It is important to note that the numerical solution of the convection- diffusion equation using finite-difference methods is plagued with a num- ber of difficulties [14] ...
Application of the Decomposition Method to the Solution of the Reaction-Convection-Diffusion Equation Eugene
Yee
Defense Research Establishment Suflield Chemical and Biological Defense Section Box 4000, Medicine Hat Alberta, Canada TlA 8K6
Transmitted by Melvin R. Scott
ABSTRACT The
nonlinear
time-dependent
reaction-convection-diffusion advection
into a prescribed sition method without
the need
chemically
ated analytically. obtained
which reactive
a means of computing
for an explicit
it is shown inert
implemented
equation,
of a chemically
flow field, is solved using the decomposition
provides
Furthermore,
and dispersion
material,
that
computer
by application
a known exact solution
discretization in the
case
the components
Consequently,
using
an approximate
material
the
released
method.
The decompo-
solution
to the problem
of the partial of the
characterizes
differential
advection
and
equation.
diffusion
of a
in the approximate
solution
can be evalu-
in this case, the decomposition
method
can be readily
algebra.
The
of the decomposition
efficacy
and accuracy
method,
are evaluated
for the case of the dispersion
of a chemically
of the
solution,
by comparison inert material
to in a
linear shear flow field.
I.
INTRODUCTION
A considerable research effort has been invested in the development of accurate numerical methods for the simulation of the advection and dispersion of chemically inert or reactive materials released into a prescribed flow field. This is not too surprising since the dynamics of advection and diffusion are important to the understanding of the physical processes that govern a large number of diverse phenomena encompassing a broad gamut of disciplines. A few examples include the dispersion of materials injected into blood APPLIED MATHEMATICS 0
1993 Canadian
AND COMPUTATZON 56:1-27 (1993)
1
Government
Published by Elsevier Science Publishing Co., Inc., 1993
0096-3003/93/$0.00
2
E. YEE
vessels 117, 81 (medicine, epidemiology, and bio-engineering), the diffusion of particles and thermal energy in host gases and liquids [4, 31 (aerosol science, chemical engineering, and petroleum engineering), the transport of radiation in optically thick media [9, 161 (pl asma physics, optics, and electro-optics) and the dispersion of contaminants (pollutants) released into the atmosphere, soil, or ocean [7, 111 (meteorology, hydrogeology, and oceanography). Indeed, the latter application has gained prominence in recent years, largely because of the marked increase more, the long-range
in the industrial use of hazardous materials. transport and ultimate fate of anthropogenic
Furtherpollution
has raised a number of contentious environmental issues such as the effect of fluorocarbons on the Earth’s protective ozone layer, the influence of the accidental release of oil and the break-up and dispersal of oil particles on the fragile marine environment, and the adverse effects of the formation of acid rain and photochemical smog on human activity. As a consequence, it is more important than ever to address the problem of the development of accurate methods for the simulation and prediction of the transport and diffusion of materials released into the Earth’s environment since these models provide the basis for many politico-environmental decisions concerning the implementation of control strategies. The dispersion of material in a shear
and/or
turbulent
flow field is an
intrinsically multidimensional process. Under these general conditions, the advection and diffusion of chemically inert materials is described by a fully three-dimensional linear convection-diffusion equation with spatially and/or temporally varying coefficients (e.g., the mean velocity field and the eddy diffusivity tensor). With the inclusion of chemically reactive materials and the concomitant
possibility of complex feedback mechanisms inherent in the equation is nonlinear and the reaction dynamics, the convection-diffusion complexity of the problem increases comparably. The construction of a solution for the fully nonlinear reaction-convection-diffusion equation that is consistent
with certain
global physical properties
such as mass conservation,
positivity, and causality is non-trivial. It is important to note that the numerical solution of the convectiondiffusion equation using finite-difference methods is plagued with a number of difficulties [14]. Firstly, finite-difference approximations of the convection-diffusion equation that are consistent with a given resolution and accuracy generally result in a computationally intensive problem from the viewpoint of both computer mass storage and total computer processing time. In this regard, it is important to note that to increase the resolution by a factor of two for a fully three-dimensional simulation, it is necessary to increase the computation or processing time 16-fold and the computer storage requirement s-fold since the number of points in the space mesh must increase by a factor of 8 (i.e., a twofold increase for each linear dimen-
Decomposition
3
Method
sion) and the time step required to satisfy the Courant-Friedrichs-Lewy (CFL) condition for numerical stability must be halved for the refined mesh. Secondly, the finite-difference method, by its very nature, is based on changing a continuous problem into a discrete problem. Along these lines, the effects on the solution arising from the neglect of the subgrid scale components that are smaller than the discrete grid spacing are difficult to quantify, especially when nonlinear terms are present in the equation. Thirdly, the finite-difference approximations for the convection-diffusion equation give rise to numerical
dispersion
and numerical
diffusion.
