# Introdução ao Firedrake

## Parte 4: Sistema Reativo

Ministrante: **Diego Tavares Volpatto (LNCC/ESSS)**

Encontro Acadêmico de Modelagem Computacional - XV EAMC (2022)

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## Sumário

Conceitos-chaves:

* Problema transiente
* Múltiplas incógnitas
* Não-linear
* Em resumo: Problema transiente, misto e não-linear (tudo que vimos)

# Modelo de Gray-Scott

## Modelo

É um para reatores de gel auto-catalisados e que formam padrões interessantes para as concentrações ao longo do tempo. É composto por duas equações:

\begin{equation}
\begin{aligned}
&\frac{\partial u}{\partial t}=\Delta u -u v^{2}+A(1-u) \\
&\frac{\partial v}{\partial t}=\delta^{2} \Delta v + u v^{2}-B v
\end{aligned}
\end{equation}

Consideraremos aqui apenas o caso 1D. O estudo que faremos é baseado [neste artigo](https://www.sciencedirect.com/science/article/pii/S0022247X15007957?via%3Dihub).

## Forma Fraca

Cada equação é "testada" por sua respectiva função-teste. Dessa forma:

\begin{equation}
\begin{aligned}
&\left(\dfrac{u - u_0}{\Delta t}, p\right)_\Omega = -(\nabla u, \nabla p)_\Omega - (u v^{2}, p)_\Omega + (A(1-u), p)_\Omega \\
&\left(\dfrac{v - v_0}{\Delta t}, q\right)_\Omega = -(\delta^{2} \nabla v, \nabla q)_\Omega + (u v^{2}, q)_\Omega - (B v, q)_\Omega
\end{aligned}
\end{equation}

Com os pares {candidato, teste}:

* $\{u, p\}$
* $\{v, q\}$

Discretização no tempo: Euler Implícito, e $u_0$ e $v_0$ sempre vão representar a solução no passo de tempo anterior, para não carregar notação.

# Implementação

## Considerações

O setup geral é baseado no artigo de referência. O objetivo é reproduzir a Fig. 3d do artigo.

## Importando as libs

```python
from firedrake import *
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import rc

plt.style.use(['seaborn-poster'])
```

## Definição da geometria

A configuração é fortemente baseada no artigo. Verifique a Seção 4.2 para detalhes.

```python
num_elements = 800
L = 100.0
x_left, x_right = 0.0, L
mesh = IntervalMesh(num_elements, x_left, x_right)
x = mesh.coordinates
```

## Espaço de Funções

Temos um problema misto e não-linear. Por conta da não-linearidade, vamos utilizar um polinômio de mais alta ordem.

```python
degree = 3
V = FunctionSpace(mesh, "CG", degree)
W = V * V
Vref = FunctionSpace(mesh, "CG", 1)  # para plotar sobre a malha
```

## Funções do problema

Seguindo as funções utilizadas na forma fraca:

```python
w = Function(W)
u, v = split(w)  # recupera funções no espaço misto
p, q = TestFunction(W)
```

## Condições Iniciais

Adotaremos as mesmas utilizadas no artigo:

\begin{equation}
u(x, 0)=1-0.5 \sin ^{100}(\pi x / L)
\end{equation}

\begin{equation}
v(x, 0)=0.25 \sin ^{100}(\pi x / L)
\end{equation}

Portanto:

```python
w0 = Function(W)
u0, v0 = w0.split()
u0.interpolate(1 - (1. / 2.) * sin(pi * x[0] / L) ** 100.)
v0.interpolate((1. / 4.) * sin(pi * x[0] / L) ** 100.)
```

    Coefficient(WithGeometry(IndexedProxyFunctionSpace(<firedrake.mesh.MeshTopology object at 0x7fce979c8eb0>, FiniteElement('Lagrange', interval, 3), name=None, index=1, component=None), Mesh(VectorElement(FiniteElement('Lagrange', interval, 1), dim=1), 1)), 9)

