时间分数阶慢扩散方程的一类有效差分方法

摘要/Abstract

摘要：

对时间分数阶慢扩散方程提出一类数值差分方法：显-隐（Explicit-Implicit，E-I）和隐-显（Implicit-Explicit，I-E）差分方法.它是将古典显式格式与古典隐式格式相结合构造出的一类有效差分格式.理论证明了格式解的存在唯一性，用傅里叶方法证明了格式的稳定性和收敛性.数值试验验证了理论分析，表明E-I格式和I-E格式在具有良好的精度且无条件稳定的情况下，计算速度比隐式格式提高了75%.从而用此格式解决分数阶慢扩散方程是可行的.

关键词: 时间分数阶慢扩散方程, 显-隐(隐-显)差分格式, 稳定性, 收敛性, 数值试验

Abstract:

In this paper, a kind of numerical difference method of explicit-implicit(E-I) scheme and implicit-explicit(I-E) scheme was constructed for solving the time fractional sub-diffusion equation. It is based on the combination of the explicit scheme and implicit scheme. Theoretical analyses have shown that the solution of E-I(I-E) scheme is uniquely solvable. At the same time the stability and convergence of the scheme were proved by the Fourier method. Numerical experiments verified the theoretical analyses and showed the computational efficiency of E-I scheme is 75% higher than the implicit scheme under the premise of unconditional stability and having better accuracy. Therefore it is feasible to use the scheme for solving the time fractional diffusion equation.

Key words: Time fractional sub-fractional equation, Explicit-implicit (implicit-explicit) difference scheme, Stability, Convergence, Numerical experiments

中图分类号:

O241.8

赵雅迪, 吴立飞, 杨晓忠, 孙淑珍. 时间分数阶慢扩散方程的一类有效差分方法[J]. 数学物理学报, 2018, 38(6): 1122-1134.

Yadi Zhao, Lifei Wu, Xiaozhong Yang, Shuzhen Sun. A Kind of Efficient Difference Method for the Time Fractional Sub-Diffusion Equation[J]. Acta mathematica scientia,Series A, 2018, 38(6): 1122-1134.

图/表 7

表 1

图 1

图 2

图 3

表 2

E-I格式的误差和空间收敛阶( $\tau=h^2$ )"

	Implicit		E-I
$m$	$L_{h, \tau}^2$	Order1	$L_{h, \tau}^2$	Order1
20	2.593343e-3	—	2.958102e-3	—
40	6.548375e-4	1.985604	6.245729e-4	2.243741
80	1.667856e-4	1.973142	1.418638e-4	2.138363
160	4.221036e-5	1.982326	3.564083e-5	1.992904
320	1.057340e-5	1.997158	8.933510e-6	1.996232

表 2

表 3

E-I格式的误差和时间收敛阶 $(m=80)$ "

		Implicit		E-I
$\alpha$	$n$	$L_{h, \tau}^2$	Order2	$L_{h, \tau}^2$	Order2
$0.7$	200	2.545864e-4	—	1.094796e-4	—
	400	1.047661e-4	1.280984	4.699933e-5	1.219950
	800	4.284855e-5	1.289853	2.031905e-5	1.209807
	1600	1.746390e-5	1.294870	8.822774e-6	1.203529
	3200	7.102732e-6	1.297930	3.842242e-6	1.199284
$0.5$	200	8.475178e-5	—	3.620591e-5	—
	400	3.023084e-5	1.487222	1.351384e-5	1.421788
	800	1.072535e-5	1.494997	5.109668e-6	1.403136
	1600	3.788677e-6	1.501259	1.950831e-6	1.389141
	3200	1.332788e-6	1.507247	7.514643e-7	1.376312
$0.4$	200	4.222321e-5	—	1.920942e-5	—
	400	1.399031e-5	1.593609	6.639797e-6	1.532603
	800	4.601493e-6	1.604254	2.352434e-6	1.496985
	1600	1.501415e-6	1.615779	8.522910e-7	1.464736
	3200	4.849225e-7	1.630497	3.161945e-7	1.430534

表 3

图 4

参考文献 18

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	0.25	0.5	0.75	1	1.25	1.5	1.75	时间/s
精确解	0.1097	0.1946	0.2356	0.2351	0.2030	0.1487	0.0787	174.6007
隐式格式解	0.1090	0.1933	0.2339	0.2334	0.2014	0.1476	0.0781	6.7294
E-I格式解	0.1100	0.1951	0.2362	0.2358	0.2036	0.1492	0.0790	3.8812
I-E格式解	0.1100	0.1952	0.2363	0.2359	0.2037	0.1493	0.0790	3.8755