	4.21E-4	1.07E-4	2.69E-5
	R_{l², τ}	-	1.95	1.98	1.99
	H_u	2.28E-3	5.90E-4	1.50E-4	3.78E-5
	R_{H¹, τ}	-	1.95	1.98	1.99
μ=0.7	M_u	4.41E-4	1.13E-4	2.87E-5	7.23E-6
	R_{l^∞, τ}	-	1.96	1.98	1.99
	L_u	9.94E-4	2.56E-4	6.50E-5	1.64E-5
	R_{l², τ}	-	1.96	1.98	1.99
	H_u	1.34E-3	3.45E-4	8.76E-5	2.21E-5
	R_{H¹, τ}	-	1.95	1.98	1.99

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表 3 不同数值方法的最大误差和CPU时间(τ=1/600)

Table 3. Maximum errors and CPU times of different numerical methods (τ=1/600)

method	h	1/2	1/4	1/8	1/16
multi-symplectic leap-frog scheme^[5]	M_u	1.902 359E-2	4.763 517E-3	1.207 190E-3	3.020 248E-4
multi-symplectic leap-frog scheme^[5]	CPU time	0.001 1	0.002 5	0.005 4	0.008 7
local energy-preserving scheme^[15]	M_u	1.902 355E-2	4.763 503E-3	1.207 182E-3	3.020 184E-4
local energy-preserving scheme^[15]	CPU time	0.071 7	0.309 2	1.659 8	8.765 4
Du Fort-Frankel scheme^[17]	M_u	1.902 337E-2	4.763 032E-3	1.205 583E-3	3.045 466E-4
Du Fort-Frankel scheme^[17]	CPU time	0.002 5	0.005 7	0.010 0	0.017 0
energy-preserving nonlinear difference scheme^[6]	M_u	1.902 357E-2	4.763 510E-3	1.207 189E-3	3.020 245E-4
energy-preserving nonlinear difference scheme^[6]	CPU time	0.002 2	0.005 1	0.009 8	0.016 1
our scheme (16)—(20)	M_u	1.364 679E-3	7.886 652E-5	4.756 628E-6	2.134 643E-7
our scheme (16)—(20)	CPU time	0.014 2	0.033 7	0.365 1	1.151 1

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