KdV-Burgers方程的一类新本性并行差分格式

潘悦悦; 杨晓忠

doi:10.21656/1000-0887.430128

KdV-Burgers方程的一类新本性并行差分格式

doi: 10.21656/1000-0887.430128

潘悦悦^1,,
杨晓忠^2, ,

1.
华北电力大学控制与计算机工程学院, 北京 102206
2.
华北电力大学数理学院信息与计算研究所, 北京 102206

基金项目:

国家自然科学基金项目 11371135

详细信息

作者简介:
潘悦悦(1995—)，女，博士生(E-mail: panyueyue@ncepu.edu.cn)

通讯作者:
杨晓忠(1965—)，男，教授，博士生导师(通讯作者. E-mail: yxiaozh@ncepu.edu.cn)

中图分类号: O241.8
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- 文章访问数: 454
- HTML全文浏览量: 139
- PDF下载量: 72
- 被引次数: 0
出版历程
- 收稿日期: 2022-04-11
- 修回日期: 2022-06-14
- 刊出日期: 2023-05-01

New Class of Difference Schemes With Intrinsic Parallelism for the KdV-Burgers Equation

PAN Yueyue^1
,,
YANG Xiaozhong^{2
, ,}

1.
School of Control and Computer Engineering, North China Electric Power University, Beijing 102206, P.R.China
2.
Institute of Information and Computing, School of Mathematics and Physics, North China Electric Power University, Beijing 102206, P.R.China

摘要

摘要: KdV-Burgers方程作为湍流规范方程，具有深刻的物理背景，其快速数值解法具有重要的实际应用价值. 针对KdV-Burgers方程，提出了一种新型的并行差分格式. 基于交替分段技术，结合经典Crank-Nicolson(C-N)格式、显格式和隐格式，构造了混合交替分段Crank-Nicolson(MASC-N)差分格式. 理论分析表明MASC-N格式是唯一可解、线性绝对稳定和二阶收敛的. 数值试验表明，MASC-N格式比C-N格式具有更高的精度和效率. 与ASE-I和ASC-N差分格式相比，MASC-N并行差分格式有最好的性能. 表明该文的MASC-N并行差分方法能有效地求解KdV-Burgers方程.
- KdV-Burgers方程 /
- MASC-N并行差分格式 /
- 线性绝对稳定性 /
- 收敛性 /
- 数值试验
Abstract: The KdV-Burgers equation as a standard equation for turbulent, has a profound physical background and its fast numerical methods are of great practical application value. A new class of parallel difference schemes were proposed for the KdV-Burgers equation. Based on the alternating segment technology, the mixed alternating segment Crank-Nicolson (MASC-N) difference scheme was constructed with the classic Crank-Nicolson (C-N) scheme, the explicit and implicit schemes. The theoretical analyses indicate that, the MASC-N scheme is uniquely solvable, linearly absolutely stable and 2nd-order convergent. Numerical experiments show that, the MASC-N scheme has higher precision and efficiency than the C-N scheme. Compared with the ASE-I and ASC-N difference schemes, the MASC-N parallel difference scheme has the best performance, and can effectively solve the KdV-Burgers equation.
- KdV-Burgers equation /
- MASC-N parallel difference scheme /
- linear absolute stability /
- convergence /
- numerical experiment

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图 1 MASC-N格式构造示意图

Figure 1. Structural schematic diagram of the MASC-N scheme

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图 2 精确解的波形

Figure 2. The waveform of the exact solution

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图 3 MASC-N格式解的波形

Figure 3. The waveform of the MASC-N scheme solution

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图 4 C-N和MASC-N格式的计算时间

Figure 4. The computing time of the C-N and MASC-N schemes

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图 5 不同CPU核数的MASC-N格式计算时间

Figure 5. The computing time of the MASC-N scheme with different numbers of CPU cores

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图 6 C-N和MASC-N格式解的SRET

Figure 6. SRET of the C-N and MASC-N scheme solutions

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图 7 C-N和MASC-N格式解的DTE

Figure 7. DTE of the C-N and MASC-N scheme solutions

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图 8 3种并行格式计算时间比较

Figure 8. Comparison on the computing time of the 3 parallel schemes

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图 9 C-N格式解的绝对误差

Figure 9. The absolute error of the C-N scheme solution

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图 10 MASC-N格式解的绝对误差

Figure 10. The absolute error of the MASC-N scheme solution

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图 11 C-N和MASC-N格式的运行时间

Figure 11. The running time of the C-N and MASC-N schemes

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表 1 两种格式解相对于解析解的绝对误差

Table 1. The absolute errors of the 2 scheme solutions relative to the analytic solution