Numerical
dispersion is the result of phase errors and inadequate damping of high wave number components of the solution. This generally leads to spurious oscillations in the solution that increase with time, leading eventually to numerical instability. Methods to nonphysical or numerical
stabilize the solution must necessarily introduce diffusion that can mask the effects of the physical or
actual diffusion. In summary, then, the development of a robust numerical scheme for the solution of the reaction-convection-diffusion equation that is simultaneously stable, efficient, In light of the deficiencies
and accurate presents a formidable challenge. inherent in a finite-difference solution, it is
desirable to develop a solution procedure for the reaction-convectiondiffusion equation that avoids the effects of linearizations and discretizations. In this regard,
one desires
the characteristics
of the solution
to reflect
the
underlying physics of the advection and diffusion process rather than the artifacts and inaccuracies introduced by the discretization process and by the resulting numerical solution. Ideally, it would be useful to develop a solution procedure for the reaction-convection-diffusion equation that explicitly provides a continuous representation of the solution and that requires little computer memory and processing time for the computation of the results. To this end, we apply the decomposition method to the construction of a solution for the nonlinear reaction-convection-diffusion equation. The paper is organized in the following way. We give a general mathematical formulation of the unsteady three-dimensional convection-diffusion equation for nonlinear chemically reactive materials in Section 2, while Section 3 is devoted to the application of the decomposition method to the construction of a solution for the convection-diffusion equation for chemically inert materials. In Section 4, we generalize this solution-scheme to the case of materials involved in nonlinear chemical reactions for both the scalar and the vector convection-diffusion equations. A comparison of the accuracy of an N-term approximation obtained from the decomposition method with a known analytical solution of the convection-diffusion equation for the case of an inert material released into a linear shear flow and subject to a gravitational force field is presented in Section 5. A conclusion is drawn in Section 6.
E. YEE
4 2.
MATHEMATICAL
FORMULATION
OF THE PROBLEM
The process of the three-dimensional advection and diffusion of a chemically reactive material in a prescribed flow field is modelled by the reactionconvection-diffusion equation (i.e., a parabolic partial differential equation)
- + $+(qgg=dC* at*
d
u.* ’
dx;
J
J
KF-
dC"
I
i Jk ax,*
+ R*(C*)
(1)
+ s*(P,t*).
Here Z* E is the mean (ensemble average) concentration of material at UC, U-f ) is the mean flow velocity field, eint P and time t* , v’* = is the rate of generation (or chemical
kinetics),
S*(P)
or loss of the material by chemical is the source term representing
t*)
the production of material by geometric sources (e.g., point, line, area, of an eddy diffusivity tensor volume, etc.), and (K$)f k= 1 are components K*. Adhering to usual convention, repeated indices in any term indicate a summation over all three values of the index, with components in the directions of the three Cartesian coordinate directions given subscripts 1, 2 and 3, respectively.
