Lattice Boltzmann Method for Two-dimensional Fractional Reaction-Diffusion Equations

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

Abstract

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

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.

Figures/Tables 9

References 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