(x, t)	analytic solution	C-N	Δ_AE	MASC-N	Δ_AE
(-40, 0.1)	-5.45×10^-8	-5.51×10^-8	5.60×10^-10	-5.50×10^-8	5.37×10^-10
(-30, 0.2)	-2.99×10^-6	-3.05×10^-6	6.16×10^-8	-3.05×10^-6	5.92×10^-8
(-20, 0.3)	-1.60×10^-4	-1.65×10^-4	4.88×10^-6	-1.64×10^-4	4.69×10^-6
(-10, 0.4)	-0.007 055	-0.007 304	0.000 250	-0.007 295	0.000 240
(0, 0.5)	-0.122 897	-0.124 388	0.001 491	-0.124 268	0.001 370
(10, 0.6)	-0.374 919	-0.373 647	0.001 272	-0.373 542	0.001 377
(20, 0.7)	-0.463 439	-0.462 955	0.000 484	-0.462 932	0.000 507
(30, 0.8)	-0.477 718	-0.477 634	8.40×10^-5	-0.477 630	8.78×10^-5
(40, 0.9)	-0.479 692	-0.479 679	1.30×10^-5	-0.479 678	1.36×10^-5

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表 2 两种格式的空间收敛阶

Table 2. The space-convergent orders of the 2 schemes

M	C-N		MASC-N
M	E_∞	S	E_∞	S
25	6.572 000×10^-3	-	7.528 900×10^-3	-
50	1.751 500×10^-3	1.907 741	1.783 904×10^-3	2.077 401
100	4.411 802×10^-4	1.989 151	4.416 614×10^-4	2.014 025
200	1.085 822×10^-4	2.022 580	1.066 623×10^-4	2.049 890
400	2.678 405×10^-5	2.019 342	2.538 697×10^-5	2.070 890

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表 3 两种格式的时间收敛阶

Table 3. The time-convergent orders of the 2 schemes

N	C-N		MASC-N
N	E_∞	T	E_∞	T
100	2.741 022×10^-3	-	2.741 022×10^-3	-
200	6.699 166×10^-4	2.032 660	6.697 803×10^-4	2.032 954
400	1.644 313×10^-4	2.026 496	1.644 313×10^-4	2.026 203
800	4.214 523×10^-5	1.964 044	4.214 494×10^-5	1.964 053
1 600	1.028 303×10^-5	2.035 104	1.028 305×10^-5	2.035 092

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表 4 两种格式运行时间的比较

Table 4. Comparison of running time between the 2 schemes

M	600	1 100	1 600	2 100	2 600
C-N	0.662 88	2.026 66	4.271 80	7.667 68	12.347 7
MASC-N	0.334 65	0.993 64	1.929 38	3.283 17	5.250 76
S_p	1.980 82	2.039 63	2.214 08	2.335 45	2.351 60
E_p	0.247 60	0.254 95	0.276 76	0.291 93	0.293 95

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表 5 3种并行格式的计算时间

Table 5. The computing time of the 3 parallel schemes

M	210	410	610	810	1 010	1 210	1 410
ASE-I^[9]	0.342 03	1.052 26	5.801 74	10.160 8	15.065 7	20.828 3	27.885 5
ASC-N^[11]	0.352 32	1.107 77	6.415 85	10.343 9	15.213 7	21.224 9	28.193 6
MASC-N	0.350 60	1.090 35	5.790 21	9.900 61	14.871 4	20.559 4	28.780 9

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表 6 数值解和解析解

Table 6. Numerical solutions and the analytic solution

x	analytic solution	ASE-I^[9]	ASC-N^[11]	MASC-N
-40	-5.663 4×10^-8	-5.584 7×10^-8	-5.584 7×10^-8	-5.585 8×10^-8
-30	-3.078 6×10^-6	-3.037 0×10^-6	-3.035 4×10^-6	-3.035 7×10^-6
-20	-1.627 7×10^-4	-1.603 2×10^-4	-1.602 3×10^-4	-1.602 4×10^-4
-10	-0.007 114 59	-0.006 941 26	-0.006 939 24	-0.006 939 28
0	-0.122 897 14	-0.118 473 11	-0.118 499 99	-0.118 499 94
10	-0.374 499 97	-0.369 167 25	-0.369 224 65	-0.369 224 71
20	-0.463 283 64	-0.462 168 85	-0.462 181 73	-0.462 181 91
30	-0.477 685 23	-0.477 524 56	-0.477 526 32	-0.477 526 41
40	-0.479 685 75	-0.479 663 82	-0.479 664 03	-0.479 664 07

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表 7 3种并行格式的比较

Table 7. Comparison of the 3 parallel schemes

parallel schemes	precision sort	time sort	weighted sort sum (0.6∶0.4)
ASE-I^[9]	3	2	2.6
ASC-N^[11]	2	3	2.4
MASC-N	1	1	1