## Condições de Contorno essenciais

Compatíveis com as CI e coletadas do artigo:

```python
boundary_value_u = 1.0
boundary_value_v = 0.0
u_bc = DirichletBC(W.sub(0), boundary_value_u, "on_boundary")
v_bc = DirichletBC(W.sub(1), boundary_value_v, "on_boundary")
```

## Parâmetros do Modelo

Também coletados do artigo (para o caso de interesse):

```python
delta_squared = Constant(0.01)
A = Constant(0.01)
B = Constant(0.053)
```

## Parâmetros temporais

Para a evolução no tempo:

```python
Total_time = 4000.
dt = 1.0
Dt = Constant(dt)
```

Note que o artigo utiliza `dt = 0.5`.

## Formulação Variacional

### Forma Fraca

\begin{equation}
\begin{aligned}
&\left(\dfrac{u - u_0}{\Delta t}, p\right)_\Omega = -(\nabla u, \nabla p)_\Omega - (u v^{2}, p)_\Omega + (A(1-u), p)_\Omega \\
&\left(\dfrac{v - v_0}{\Delta t}, q\right)_\Omega = -(\delta^{2} \nabla v, \nabla q)_\Omega + (u v^{2}, q)_\Omega - (B v, q)_\Omega
\end{aligned}
\end{equation}

### Implementação

```python
# ** U part **
F = inner((u - u0) / Dt, p) * dx + inner(grad(u), grad(p)) * dx + u * v * v * p * dx - A * (1.0 - u) * p * dx
# ** V part **
F += inner((v - v0) / Dt, q) * dx + inner(delta_squared * grad(v), grad(q)) * dx - u * v * v * q * dx + B * v * q * dx
```

## Problema Variacional e Inicialização do Solver

Mesmo procedimento para o que vimos na Eq. do Calor:

```python
solver_parameters = {
    'mat_type': 'aij',
    'snes_tyoe': 'newtonls',
    'pc_type': 'lu'
}
problem = NonlinearVariationalProblem(F, w)
solver = NonlinearVariationalSolver(problem, solver_parameters=solver_parameters)
```

## Iterando no tempo

Vamos definir algumas quantidades para armazenar nossas soluções:

```python
u_values = []
v_values = []
u_values_deg1 = []
v_values_deg1 = []
```

Algumas quantidades auxiliares:

```python
usol_deg1 = Function(Vref)  # para plotar sobre a malha
vsol_deg1 = Function(Vref)
x_values = mesh.coordinates.vector().dat.data
```

Por fim, o laço no tempo:

```python
step = 0
t = 0.0
while t < Total_time:
    step += 1
    print('============================')
    print(f'\ttime = {t:.3e}')
    print(f'\tstep = {step}')
    print('============================')

    solver.solve()
    w0.assign(w)

    usol, vsol = w.split()  # a solução no passo de tempo

    # Projeções auxiliares
    usol_deg1.project(usol)
    vsol_deg1.project(vsol)

    # Armazenamentos
    u_vec = np.array(usol.vector().dat.data)
    u_values.append(u_vec)

    u_vec_deg1 = np.array(usol_deg1.vector().dat.data)
    u_values_deg1.append(u_vec_deg1)

    v_vec = np.array(vsol.vector().dat.data)
    v_values.append(v_vec)

    v_vec_deg1 = np.array(vsol_deg1.vector().dat.data)
    v_values_deg1.append(v_vec_deg1)

    t += dt
```