It is noteworthy
R*(C*) is the result of the fomulation mechanism for the material functional form.
considered
to point out that the reaction term of an appropriate chemical kinetic and generally
leads to a nonlinear
To facilitate the construction of the solution of the nonlinear reactionit is convenient to write the equation in convection-diffusion equation, nondimensional form. Towards this objective, let L and V denote physically meaningful characteristic length and velocity scales for the flow and let C denote a typical concentration scale. Now, consider the following transformations:
*
x=“,y_z=~ L
Yh L
* L'
t=-
i7t* L
)
(qj=2 1
(p 1 v
’
(2a>
5
Decomposition Method Observe
that a starred superscript
nondimensional tion-diffusion
variable. equation
dC x
+
qg
+
I In the
3.1.
from
reaction-convec-
can be written in the form
(0,; =;
+R(C) +S(x’,t).
FOR
reaction-convection-diffusion
CONVECTION-DIFFUSION
Decomposition method In this section, we describe
(2b)
J
J
of the paper, the dimensionless will be used unless otherwise stated.
SOLUTION
the dimensional
dimensionless
rest
equation 3.
is used to distinguish
The corresponding
the application
EQUATION
of the decomposition
method
to the construction of a solution to (2b) for the case of chemically inert materials (viz., for R(C) = 0). The decomposition method was introduced and developed by Adomian and is well documented in two monographs [2, 11, which provide a systematic survey of the application of the method nonlinear and/or stochastic differential and partial differential equations well as to algebraic and matrix equations. To begin, it is convenient
K-
d;t
to rewrite (2b) in the following form:
d2C
dC
=
dXjdXj
-qg
-
J
(qjg I
-
Kg +S(x’,t), 3
where
K
is some prescribed
to as
constant
(3)
J
which can be interpreted
as a dimen-
sionless diffusivity. To fix ideas, let us assume the source term S( x’, t) is a point source at ?’ = G consisting of a unit mass of material released instantaneously at t = 0, viz.,
S(x’,t)
= qqiqt>
= S(r)6(y)6(z)6(t),
(4
where SC.1 denotes the Dirac delta function. It should be noted that the point source of (4) is generic in the sense that a more complex source term can be formed from the point source by the principle of linear superposition.
E. YEE
6
Now, keeping operator form:
(3)
in view,
let us rewrite
-(921 +q
_5? t,x,y,zC =
the
equation
in the
following
+ S(x’,t),
+9$
(5)
with
27t,x,y,z
a
a2
at
dxjaxj
s_--_K-
’
and
a2 +/C-
Now, assuming
that the inverse
operator
qi,
dXjdXj
.
y, x exists, it is clear that the
solution of (5) can be written formally as
c
=
--qZ,,,.(%
+92
+%)c
+~*“,,,*ww.
To proceed further, it is convenient to introduce write (6) in the following parametrized form:
c = -K,:,y,*
(3
+92
+%>c
+-Qc,,,AW.
