时间分数阶慢扩散方程的一类有效差分方法

Abstract

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

CLC Number:

O241.8

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.

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Figures/Tables 7

References 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

	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

		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

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A Kind of Efficient Difference Method for the Time Fractional Sub-Diffusion Equation

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