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MECA2120

Introduction aux éléments finis

Evolution of a pollutant’s concentration in the Atlantic Ocean using Finite
Element Method

Janssens de Bisthoven Damien, Department of Civil Engineering, Ecole Polytechnique de Louvain, UCL (Belgium)
Stoupy Pierre, Department of Mechanical Engineering, Ecole Polytechnique de Louvain, UCL (Belgium)

Keywords
Stommel stream function,
Petrov-Galerkin,
Runge-Kutta,

1

Abstract
This fourth and last step of the project is to evaluate the change of the concentration
of a pollutant in the Atlantic Ocean, whose flow is carried by the Gulf Stream. For
this, the advection-diffusion equation was evaluated with a Petrov-Galerkin model and
integrated in time with the 4-order Runge-Kutta method. The programming language
C was used to make all the computation and the numerical computing environment
Matlab to plot the results.

Introduction
ψ(x, y) = sin(πy)(pexz+ + qexz− − 1)

The advection-diffusion equation related to the problem is
∂c
+ u · ∇c = α∇2 c
∂t

 z±
where
p

q

(1)

Where c(x, t) is the concentration at position x = (x, y)
at time t, u(x, y, t) is the velocity (u, v) and α is the inverse
of the Peclet number (α = 10−3 ).
The aim was to compute the evolution of c in time considering that the boundary is perfectly impermeable for the
pollutant (Neumann’s conditions, the flow is zero at the
boundary). The velocity is given by the following stream
function of Stommel (simplified model of recirculation loop
of the Gulf Stream) ψ(x, y):

(2)

= −5 ± 25 + π 2
z−
= ez1−e
+ −ez−
= 1−p

The discretization of the problem was the first step, using Galerkin and then Petrov-Galerkin Method. Once the
discretization performed, the 4-order Runge-Kutta method
was used to integrate from t = 0 to t = 2 with a step estimated to be the largest one (while ensuring stability). After
the programming itself, an analyse of the convergence was
done in order to determine the best values of the parameters
of the problem.
1

2

Discretization and Galerkin Method
∂c
= f (c)
∂t

The concentration is discretized by
X
c(x, y; t) ∼
Cj (t)τj (x, y)
= ch (x, y; t) =

(7)

where f (c) is time-independant. The 4-order RungeKutta method for the general problem (7) is find in [1] page
111. The discrete problem (3) can be written as

j

and so (1) with the Galerkin method becomes
∂C
M
+ (A + J)C = 0
∂t

M

(3)

=

&lt; τi τj &gt;
&lt; ∇τi · ∇τj &gt;
&lt; τi uh · ∇ch &gt;

P
P
∂τ
∂τ
&lt; τi uh j Cj ∂xj + v h j Cj ∂yj &gt;

MK1
MK2
MK3
MK4

(4)

Some modifications were were carried in the C-code in

3

=
=
=
=

−(A + J)C n
n
−(A + J + ∆t
2 K1 )C
∆t
−(A + J + 2 K2 )C n
−(A + J + ∆tK3 )C n

(9)

Hence, the computing of each Ki will require the resolution of a linear system, which will be performed by the
algorith of Chebyshev (the reader is referred to [2] for more
details).

Petrov-Galerkin Method

As the Galerkin method converges slowly, it became essential to prefer the Petrov improvement. This was done by
replacing the advection term in (4) by


ζh uh
· ∇τi uh · ∇ch &gt;
(5)
&lt; τi +
2 kuk

 h

h
where ζ = γ coth P2e − P2eh , P eh = uαh and h is
the smallest edge of each element. The term γ is yet to be
determined by the try-and-error method. The velocity uh
for each element i was approach by
v
!2
!2
u P3
P3
u
t
h
k=1 Uk,i
k=1 Vk,i
u =
+
(6)
3
3

5

Initial concentration

The first idea to impose an initial condition is to take

4
Ci (0) = ch (Xi , Yi ; 0) = sin πXi2 sin (πYi ) .

(10)

An another idea (more accurate) is to take the LSE
(Least Squares Estimation) of (10). By doing this, the initial
solution is already in the space of functions that approach
the solution. Although this solution is better, it was not
further explored due to time.

References

where the Uk,i and de Vk,i are the nodal values for the
element i computed in step 2.

4

(8)

By applying [1] to (1), K1 is just the time-derivate of the
concentration. Thus,

Where

M =
A =
J =

C n+1 − C n
= −(A + J)C n
∆t

[1] Vincent Legat, Mathématiques et méthodes numériques,
ou les aspects facétieux du calcul sur un orginateur. Ecole
Polytechnique de Louvain, LLN, Version 6.4, 2010.

4-order Runge-Kutta Method

[2] Suetin P.K, Chebyshev polynomials. Encyclopaedia of
Mathematics, Springer, 2001.

The first step is to observe that the advection-diffusion equation (1) is the differential equation

2

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