(6)
the parameter
h and
(7)
Obviously, the required solution is regained when h is set to unity. At this point, it should be remarked that A is introduced here only for the purpose of grouping terms, a process that will be clarified presently. In other words, the parameter h will play the role of a “placeholder” and its presence in the following analysis will be removed once this role has been fulfilled. Now, following Adomian [Z, I], let us decompose the solution into the following form with the understanding that the required solution is obtained as A -+ 1:
c =
e
AT,. T&=0
(8)
Decomposition Method Now, by direct substitution
of (8) into (7), there then follows
Equating sequence
same
terms with the of equations:
power
of
h now leads
to the
following
or, in general,
Observe that the component C, is calculable since it depends only on the source term S(x’, t) and on the initial and boundary conditions, both of which are assumed to be known. Furthermore, C, can be calculated from Co, C, can be calculated from C, and, in general, C, can be calculated from C, _ r. Consequently, (9) provides a useful recursion in the sense that each component C, of the solution C (cf. (8)) can be calculated solely from knowledge of the previous component C,_ r. However, it should be noted that except for certain special cases, it is impossible to write the general component C, of the solution in a closed form and to evaluate the infinite series of (8) exactly. Nevertheless, an N-term approximation of the solution can be easily computed using the recursion displayed in (9). Furthermore, the accuracy of the solution can be improved by simply incorporating further terms or components. 3.2. Evaluation of inverse operator Next, we provide an explicit form for the inverse operator fact of basic importance is that the form of the operator
.LZ~T~, y, Z. The g, x, y, z. was
purposely chosen so that it is invertible. The inverse of this operator can be easily evaluated if one replaces the partial differential operator with an equivalent integral operator whose kernel G, is determined as the Green’s
E. YEE
8 function
for the following partial differential
equation:
dG,(x',t;i?',t') 9
t,x,y,zG3(Kt; x”,t’) = x’,t;
d2G3( -K
defined
z,
ax,dXj
over a three-dimensional
dt
t’) =
6(2-
Euclidean
qqt
- t’),
space R3. Naturally,
(10) the precise
form of G, depends on the choice of the boundary conditions. In the interest of clarity and for simplicity of exposition, we consider only unbounded flow domains and, in this regard, the solution to the preceding satisfy homogeneous m). Generalizations incorporated
equation
is taken to
conditions at infinity (viz., G,( x’, t;St', t '> + 0 as 11?I] + to more complex boundary conditions can be readily
within the present
framework.
Physically, the solution G3( 2, t; x”, t ‘) corresponds to the concentration distribution of a unit mass of material with (dimensionless) diffusivity K that is instantaneously released at time t = t’ from a point source located at x’= x”. The important point to note is that since K is a constant, (10) corresponds to the “classical” diffusion equation whose solution is known to be a Gaussian with the following form for t > t ’ [6]:
G,(x’,t;
Z’,t’)
=
It is clear that the solution of (11) has the following properties: (1) G, satisfies the homogeneous partial differential equation q, r, y, ZG, = 0 everywhere except at the space-time point (2, t> = (S’, t’>; (2) G, verifies the homogeneous boundary conditions at infinity and possesses a characteristic singularity at the source point x’ = x”, t = t’; and, (3) the total mass of material at any time t is /Gs( x’, t; 7, t’)& = 1, which is equal to the mass of material released at the point source. Given the Green’s function associated with the operator Tt, x, y, z, it is straightforward to write an explicit form for the corresponding inverse operator
q,’ . , y. z as follows:
~&&)
= jotj-gG3( 2”,t; ,_‘I, t”)u(
Z”, t”)d?‘dt”.
9
Decomposition Method With this in view, the symbolic
form for the components
C, of C displayed
in (9) can be written explicitly in the following manner:
C,( x’, t) = /d/R3G,( x’, t; 2”) t”)S’( ?‘, t”)dx”‘dt”,
C,( 2, t) =
-(/,3G,(2, t; i?‘, t”)(S’,
C,( Z, t) = -&G3(
Z, t; x”‘, t”)(S,
( =a)
+.B$ +c%‘;)C,( x”‘, t”)&“dt”,
2”) t”)&“‘dt”,
+c%$ +9JC,(
or, in general,
Observe that in writing (12a), it is implicitly assumed that there was no material in the flow field before the source function S was “turned on”, viz., the material from the source was released into an initially clean flow. Note that with the source term S(x’, t) of (4), the first compone?t C,,(x’, t) (cf. (12a)) of the solution is easily seen to be C,(x’, t) = G3( 2, t; 0,0). 3.3. Analytic evaluation of integrals At this point, it should be noted
that the various
solution C(x’, t) can be computed numerically quadrature formulae. However, it is noteworthy
components
of the
by application of standard that the integrations in (12)
can be performed analytically for certain choices of the functional form for the mean velocity field and the diffusivity tensor. To this purpose, it is assumed that the mean velocity field and the components of the eddy diffusivity tensor are in the form of multivariate polynomials in x, y, z and t:
q( x, y, z, t) = C C C CaiklmnXkylzmtn, klmn
( 13a)
E. YEE
10
and I$(
x, y, 72,t) = c
c
c
~b;&&jvY,
( 13b)
klmn
where k, I, m, and n are non-negative integers. It is important to emphasize that the adoption of these functional forms for the velocity field and the diffusivity tensor do not limit the applicability of the present methodology since all continuous functions can be adequately approximated by multivariate polynomials of the form displayed in (IS). With these forms for Uj and Kij and with a source term of the form displayed in (4, it is apparent that the computation of the components C,( 2, t> of the solution integrals
C( 2, t> reduces
to the problem
of the evaluation
of
of the type
x d’G,( ii?’ , t” ; ti, 0) d x;
dx’” dt ” ,
where j, k, 1, m, n, and r are non-negative
(14)
integers.
