<b>A General Boundary Condition with Linear Flux for Advection-Diffusion Models</b>

MIYAOKA, T.Y.; MEYER, J.F.C.A.; SOUZA, J.M.R.

doi:10.5540/tema.2017.018.02.0253

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Articles • TEMA (São Carlos) 18 (2) • Aug 2017 • https://doi.org/10.5540/tema.2017.018.02.0253 copy

A General Boundary Condition with Linear Flux for Advection-Diffusion Models ^† † This work was supported by CAPES.

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

Advection-diffusion equations are widely used in modeling a diverse range of problems. These mathematical models consist in a partial differential equation or system with initial and boundary conditions, which depend on the phenomena being studied. In the modeling, boundary conditions may be neglected and unnecessarily simplified, or even misunderstood, causing a model not to reflect the reality adequately, making qualitative and/or quantitative analyses more difficult. In this work we derive a general linear flux dependent boundary condition for advection-diffusion problems and show that it generates all possible boundary conditions, according to the outward flux on the boundary. This is done through an integral formulation, analyzing the total mass of the system. We illustrate the exposed cases with applications willing to clarify their meanings. Numerical simulations, by means of the Finite Difference Method, are used in order to exemplify the different boundary conditions’ impact, making it possible to quantify the flux along the boundary. With qualitative and quantitative analysis, this work can be useful to researchers and students working on mathematical models with advection-diffusion equations.

Keywords:
boundary conditions; partial differential equations; mathematical models; computer simulation

Condition		Kind
u = 0	homogeneous	Dirichlet or first kind
u = f( $\vec{x}$ , t)	non homogeneous	Dirichlet or first kind
$\frac{\partial u}{\partial \vec{n}} = 0$	homogeneous	Neumann or second kind
$\frac{\partial u}{\partial \vec{n}} = g (\vec{x}, t)$	non homogeneous	Neumann or second kind
$a u + b \frac{\partial u}{\partial \vec{n}} = 0$	homogeneous	Robin or third kind
$a u + b \frac{\partial u}{\partial \vec{n}} = h (\vec{x}, t)$	non homogeneous	Robin or third kind

Condition	Equation
$- α \frac{\partial u}{\partial \vec{n}} + V \cdot \vec{n} u = γ c$	(2.7)
$- α \frac{\partial u}{\partial \vec{n}} + V \cdot \vec{n} u = 0$	(2.8)
$\frac{\partial u}{\partial \vec{n}} = γ c$	(2.9)
$\frac{\partial u}{\partial \vec{n}} = 0$	(2.10)
$- α \frac{\partial u}{\partial \vec{n}} + V \cdot \vec{n} u = β (u - c)$	(2.11)
$- α \frac{\partial u}{\partial \vec{n}} + V \cdot \vec{n} u = β u + (γ - β) c$	(2.12)
$- α \frac{\partial u}{\partial \vec{n}} + V \cdot \vec{n} u = β u$	(2.13)
$- α \frac{\partial u}{\partial \vec{n}} = β (u - c)$	(2.14)
$- α \frac{\partial u}{\partial \vec{n}} = β (u - c) + γ c$	(2.15)
$- α \frac{\partial u}{\partial \vec{n}} = β u$	(2.16)
u = c	(2.17)
u = 0	(2.18)

n	Error	Relative error	Order
8
16	8.86 × 10^-1	5.35 × 10^-2
32	2.45 × 10^-1	1.50 × 10^-2	1.8565
64	6.16 × 10^-2	3.79 × 10^-3	1.9889
128	1.54 × 10^-2	9.50 × 10^-4	1.9988
256	3.86 × 10^-3	2.38 × 10^-4	1.9998

n	Error	Relative error	Order
16
32	2.40 × 10^-7	2.40 × 10^-7
64	5.95 × 10^-8	5.95 × 10^-8	2.0098
128	1.49 × 10^-8	1.49 × 10^-8	2.0024
256	3.72 × 10^-9	3.72 × 10^-9	2.0006
512	9.29 × 10^-10	9.29 × 10^-10	2.0002
1024	2.32 × 10^-10	2.32 × 10^-10	2.0001
2048	5.80 × 10^-11	5.80 × 10^-11	2.0021
4096	1.45 × 10^-11	1.45 × 10^-11	2.0022

Sociedade Brasileira de Matemática Aplicada e Computacional Rua Maestro João Seppe, nº. 900, 16º. andar - Sala 163 , 13561-120 São Carlos - SP, Tel. / Fax: (55 16) 3412-9752 - São Carlos - SP - Brazil
E-mail: sbmac@sbmac.org.br

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[1] *Corresponding author: Tiago Yuzo Miyaoka - E-mail: tiagoyuzo@gmail.com

Data: Parameters: α, v, w, β, γ , c.
Result: solutions: u ^k , M(t).
Define initial condition u ⁰;
Calculate A, B, f as in (4.1) and (4.2);
Factorize A as A = L _A U _A ;
fortemporal iteration = 1, . . . , _^nt do
	Solve L _A y = Bu ^k + f;
	Solve U _A u ^k ⁺¹ = y;
	Calculate M(_^tk );
end

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A General Boundary Condition with Linear Flux for Advection-Diffusion Models † † This work was supported by CAPES.

ABSTRACT

A General Boundary Condition with Linear Flux for Advection-Diffusion Models ^† † This work was supported by CAPES.