二维空间分数阶反应扩散方程组的格子Boltzmann方法

doi:10.19596/j.cnki.1001-246x.8495

摘要/Abstract

摘要：

提出一种二维空间分数阶反应扩散方程组的格子Boltzmann方法, 对方程组的分数阶积分项进行离散化处理。通过Chapman-Enskog多尺度技术和Taylor展开技术, 从所建立的模型中恢复出二维空间分数阶反应扩散方程组, 推导出各个速度方向上的平衡态分布函数。通过数值算例对所构造的LB模型进行检验, 数值计算结果与算例的精确解吻合较好。

关键词: Riemann-Liouville分数阶导数, 反应扩散方程组, 格子Boltzmann方法, Chapman-Enskog多尺度技术

Abstract:

A lattice Boltzmann method(LBM) for two-dimensional fractional reaction-diffusion equations involving Riemann-Liouville derivatives is proposed. Firstly, fractional integral terms in equations are discretized. Then, with Chapman-Enskog analysis and Taylor-expansion technology two-dimensional fractional reaction-diffusion equations are recovered from the LB model. Equilibrium state of distribution function in each velocity direction is derived. Finally, the LB model is tested with numerical examples. It was found that numerical results agree well with analytical solutions.

Key words: Riemann-Liouville fractional derivative, reaction-diffusion equations, lattice Boltzmann method, Chapman-Enskog multi-scale analysis

冯舒婷, 戴厚平, 宋通政. 二维空间分数阶反应扩散方程组的格子Boltzmann方法[J]. 计算物理, 2022, 39(6): 666-676.

Shuting FENG, Houping DAI, Tongzheng SONG. Lattice Boltzmann Method for Two-dimensional Fractional Reaction-Diffusion Equations[J]. Chinese Journal of Computational Physics, 2022, 39(6): 666-676.

图/表 9

参考文献 36

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T	GRE(N, M, Δt, u)	GRE(N, M, Δt, v)	CPU/s
0.2	2.052 3 × 10^-4	2.960 8 × 10^-4	14.1
0.4	5.730 8 × 10^-5	2.115 2 × 10^-4	25.5
0.6	1.686 1 × 10^-4	1.265 0 × 10^-4	35.8
0.8	3.499 2 × 10^-4	6.430 0 × 10^-5	47.1
1	5.496 1 × 10^-4	7.912 7 × 10^-5	58.9

T	GRE(N, M, Δt, u)	GRE(N, M, Δt, v)	CPU/s
0.2	2.052 3 × 10^-4	2.960 8 × 10^-4	14.1
0.4	5.730 8 × 10^-5	2.115 2 × 10^-4	25.5
0.6	1.686 1 × 10^-4	1.265 0 × 10^-4	35.8
0.8	3.499 2 × 10^-4	6.430 0 × 10^-5	47.1
1	5.496 1 × 10^-4	7.912 7 × 10^-5	58.9

α	N	GRE(N, M, Δt, u)	Order₁(u)	GRE(N, M, Δt, v)	Order₁(v)
1.1	4	6.183 8 × 10^-2		6.434 2 × 10^-2
	8	1.970 4 × 10^-2	1.650 0	2.175 6 × 10^-2	1.564 3
	16	4.885 2 × 10^-3	2.012 0	6.162 1 × 10^-3	1.819 9
	32	8.444 8 × 10^-4	2.532 3	1.560 9 × 10^-3	1.981 0
	64	2.409 2 × 10^-4	1.809 5	3.051 2 × 10^-4	2.354 9
1.3	4	6.353 8 × 10^-2		6.438 9 × 10^-2
	8	2.097 4 × 10^-2	1.599 0	2.183 6 × 10^-2	1.560 1
	16	5.581 2 × 10^-3	1.910 0	6.246 6 × 10^-3	1.805 6
	32	1.157 5 × 10^-3	2.269 6	1.633 2 × 10^-3	1.935 4
	64	1.994 9 × 10^-4	2.536 6	3.606 0 × 10^-4	2.179 2
1.5	4	6.448 4 × 10^-2		6.442 7 × 10^-2
	8	2.184 1 × 10^-2	1.561 9	2.192 5 × 10^-2	1.555 1
	16	6.174 5 × 10^-3	1.822 6	6.359 5 × 10^-3	1.785 6
	32	1.508 5 × 10^-3	2.033 2	1.746 7 × 10^-3	1.864 3
	64	2.393 2 × 10^-4	2.656 1	4.619 3 × 10^-4	1.918 9

α	N	GRE(N, M, Δt, u)	Order₁(u)	GRE(N, M, Δt, v)	Order₁(v)
1.1	4	6.183 8 × 10^-2		6.434 2 × 10^-2
	8	1.970 4 × 10^-2	1.650 0	2.175 6 × 10^-2	1.564 3
	16	4.885 2 × 10^-3	2.012 0	6.162 1 × 10^-3	1.819 9
	32	8.444 8 × 10^-4	2.532 3	1.560 9 × 10^-3	1.981 0
	64	2.409 2 × 10^-4	1.809 5	3.051 2 × 10^-4	2.354 9
1.3	4	6.353 8 × 10^-2		6.438 9 × 10^-2
	8	2.097 4 × 10^-2	1.599 0	2.183 6 × 10^-2	1.560 1
	16	5.581 2 × 10^-3	1.910 0	6.246 6 × 10^-3	1.805 6
	32	1.157 5 × 10^-3	2.269 6	1.633 2 × 10^-3	1.935 4
	64	1.994 9 × 10^-4	2.536 6	3.606 0 × 10^-4	2.179 2
1.5	4	6.448 4 × 10^-2		6.442 7 × 10^-2
	8	2.184 1 × 10^-2	1.561 9	2.192 5 × 10^-2	1.555 1
	16	6.174 5 × 10^-3	1.822 6	6.359 5 × 10^-3	1.785 6
	32	1.508 5 × 10^-3	2.033 2	1.746 7 × 10^-3	1.864 3
	64	2.393 2 × 10^-4	2.656 1	4.619 3 × 10^-4	1.918 9

T	α=1.1	α=1.3	α=1.5	α=1.7	α=1.9
0.2	5.826 5 × 10^-4	5.879 4 × 10^-4	5.958 1 × 10^-4	6.088 1 × 10^-4	6.329 9 × 10^-4
0.4	5.136 5 × 10^-4	5.243 1 × 10^-4	5.400 8 × 10^-4	5.660 8 × 10^-4	6.144 6 × 10^-4
0.6	4.285 4 × 10^-4	4.445 7 × 10^-4	4.681 5 × 10^-4	5.069 4 × 10^-4	5.792 2 × 10^-4
0.8	3.266 8 × 10^-4	3.481 1 × 10^-4	3.794 1 × 10^-4	4.307 6 × 10^-4	5.265 5 × 10^-4
1.0	2.515 0 × 10^-4	2.608 2 × 10^-4	2.818 1 × 10^-4	3.364 5 × 10^-4	4.562 3 × 10^-4