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表 8 数值解和解析解

Table 8. Numerical solutions and the analytic solution

x	analytic solution	C-N	Δ_AE	MASC-N	Δ_AE
-40	-0.816 495	-0.816 495	1.15×10^-7	-0.816 495	1.46×10^-7
-30	-0.816 461	-0.816 458	3.22×10^-6	-0.816 457	4.10×10^-6
-20	-0.815 496	-0.815 407	8.99×10^-5	-0.815 382	1.15×10^-4
-10	-0.789 361	-0.787 123	0.002 237	-0.786 488	0.002 873
0	-0.415 808	-0.409 564	0.006 244	-0.405 151	0.010 656
10	-0.029 148	-0.029 735	0.000 588	-0.029 105	4.25×10^-5
20	-0.001 077	-0.001 104	2.70×10^-5	-0.001 080	2.68×10^-6
30	-3.80×10^-5	-3.90×10^-5	9.73×10^-7	-3.90×10^-5	1.01×10^-7
40	-1.37×10^-6	-1.41×10^-6	3.47×10^-8	-1.38×10^-6	3.63×10^-9

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参考文献(27)

[1]	李志斌. 非线性数学物理方程的行波解[M]. 北京: 科学出版社, 2007. LI Zhibin. Traveling Wave Solutions of Nonlinear Mathematical Physical Equations[M]. Beijing: Science Press, 2007. (in Chinese)
[2]	程建春. 数学物理方程及其近似方法[M]. 2版. 北京: 科学出版社, 2017. CHENG Jianchun. Mathematical Physics Equations and Theirs Approximate Methods[M]. 2nd ed. Beijing: Science Press, 2017. (in Chinese)
[3]	WAZWAZ A M. Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations[J]. Applied Mathematics and Computation, 2005, 169(1): 639-656. doi: 10.1016/j.amc.2004.09.081
[4]	CHUGAINOVA A P, SHARGATOV V A. Stability of nonstationary solutions of the generalized KdV-Burgers equation[J]. Computational Mathematics and Mathematical Physics, 2015, 55(2): 251-263. doi: 10.1134/S0965542515020074
[5]	RASHIDINIA J, RASOULIZADEH M N. Numerical methods based on radial basis function generated finite difference (RBF-FD) for solution of GKdVB equation[J]. Wave Motion, 2019, 90: 152-167. doi: 10.1016/j.wavemoti.2019.05.006
[6]	EVANS D J, ABDULLAH A R B. Group explicit methods for parabolic equations[J]. International Journal of Computer Mathematics, 1983, 14(1): 73-105. doi: 10.1080/00207168308803377
[7]	张宝琳, 谷同祥, 莫则尧. 数值并行计算原理与方法[M]. 北京: 国防工业出版社, 1999. ZHANG Baolin, GU Tongxiang, MO Zeyao. Principles and Methods of Numerical Parallel Computation[M]. Beijing: National Defense Industry Press, 1999. (in Chinese)
[8]	周毓麟. 拟线性抛物组具并行本性的差分格式[J]. 中国科学(A辑), 1997, 27(1): 43-48. https://www.cnki.com.cn/Article/CJFDTOTAL-JAXK199701005.htm ZHOU Yulin. Difference schemes with intrinsic parallelism for quasi-linear parabolic systems[J]. Science in China (Series A), 1997, 27(1): 43-48. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-JAXK199701005.htm
[9]	曲富丽, 王文洽. 三阶非线性KdV方程的交替分段显-隐差分格式[J]. 应用数学和力学, 2007, 28(7): 869-876. doi: 10.3321/j.issn:1000-0887.2007.07.014 QU Fuli, WANG Wenqia. Alternating segment explicit-implicit scheme for nonlinear third-order KdV equation[J]. Applied Mathematics and Mechanics, 2007, 28(7): 869-876. (in Chinese) doi: 10.3321/j.issn:1000-0887.2007.07.014
[10]	SHENG Z Q, YUAN G W, HANG X D. Unconditional stability of parallel difference schemes with second order accuracy for parabolic equation[J]. Applied Mathematics and Computation, 2007, 184(2): 1015-1031. doi: 10.1016/j.amc.2006.07.003
[11]	王文洽. KdV方程的一类本性并行差分格式[J]. 应用数学学报, 2006, 29(6): 995-1003. doi: 10.3321/j.issn:0254-3079.2006.06.004 WANG Wenqia. Difference schemes with intrinsic parallelism for the KdV equation[J]. Acta Mathematicae Applicatae Sinica, 2006, 29(6): 995-1003. (in Chinese) doi: 10.3321/j.issn:0254-3079.2006.06.004
[12]	YUAN G W, SHENG Z Q, HANG X D. The unconditional stability of parallel difference schemes with second order convergence for nonlinear parabolic system[J]. Journal of Partial Differential Equations, 2007, 20(1): 45-64. http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=A312265B9157D52B9F01259BD372F6B9?doi=10.1.1.329.9741&rep=rep1&type=pdf
