分数阶对流方程的全离散间断Galerkin方法

李晓婷; 王震

doi:10.21656/1000-0887.450341

分数阶对流方程的全离散间断Galerkin方法

doi: 10.21656/1000-0887.450341

李晓婷^1,,
王震^2, ,

1.
江苏大学京江学院，江苏镇江 212013
2.
江苏大学数学科学学院，江苏镇江 212013

基金项目:

国家自然科学基金 12101266

详细信息

作者简介:
李晓婷(1987—)，女，讲师，硕士(E-mail: lixiaoting@ujs.edu.cn)

通讯作者:
王震(1992—)，男，副教授，博士，硕士生导师(通讯作者. E-mail: wangzhen@ujs.edu.cn)

中图分类号: O241.82
计量
- 文章访问数: 126
- HTML全文浏览量: 36
- PDF下载量: 25
- 被引次数: 0
出版历程
- 收稿日期: 2024-12-30
- 修回日期: 2025-02-17
- 刊出日期: 2025-12-01

A Fully Discrete Discontinuous Galerkin Method for Fractional Convection Equations

LI Xiaoting^1
,,
WANG Zhen^{2
, ,}

1.
Jingjiang College, Jiangsu University, Zhenjiang, Jiangsu 212013, P.R.China
2.
School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu 212013, P.R.China

摘要

摘要: 分数阶导数因其在描述自然界中的反常现象方面具有优势而受到广泛关注. 研究了一类含时间Caputo-Hadamard分数阶导数的对流方程的数值解法, 采用L1方法近似时间导数, 运用间断Galerkin有限元方法对空间方向进行逼近, 进而得到该方程的全离散数值格式. 借助离散的Gronwall不等式分析了格式的稳定性、收敛性及误差估计, 最后通过数值例子验证了理论分析的正确性.
- 分数阶对流 /
- 间断Galerkin方法 /
- 全离散 /
- 稳定性 /
- 收敛性
Abstract: Fractional derivatives have received extensive attention due to their advantages in describing anomalous phenomena in nature. The numerical solutions to a class of convection equations containing temporal Caputo-Hadamard fractional derivatives were studied. The L1 method was adopted to approximate the time derivative, and the discontinuous Galerkin finite element method was used to approximate the spatial direction, thus to obtain the fully discrete numerical scheme for the equations. With the discrete Gronwall inequality, the stability, convergence and error estimates of the scheme were analyzed. Finally, numerical examples verify the correctness of the proposed theoretical method.
- fractional convection /
- discontinuous Galerkin method /
- fully discrete /
- stability /
- convergence

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图 1 数值解和精确解的对比(例1)

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 1. The comparison chart of numerical solution and exact solution (example 1)

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图 2 不同时刻的数值解(例1)

Figure 2. The contours of numerical solutions at different time points (example 1)

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图 3 数值解和精确解的对比(例2)

Figure 3. The comparison chart of numerical solution and exact solution (example 2)

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图 4 不同时刻的数值解(例2)

Figure 4. The contours of numerical solutions at different time points (example 2)

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表 1 时间方向上的L²误差和收敛阶(例1), r=1

Table 1. L² errors and convergence orders in the time direction (example 1), r=1

	M	α=0.4		α=0.6		α=0.8
	M	error	order	error	order	error	order
L²-norm	512	9.73E-3	*	2.77E-3	*	5.76E-4	*
	1 024	7.58E-3	0.36	1.83E-3	0.60	3.32E-4	0.79
	2 048	5.85E-3	0.38	1.21E-3	0.60	1.91E-4	0.80
	4096	4.48E-3	0.39	7.99E-4	0.60	1.10E-4	0.80
L^∞-norm	512	1.37E-2	0.34	3.92E-3	0.59	8.14E-4	0.79
	1 024	1.07E-2	0.36	2.59E-3	0.60	4.70E-4	0.79
	2 048	8.26E-3	0.37	1.71E-3	0.60	2.70E-4	0.80
	4 096	6.33E-3	0.38	1.13E-3	0.60	1.55E-4	0.80
	theory		0.40		0.60		0.80

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表 2 时间方向上的L²误差和收敛阶(例1), r=(2－α)/(2α)

Table 2. L² errors and convergence orders in the time direction (example 1), r=(2－α)/(2α)

	M	α=0.4		α=0.6		α=0.8
	M	error	order	error	order	error	order
L²-norm	64	4.48E-3	*	6.23E-3	*	5.55E-3	*
	128	2.60E-3	0.79	3.90E-3	0.68	3.90E-3	0.51
	256	1.50E-3	0.80	2.41E-3	0.69	2.66E-3	0.55
	512	8.60E-4	0.80	1.49E-3	0.70	1.79E-3	0.57
L^∞-norm	64	6.34E-3	*	8.79E-3	*	7.84E-3	*
	128	3.67E-3	0.79	5.51E-3	0.68	5.51E-3	0.51
	256	2.12E-3	0.80	3.41E-3	0.69	3.76E-3	0.55
	512	1.22E-3	0.80	2.11E-3	0.70	2.53E-3	0.57
	theory		0.80		0.70		0.60

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表 3 时间方向上的L²误差和收敛阶(例1), r=(2－α)/α

Table 3. L² errors and convergence orders in the time direction (example 1), r=(2－α)/α

	M	α=0.4		α=0.6		α=0.8
	M	error	order	error	order	error	order
L²-norm	64	4.17E-4	*	7.28E-4	*	9.85E-4	*
	128	1.43E-4	1.55	2.88E-4	1.34	4.75E-4	1.05
	256	4.85E-5	1.56	1.12E-4	1.36	2.23E-4	1.09
	512	1.64E-5	1.56	4.30E-5	1.38	1.03E-4	1.11
L^∞-norm	64	5.97E-4	*	1.03E-3	*	1.40E-3	*
	128	2.02E-4	1.57	4.07E-4	1.34	6.73E-4	1.05
	256	6.85E-5	1.56	1.58E-4	1.36	3.17E-4	1.09
	512	2.32E-5	1.56	6.09E-5	1.38	1.46E-4	1.11
	theory		1.60		1.40		1.20

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表 4 空间方向上的L²误差和收敛阶(例1), M=500, k=1

Table 4. L² errors and convergence orders in the spatial direction (example 1), M=500, k=1

	M	α=0.4		α=0.6		α=0.8
	M	error	order	error	order	error	order
L²-norm	4	1.03E-1	*	9.97E-2	*	9.73E-2	*
	8	2.54E-2	2.02	2.44E-2	2.03	2.36E-2	2.05
	16	6.24E-3	2.02	5.97E-3	2.03	5.72E-3	2.04
	32	1.54E-3	2.02	1.47E-3	2.02	1.39E-3	2.04
L^∞-norm	64	1.07E-1	*	1.05E-1	*	1.03E-1	*
	128	3.35E-2	1.69	3.23E-2	1.70	3.13E-2	1.72
	256	8.69E-3	1.95	8.34E-3	1.95	8.03E-3	1.96
	512	2.18E-3	2.00	2.08E-3	2.00	1.99E-3	2.01
	theory		2.00		2.00		2.00

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表 5 当α→1^－时鲁棒性测试，N=500, M=32

Table 5. The robustness test for α→1^－, N=500, M=32

α	error
0.8	0.001 903 142 890 661
0.9	0.002 505 224 877 617
0.99	0.004 017 895 756 442
0.999	0.004 298 546 559 962

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