    ============================
    	time = 0.000e+00
    	step = 1
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    	time = 3.751e+03
    	step = 3752
    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    	time = 3.761e+03
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    ============================
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    	time = 3.762e+03
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    ============================
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    	time = 3.763e+03
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    ============================
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    ============================
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    	time = 3.765e+03
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    ============================
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    	time = 3.766e+03
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    ============================
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    	time = 3.767e+03
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    ============================
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    	time = 3.768e+03
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    ============================
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    	time = 3.769e+03
    	step = 3770
    ============================
    ============================
    	time = 3.770e+03
    	step = 3771
    ============================
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    	time = 3.771e+03
    	step = 3772
    ============================
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    	time = 3.772e+03
    	step = 3773
    ============================
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    	time = 3.773e+03
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    ============================
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    	time = 3.774e+03
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    ============================
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    	time = 3.775e+03
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    ============================
    ============================
    	time = 3.776e+03
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    ============================
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    	time = 3.777e+03
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    ============================
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    	time = 3.778e+03
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    ============================
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    	time = 3.779e+03
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    ============================
    ============================
    	time = 3.780e+03
    	step = 3781
    ============================
    ============================
    	time = 3.781e+03
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    ============================
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    	time = 3.782e+03
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    ============================
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    	time = 3.783e+03
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    ============================
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    ============================
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    	time = 3.785e+03
    	step = 3786
    ============================
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    	time = 3.786e+03
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    ============================
    ============================
    	time = 3.787e+03
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    ============================
    ============================
    	time = 3.788e+03
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    ============================
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    	time = 3.789e+03
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    ============================
    ============================
    	time = 3.790e+03
    	step = 3791
    ============================
    ============================
    	time = 3.791e+03
    	step = 3792
    ============================
    ============================
    	time = 3.792e+03
    	step = 3793
    ============================
    ============================
    	time = 3.793e+03
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    ============================
    ============================
    	time = 3.794e+03
    	step = 3795
    ============================
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    	time = 3.795e+03
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    ============================
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    ============================
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    	time = 3.797e+03
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    ============================
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    ============================
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    ============================
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    ============================
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    	time = 3.801e+03
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    ============================
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    	time = 3.802e+03
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    ============================
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    ============================
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    ============================
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    ============================
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    	time = 3.806e+03
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    ============================
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    ============================
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    	time = 3.808e+03
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    ============================
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    ============================
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    	time = 3.810e+03
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    ============================
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    	time = 3.811e+03
    	step = 3812
    ============================
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    	time = 3.812e+03
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    ============================
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    	time = 3.813e+03
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    ============================
    ============================
    	time = 3.814e+03
    	step = 3815
    ============================
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    	time = 3.815e+03
    	step = 3816
    ============================
    ============================
    	time = 3.816e+03
    	step = 3817
    ============================
    ============================
    	time = 3.817e+03
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    ============================
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    	time = 3.818e+03
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    ============================
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    	time = 3.819e+03
    	step = 3820
    ============================
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    	time = 3.820e+03
    	step = 3821
    ============================
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    ============================
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    	time = 3.822e+03
    	step = 3823
    ============================
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    ============================
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    ============================
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    	time = 3.825e+03
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    ============================
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    	step = 3827
    ============================
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    	step = 3828
    ============================
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    	step = 3829
    ============================
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    	step = 3830
    ============================
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    	time = 3.830e+03
    	step = 3831
    ============================
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    	step = 3832
    ============================
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    	time = 3.832e+03
    	step = 3833
    ============================
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    	time = 3.833e+03
    	step = 3834
    ============================
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    	time = 3.834e+03
    	step = 3835
    ============================
