LUGON JUNIOR, JADER; RODRIGUES, PEDRO PAULO GOMES WATTS; BEVILACQUA, LUIZ; MARINHO, GISELE MORAES; KNUPP, DIEGO CAMPOS; VASCONCELLOS, JOÃO FLÁVIO VIEIRA; SILVA NETO, ANTÔNIO J.

doi:10.5540/tema.2020.021.02.0002290229

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Articles • TEMA (São Carlos) 21 (2) • May-Aug 2020 • https://doi.org/10.5540/tema.2020.021.02.0002290229 copy

Estudo do Coeficiente de Difusão Secundária em Problema de Difusão com Fluxo Bimodal

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

A recent formulation for anomalous diffusion problems, which envolves a fourth-order differential term, has presented negative values in its solution for certain cases. This paper presents a study of the effect of secondary diffusion coefficient in order to contribute with the understanding of the solutions behavior in these situations. A function was implemented to represent the variation in the parcel subject to primary and secondary diffusion, according to the amount of the property in diffusion. The formulation for the solution of this new model by the Finite Difference Method is presented. The results obtained are compatible with those presented in previous works in the literature.

Keywords:
anomalous difusion; Finite Difference Method; fourth-order differential equation

∆t	\|\|ε\|\|_∞	Ordem	\|\|ε\|\|₂	Ordem	CPU (s)
4 × 10⁻²	2,12 × 10⁻²	−	4,24 × 10⁻¹	−	9,09
2 × 10⁻²	1,09 × 10⁻²	0.9499	2.24 × 10⁻¹	0.9195	14,09
1 × 10⁻²	5.52 × 10⁻³	0.9892	1.12 × 10⁻¹	1.0050	22.95
5 × 10⁻³	2.88 × 10⁻³	0.9401	5.81 × 10⁻²	0.9437	26.82

h	\|\|ε\|\|_∞	Ordem	\|\|ε\|\|₂	Ordem	CPU (s)
2 × 10⁻¹	2.03 × 10⁻¹	−	1.78522	−	0.079
1 × 10⁻¹	3.83 × 10⁻²	2.40866	3.43 × 10⁻¹	2.37931	0.172
5 × 10⁻²	9.45 × 10⁻³	2.01759	8, 38 × 10⁻²	2.03382	0.478
2.5 × 10⁻²	2.35 × 10⁻³	2.01004	2.24 × 10⁻²	1.94384	4.365

∆t	\|\|ε\|\|_∞	Ordem	\|\|ε\|\|₂	Ordem	CPU (s)
4 × 10⁻²	1.91 × 10⁻²	−	−3.90 × 10⁻¹	−	12,05
2 × 10⁻²	9.71 × 10⁻³	0.9805	2.03 × 10⁻¹	0.9401	22,43
1 × 10⁻²	5.00 × 10⁻³	0.9584	1.13 × 10⁻¹	0.8443	32,3
5 × 10⁻³	2.57 × 10⁻³	0.9585	5.81 × 10⁻²	0.9229	43,24

h	\|\|ε\|\|_∞	Ordem	\|\|ε\|\|₂	Ordem	CPU (s)
1 × 10⁻¹	2.17 × 10⁻¹	−	2.1754	−	0.062
1 × 10⁻¹	4.04 × 10⁻²	2.4204	4.17 × 10⁻¹	2.3811	0.114
5 × 10⁻²	9.98 × 10⁻³	2.0188	1.05 × 10⁻¹	1.9875	0.453
2.5 × 10⁻²	2.50 × 10⁻³	1.9983	2.50 × 10⁻²	2.0036	6.641

Variável	Caso 1	Caso 2	Caso 3	Caso 4	Caso 5	Caso 6
Problema	A	A	A	B	B	B
κ₂	10⁻¹	10⁻¹	10⁻¹	10⁻³	10⁻³	10⁻³
κ₄	0	5 × 10⁻²	5 × 10⁻²	10⁻⁵	10⁻⁵	10⁻⁵
β	0.8	0.8	β(ϕ)	0.2	β(ϕ)	β(ϕ)
β_min	−	−	0.8	−	0.2	0.2
γ	−	−	2500	−	1000	50
ϕ_min	−	−	10⁻³	−	10⁻²	0.5
L	4.0	4.0	4.0	1.0	1.0	1.0
t _f	1.0	1.0	1.0	10.0	10.0	10.0
h	8 × 10⁻³	8 × 10⁻³	8 × 10⁻³	10−2	10−2	10−2
∆t	10⁻³	10⁻³	10⁻³	10⁻²	10⁻²	10⁻²

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[1] *Autor correspondente: Jader Lugon Junior - E-mail: jljunior@iff.edu.br