Appl. Math. L&t. Vol. 6, No. 5, pp. 35-38, 1993
Printed in Great Britain. All rights reserved
NONLINEAR
Copyright@
TRANSPORT
IN MOVING
08939659/93
$6.00 + 0.00
1993 Pergamon Press Ltd
FLUIDS
G. ADOMIAN
General Analytics
Corp.
155 Clyde Road, Athens, GA 30605, U.S.A.
R.
MEYEBS
U.S. Army Research Laboratory,
(Received
White Sands Missile Range 88002
and accepted May 1993)
Abstract-The
timedependent
spread of contaminants in moving fluids is normally studied by
computer-intensive
discretized procedures which have some disadvantages.
Application of the decomposition method allows a continuous, convenient, accurate procedure which works and extends
to nonlinear and stochastic partial differential equations ss well.
ONE-DIMENSIONAL
ADVECTION
EQUATION
The simplest case is modelled by
du
a&L
~+~=O’
By decomposition,
O<t<T,
u (x70)= f(x),
O<a:Il,
u (G t) = g(t),
0 > 0.
we have
(
u=u(x,O)-L;lcY
u
where L,' = &, L;l = J; (*)&,
differentiable as necessary. Then,
-
d
8X
O”
c
)
= c,"=,?h,
u (x,
0)is identified as 2~g,and f(x) is assumed
un = (-l)n
(
7
>
f’“‘(z),
so that
u =
UT%,
n=O
2
(-1)n
($)
n=O
35
f’“‘(z).
G. ADOMIAN, R. MEYERS
36
Then,
b&+1
= 2
(-v
(7)
f’“‘W
(1)
n=O
is an (m + 1)-term approximation to u satisfying the equation and condition at t = 0 using the
t-dimension“partia1 solution” [l]. The z-dimension partial solution is derived by
a&L
dx
du
=-dt’
Consequently,
m-1
c
(-1)n
n=O
Either (1) or (2) represents the solution.
4 m+l
a-n
=
(
$
)
g(n)(t).
ADVECTION-DIFFUSION
Let < (x, y, z, t) represent concentration. Let the fluid velocity be ?i with components u, w, w
in Rs and assume an incompressible fluid [2,3]
where D is the diffusion constant (which is a constant for a particular fluid or contaminant, temperature and pressure), E (x, y, z, 0) is a given initial condition and various boundary conditions
are possible, e.g., ,$ --+ 0 as x,y, z + 00, or c(t) is specified on a boundary r, or, we have a
preassigned flux at l?. We have
!%+!%+z
dY2
By decomposition
[4,5], using L = 8 and L-l
-~_““E____WX
(3)
dY
= s,’ (.) dt,
Thus,
Eo=~(t=O)=f(GY,z),
<m+l = DL-’
V2 &, - L-%iVJ,,,,
for m 2 0. Now all components are determined and we can write IN
approximation to < improving as iV increases.
= CzIt
cm as an
Moving fluids
STOCHASTIC
37
CASE
If we have turbulent motion of the fluid, we can have random fluctuations of the concentration
and hydrodynamical variables, hence statistical characteristics, would become necessary. #N(<)
becomes a series of stochastic terms and we form (&(c)) to get the expectation as functions
of average velocity components. The customary treatments of turbulent motion lead to a lack
of closure and untenable assumptions which are avoided by using decomposition. Thus, in the
above method, u, 21,zu, and < are replaced by their corresponding steady states plus quantities
representing fluctuations from the steady states. Thus,
t=Y$+Q,
u=ii+u’,
v=5+v’,
w=‘ii?+w’.
Statistical averaging causes terms such as
etc. to vanish. We then have
The last three correlation terms involve correlations of velocities and concentration which are
unknown. Then the procedure is to let ui for i = 1, 2, 3 denote u, 21,w, and zj for j = 1, 2, 3
represent x, y, z and write terms as being proportional to a mean gradient of the concentration
in terms of a “turbulent diffusion tensor” -Kij (zj, t) $$. TO clarify the difficulty, consider
the operator format Lu + Ru = g or u = L-‘g - L-l Ru. If we average, we have (u) =
L-l (g) - L-’ (Ru). W e can think of g as an input to a system containing R. The output u
can be statistically independent of g, but not of R. By decomposition, one writes u = L-‘g L-lR C u, = L-‘g - L-‘R L-lg + L-lR L-‘R L-lg - . . . . Averaging is no problem since g
is statistically independent of R.
NONLINEAR
Let’s consider the equation Lt + q
L=
-$
TRANSPORT
+ NJ = g where
NC = f(5),
R=;ii.V-DV2.
Let t = C,“=, tn and NC = CT=, A,. Then,
J=L,lg-L,lR
f&&A,,
n=O
n=O
where
Q = L,lg
&,_i
= -L,’
R&,, - L;'
A,,,,
for m 2 0. Then &+I = Cr=, 4 which converges to 5 = C,“=, cm. Further generalizations are
straightforward. We can, for example, consider Fu = g where
Fu=Ltu+L,u+Lyu+Lzu+Ru+Nu=g,
38
G. ADOMIAN, R. MEYERS
and solve for Ltu, Lzu, L,u or L,u which would simply treat the other operator terms as the
remainder operator R and would require the appropriate given boundary conditions [6]. The
case of stochastic 5 or stochastic processes in the R term leads to a stochastic 4~ which can be
averaged or from which expectations and covariances can be found. Convergence has been proven
by Cherruault [7,8]. The solutions are verifiable by checking that the original equation and the
given conditions are satisfied.
Since the concern here is solution of physical systems, unbounded or pathological inputs and
conditions are of no interest. If the model equation and the conditions are physically correct
and consistent, a solution is obtained which is unique and accurate.
If numerical results are
calculated, one sees the approach to a stable solution for the desired number of decimal places. If
conditions on one variable are better known than the others, we consider the appropriate operator
equation which can yield the solution.
REFERENCES
1. G. Adomian and R. Bach, Equality of partial solutions in the decomposition method for linear or nonlinear
partial differential equations, Computers Math. Applic. 10 (12), 9-12 (1990).
2. R.V. Ozmidov, D@sion
of Contaminants in the Ocean, Kluwer, Dordrecht, (1991).
3. J. Noye, Some explicit threelevel finite-difference simulations of advection, Math. and Comput. in Simulation
32, 359372
(1990).
4. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, New York, (1986).
5. G. Adomian, R. Bach and R. Meyers, An efficient methodology for the physical sciences, Kybernetes 20 (6),
104-113 (1991).
6. G. Adomian, A review of the decomposition method and some recent results for nonlinear equations,
Computers Math. Applic. 21 (5), 101-127 (1991);
(Errata in Mathl. Comput. Modelling, in press).
7. Y. Cherruault, Convergence of Adomian’s method, Kybernetes 18 (2), 31-38 (1989).
8. Y. Cherruault, G. Saccamondi and B. Some, New results for convergence of Adomian’s method applied to
integral equations, Mathl. Comput. Modelling 16 (2), 85-93 (1992).