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    	step = 3836
    ============================
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    	step = 3837
    ============================
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    	step = 3838
    ============================
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    	time = 3.838e+03
    	step = 3839
    ============================
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    	time = 3.839e+03
    	step = 3840
    ============================
    ============================
    	time = 3.840e+03
    	step = 3841
    ============================
    ============================
    	time = 3.841e+03
    	step = 3842
    ============================
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    	time = 3.842e+03
    	step = 3843
    ============================
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    	step = 3844
    ============================
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    	step = 3845
    ============================
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    	time = 3.845e+03
    	step = 3846
    ============================
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    	time = 3.846e+03
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    ============================
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    	time = 3.847e+03
    	step = 3848
    ============================
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    	time = 3.848e+03
    	step = 3849
    ============================
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    	time = 3.849e+03
    	step = 3850
    ============================
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    	time = 3.850e+03
    	step = 3851
    ============================
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    	time = 3.851e+03
    	step = 3852
    ============================
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    	time = 3.852e+03
    	step = 3853
    ============================
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    	time = 3.853e+03
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    ============================
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    	time = 3.854e+03
    	step = 3855
    ============================
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    	time = 3.855e+03
    	step = 3856
    ============================
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    	time = 3.856e+03
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    ============================
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    	time = 3.857e+03
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    ============================
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    	time = 3.858e+03
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    ============================
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    	time = 3.859e+03
    	step = 3860
    ============================
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    	time = 3.860e+03
    	step = 3861
    ============================
    ============================
    	time = 3.861e+03
    	step = 3862
    ============================
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    	time = 3.862e+03
    	step = 3863
    ============================
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    	time = 3.863e+03
    	step = 3864
    ============================
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    	time = 3.864e+03
    	step = 3865
    ============================
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    	time = 3.865e+03
    	step = 3866
    ============================
    ============================
    	time = 3.866e+03
    	step = 3867
    ============================
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    	time = 3.867e+03
    	step = 3868
    ============================
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    	time = 3.868e+03
    	step = 3869
    ============================
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    	time = 3.869e+03
    	step = 3870
    ============================
    ============================
    	time = 3.870e+03
    	step = 3871
    ============================
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    	time = 3.871e+03
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    ============================
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    	time = 3.872e+03
    	step = 3873
    ============================
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    	time = 3.873e+03
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    ============================
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    	time = 3.874e+03
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    ============================
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    	time = 3.875e+03
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    ============================
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    	time = 3.876e+03
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    ============================
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    	time = 3.877e+03
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    ============================
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    	time = 3.878e+03
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    ============================
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    	time = 3.879e+03
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    ============================
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    	time = 3.880e+03
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    ============================
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    	time = 3.881e+03
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    ============================
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    	time = 3.882e+03
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    ============================
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    	time = 3.883e+03
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    ============================
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    	time = 3.884e+03
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    ============================
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    ============================
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    	time = 3.886e+03
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    ============================
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    	time = 3.887e+03
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    ============================
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    	time = 3.888e+03
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    ============================
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    ============================
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    ============================
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    	time = 3.891e+03
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    ============================
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    ============================
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    ============================
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    ============================
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    	time = 3.895e+03
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    ============================
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    	time = 3.920e+03
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    ============================
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    ============================
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    ============================
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    ============================
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    	time = 3.924e+03
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    ============================
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    	time = 3.925e+03
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    ============================
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    	time = 3.926e+03
    	step = 3927
    ============================
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    	time = 3.927e+03
    	step = 3928
    ============================
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    	time = 3.928e+03
    	step = 3929
    ============================
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    	time = 3.929e+03
    	step = 3930
    ============================
    ============================
    	time = 3.930e+03
    	step = 3931
    ============================
    ============================
    	time = 3.931e+03
    	step = 3932
    ============================
    ============================
    	time = 3.932e+03
    	step = 3933
    ============================
    ============================
    	time = 3.933e+03
    	step = 3934
    ============================
