Kouli_etal_2008_Groundwater modelling_BOOK.pdf - Pantelis ...

Analytical and Numerical Solutions... 275
where c 0 > 0 is the initial concentration and ǫ > 0 is the length of the triangular impulse.
The initial conditions are transferred as follows:
2 (1 − exp(− ǫ 2
û 1 (s,0) = c s))2
0
ǫ s 2 (68)
The general situation with piecewise polynomial initial conditions could derive and are
shown in [18]. We will only derive the triangular impulse for the benchmark problem.
The transferred initial conditions given by the equation (68) are inserted in the Laplacetransformed
equation, see [18]. We did the re transformation and the solution is given by:
⎧
2
⎪⎨
u 1 (x, t) = c 0
ǫ exp(−λ 1t)
⎪⎩
∏i−1
u i (x, t) =
j=1
⎛
⎜
λ j ⎝
i∑
i∏
exp(−λ j t)(
j=1
with i = 2, . . .,n
i∏ λ jl
Λ jk =
λ jl − λ jk
l=1
l≠k
l≠j
⎧
⎪⎩
0 0 ≤ x < v 1 t
x − v 1 t + b v 1 t ≤ x < v 1 t + ǫ 2
v 1 t + ǫ − x v 1 t ≤ x < v 1 t + ǫ
0 v 1 t + ǫ < x
k=1
k≠j
1
λ k − λ j
)
(69)
i∑
A jk Λ jlk (70)
k=1
k≠j
0 0 ≤ x < v j t
x − v j t + 1
λ jk
(−1 + exp(−λ jk (x − v j t))) v j t ≤ x < v j t + ǫ 2
v j t + ǫ − x + 1
λ jk
(exp(−λ jk (x − v j t))
⎪⎨
A jk = −2 exp(−λ jk (x − (v j t + ǫ 2
))) + 1) v
with
jt + ǫ 2 < x < v jt + ǫ
1
λ jk
(exp(−λ jk (x − v j t))
−2 exp(−λ jk (x − (v j t + ǫ 2 )))
+ exp(−λ jk (x − (v j t + ǫ)))) v j t + ǫ 2 < x < v jt + ǫ
λ jk = λ kj
5.4. Generalization of the Analytical Methods to Characteristics
In this subsection we generalize the one-dimensional analytical solutions for the
convection-reaction equations for piecewise polynomial characteristics in two dimensions.
The contribution here is the transformation of the two-dimensional characteristics to a onedimensional
arc-length which has piecewise constant velocities.
For the characteristics we assume a constant velocity between the start- and end-points
of the line. The piecewise constant velocities are given as:
v =
( f(x(t), y(t), t)
g(x(t), y(t), t)
)
, (71)