Environmental Engineering Reference
In-Depth Information
3. Diffusion Algorithms
There are several well-known finite-difference substitutes for the diffusion
term,
∂ ϕμ , which are usually of second order of accuracy and differ
one from another in methods for calculating the values
/
∂
x
(
∂
/
∂
x
)
i+
μ corresponding to
“half-integer” grid points. Some of them do not work well in dispersion appli-
cations because of strong variations of the eddy diffusivity μas a function of x
(especially, in case of vertical diffusion). In particular, μ could be close to zero
near the underlying surface and inside the layers with very stable thermal
stratifications. As a result, derivatives of ϕ and the error of approximation
equation (1) with finite differences could be very high. To avoid corresponding
problems, Marchuck (1961), Samarski (1971), and independently (actually, his
work was published in Russian in 1961) Berlyand (1982) introduced a so called
“balance method for constructing finite-difference schemes”. Practical applications
of this method are discussed by Genikhovich et al. (1995).
Using this method, one can derive the following expressions for
1
/
2
μ
:
i±
1
/
2
x
x
i
+
1
i
∫
∫
μ
=
(
x
−
x
)
/
dx
/
μ
;
μ
=
(
x
−
x
)
/
dx
/
μ
;
(2)
i
+
1
/
2
i
+
1
i
i
−
1
/
2
i
i
−
1
x
x
i
i
−
1
It was proven by Samarski (1961) that, when using Eq. 2, the solution of the
finite-difference scheme approaches the exact solution of Eq. 1 even for sharply
varying μ. In addition, Marchuck (1961) introduced the following discrete
approximation for the right-hand term, E, in Eq. 1:
x
x
x
x
i
+
1
i
i
μ
1
μ
1
i
+
1
/
2
i
−
1
/
2
∫∫
∫∫
E
=
2
[
Edtdx
+
Edtdx
]
/(
x
−
x
]
(3)
i
i
+
1
i
−
1
x
−
x
μ
x
−
x
μ
i
+
1
i
i
i
−
1
x
x
x
x
i
i
i
−
1
As shown by Marchuck, asymptotically the finite-difference scheme, which
uses Eqs. 2, 3, has an “infinite order of approximation”, i.e. its solution coincide
with the solution of Eq. 1 (assuming a = 0). It should be noted that, with certain
corrections, a similar approach can be used in the case with a ≠ 0 (e.g. to account
for the gravitational settling). Actually, Eq. 2 provides expressions for the
“turbulent resistance”. In this sense, this finite-difference scheme is analogues to
the schemes introduced later by Erisman et al
.
(1994) and Sofiev (2002).
Acknowledgments
This paper is dedicated to the memory of the outstanding scientist, our close
friend and long-lasting colleague, late Prof. Michael Galperin.