    ============================
    	time = 3.934e+03
    	step = 3935
    ============================
    ============================
    	time = 3.935e+03
    	step = 3936
    ============================
    ============================
    	time = 3.936e+03
    	step = 3937
    ============================
    ============================
    	time = 3.937e+03
    	step = 3938
    ============================
    ============================
    	time = 3.938e+03
    	step = 3939
    ============================
    ============================
    	time = 3.939e+03
    	step = 3940
    ============================
    ============================
    	time = 3.940e+03
    	step = 3941
    ============================
    ============================
    	time = 3.941e+03
    	step = 3942
    ============================
    ============================
    	time = 3.942e+03
    	step = 3943
    ============================
    ============================
    	time = 3.943e+03
    	step = 3944
    ============================
    ============================
    	time = 3.944e+03
    	step = 3945
    ============================
    ============================
    	time = 3.945e+03
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    ============================
    ============================
    	time = 3.946e+03
    	step = 3947
    ============================
    ============================
    	time = 3.947e+03
    	step = 3948
    ============================
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    	time = 3.948e+03
    	step = 3949
    ============================
    ============================
    	time = 3.949e+03
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    ============================
    ============================
    	time = 3.950e+03
    	step = 3951
    ============================
    ============================
    	time = 3.951e+03
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    ============================
    ============================
    	time = 3.952e+03
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    ============================
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    	time = 3.953e+03
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    ============================
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    	time = 3.954e+03
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    ============================
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    	time = 3.955e+03
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    ============================
    ============================
    	time = 3.956e+03
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    ============================
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    	time = 3.957e+03
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    ============================
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    	time = 3.958e+03
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    ============================
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    	time = 3.959e+03
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    ============================
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    	time = 3.960e+03
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    ============================
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    	time = 3.961e+03
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    ============================
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    	time = 3.962e+03
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    ============================
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    	time = 3.963e+03
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    ============================
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    	time = 3.964e+03
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    ============================
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    	time = 3.965e+03
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    ============================
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    	time = 3.966e+03
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    ============================
    ============================
    	time = 3.967e+03
    	step = 3968
    ============================
    ============================
    	time = 3.968e+03
    	step = 3969
    ============================
    ============================
    	time = 3.969e+03
    	step = 3970
    ============================
    ============================
    	time = 3.970e+03
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    ============================
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    	time = 3.971e+03
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    ============================
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    	time = 3.972e+03
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    ============================
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    	time = 3.973e+03
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    ============================
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    	time = 3.974e+03
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    ============================
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    	time = 3.975e+03
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    ============================
    ============================
    	time = 3.976e+03
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    ============================
    ============================
    	time = 3.977e+03
    	step = 3978
    ============================
    ============================
    	time = 3.978e+03
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    ============================
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    	time = 3.979e+03
    	step = 3980
    ============================
    ============================
    	time = 3.980e+03
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    ============================
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    	time = 3.981e+03
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    ============================
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    	time = 3.982e+03
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    ============================
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    	time = 3.983e+03
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    ============================
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    	time = 3.984e+03
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    ============================
    ============================
    	time = 3.985e+03
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    ============================
    ============================
    	time = 3.986e+03
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    ============================
    ============================
    	time = 3.987e+03
    	step = 3988
    ============================
    ============================
    	time = 3.988e+03
    	step = 3989
    ============================
    ============================
    	time = 3.989e+03
    	step = 3990
    ============================
    ============================
    	time = 3.990e+03
    	step = 3991
    ============================
    ============================
    	time = 3.991e+03
    	step = 3992
    ============================
    ============================
    	time = 3.992e+03
    	step = 3993
    ============================
    ============================
    	time = 3.993e+03
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    ============================
    ============================
    	time = 3.994e+03
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    ============================
    ============================
    	time = 3.995e+03
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    ============================
    ============================
    	time = 3.996e+03
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    ============================
    ============================
    	time = 3.997e+03
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    ============================
    ============================
    	time = 3.998e+03
    	step = 3999
    ============================
    ============================
    	time = 3.999e+03
    	step = 4000
    ============================

## Visualizando as Concentrações

Reprodução do resultado das concentrações (Fig. 3). Selecione o momento no tempo de interesse:

```python
time = 4000
```

Gráfico nesse momento:

```python
fig = plt.figure(figsize=(8, 6))
ax = plt.subplot(111)

# Plotting
ax.plot(x_values, u_values_deg1[time-1], '--', label='U')
ax.plot(x_values, v_values_deg1[time-1], label='V')

# Setting the xy-labels
plt.xlabel('x [L]')
plt.ylabel('concentration [adim]')
plt.xlim(x_values.min(), x_values.max())

plt.tight_layout()
plt.show()
```

## Visualizando o padrão no tempo

Baseado na Figura 4 do artigo. Note que é necessário mudar os parâmetros e o tempo de simulação para coincidir.

```python
fig = plt.figure(figsize=(8, 6))
Vplot = np.array(v_values_deg1)
p = plt.imshow(Vplot, origin="lower", aspect='auto', cmap='jet')
clb = plt.colorbar(p, label="v-component concentration")
plt.xlabel('x')
plt.ylabel('t')
plt.show()
```