[13]	BORHANIFAR A, ABAZARI R. An unconditionally stable parallel difference scheme for telegraph equation[J]. Mathematical Problems in Engineering, 2009, 2009: 969610. http://www.xueshufan.com/publication/2012168606
[14]	GUO G Y, LIU B. Unconditional stability of alternating difference schemes with intrinsic parallelism for the fourth-order parabolic equation[J]. Applied Mathematics and Computation, 2013, 219(14): 7319-7328. doi: 10.1016/j.amc.2012.12.010
[15]	NAMJOO M, ZEINADINI M, ZIBAEI S. Nonstandard finite-difference scheme to approximate the generalized Burgers-Fisher equation[J]. Mathematical Methods in Applied Sciences, 2018, 41(17): 8212-8228. doi: 10.1002/mma.5283
[16]	XUE G Y, FENG H. An alternating segment explicit-implicit scheme with intrinsic parallelism for Burgers' equation[J]. Journal of Computational and Theoretical Transport, 2020, 49(1): 15-30. doi: 10.1080/23324309.2019.1709081
[17]	袁光伟, 盛志强, 杭旭登, 等. 扩散方程计算方法[M]. 北京: 科学出版社, 2015. YUAN Guangwei, SHENG Zhiqiang, HANG Xudeng, et al. Calculation Methods of Diffusion Equations[M]. Beijing: Science Press, 2015. (in Chinese)
[18]	FERZIGER J H, PERI¢ M, STREET R L. Computational Methods for Fluid Dynamics[M]. Berlin: Springer, 2020.
[19]	XUE G Y, FENG H. New parallel algorithm for convection-dominated diffusion equation[J]. East Asian Journal on Applied Mathematics, 2018, 8(2): 261-279. doi: 10.4208/eajam.240817.201217a
[20]	ZHANG W G, ZHAO Y, TENG X Y. Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates[J]. Acta Mathematicae Applicatae Sinica (English Series), 2012, 28(2): 305-324. doi: 10.1007/s10255-012-0147-5
[21]	ISMAIL H N A, RAGEH T M, SALEM G S E. Modified approximation for the KdV-Burgers equation[J]. Applied Mathematics and Computation, 2014, 234: 58-62. doi: 10.1016/j.amc.2014.01.167
[22]	汪精英, 翟术英. 分数阶Cahn-Hilliard方程的高效数值算法[J]. 应用数学和力学, 2021, 42(8): 832-840. doi: 10.21656/1000-0887.420008 WANG Jingying, ZHAI Shuying. An efficient numerical algorithm for fractional Cahn-Hilliard equations[J]. Applied Mathematics and Mechanics, 2021, 42(8): 832-840. (in Chinese) doi: 10.21656/1000-0887.420008
[23]	魏剑英, 葛永斌. 一种求解三维非稳态对流扩散反应方程的高精度有限差分格式[J]. 应用数学和力学, 2022, 43(2): 187-197. doi: 10.21656/1000-0887.420151 WEI Jianying, GE Yongbin. A high-order finite difference scheme for 3D unsteady convection diffusion reaction equations[J]. Applied Mathematics and Mechanics, 2022, 43(2): 187-197. (in Chinese) doi: 10.21656/1000-0887.420151
[24]	PAN Y Y, WU L F, YANG X Z. A difference method with intrinsic parallelism for the variable-coefficient compound KdV-Burgers equation[J]. Applied Numerical Mathematics, 2021, 169: 201-220. doi: 10.1016/j.apnum.2021.06.009
[25]	LU D C, CHEN C X. Computable analysis of a boundary-value problem for the generalized KdV-Burgers equation[J]. Mathematical Methods in the Applied Sciences, 2015, 38(11): 2243-2249. doi: 10.1002/mma.3218
[26]	ZHANG W G, SUN Y J, LI Z M. Bounded traveling wave solutions for MKdV-Burgers equation with the negative dispersive coefficient[J]. Discrete and Continuous Dynamical Systems B, 2016, 21(8): 2883-2903. doi: 10.3934/dcdsb.2016078
[27]	包立平, 李瑞翔, 吴立群. 一类KdV-Burgers方程的奇摄动解与孤子解[J]. 应用数学和力学, 2021, 42(9): 948-957. doi: 10.21656/1000-0887.420011 BAO Liping, LI Ruixiang, WU Liqun. Singularly perturbed and soliton solutions to a class of KdV-Burgers equations[J]. Applied Mathematics and Mechanics, 2021, 42(9): 948-957. (in Chinese) doi: 10.21656/1000-0887.420011