To evaluate
1, first note
that the Green’s function G,(x’, t; 3, t ‘1 can be expressed as the product three factors each of which depends on only one spatial coordinate, viz., G,(x’,t;x”,t’)
of
=G,(x,,t;x;,t’)G,(x,,t;x6,t’)G,(x,,t;rj,t’), (Isa)
where 1 G,(xj,t;
x;,t’)
l
, _ ('j -
= 2(TK(t
-
tr))1’2exp
4K(t
‘iI2 -t’)
(15b)
of (14) and (15) suggests that it is usefi 11 to define
Now, a careful examination the following function:
$r(
xj,
q
E
(
_2>r
drG1(;;J;o,o) . 3
(16)
11
Decomposition Method It should be pointed out that the factor ( - 2>r in the definition included
only for normalizatio+n purposes.
Furthermore,
for I,!J,.(xj, t ) is
note that
&,(xj,
t)
= Gr(xj> t; 0,O) and Ga(x’, t; 0,O) = &lo(ri, t)$,Jxs, t)&Jxs, t). Before proceeding to the evaluation of I, it is useful to examine certain properties of I,!+,(xj, t). Note that I&:( xj, t) has a finite “energy” in the sense that it is square-integrable and, as such, corresponds to a transient waveform as time progresses and/or spatial position increases. Since the result of r differentiations of G,(xj, t; 0, 0) with respect to xj is a polynomial in xj/~t and - l/~t times G,(xj, t; 0, O), it is clear that &(xj, t) can be expressed as
where
F,. is a polynomial
in xj/Kt and - l/Kt. A few of these polynomials 1 for reference purposes. Observe that for a f=ed t,
are listed in Table
&(xj,
t> is an even
associated
or odd function of xj depending on whether order r of the function is even or odd, respectively.
With the definition for I,!+(xj, t) displayed in (16), it is straightforward show that this function verifies the following two recurrence relations:
aGr(xjY
t,
dXj
=
the to
(17)
-LIL,+l(xj,t),
2
and ‘j’I’r(xj,t)
=
Kt’J’r+l(xj>t‘)
f
(18)
zrlCI,_,(~j,t).
TABLE 1 THE POLYNOMIALS
0 1 2 3 4 5 6 7 8 9
F,.( 5, p)
c6 + c$? + 5’ + 565% ts + 7255 +
WHERE 5 = Xi/Kt
AND p =
- l/Kt
t2 + 2P t3 + 65~ (4 + 125% + 12p2 tJ5 + 2053p + 605~’ 305% + 18052p2 + 120~~ 425% + 44053p2 + 8405~~ + 840E4p2 + 3360.$2p3 + 1680~~ 1512i$5p2 + 1008053p3 + 15120(p4
E. YEE
12
The importance of these recurrence relations to the evaluation of I can be summarized as follows. Repeat:d application of (17) allows one to remove the spatial derivatives of G,( x’, t; 0,O) f rom the integrand and to replace them with products of the functions r,!q(xj, t). Next, products of the spatial coordinates xj with the associated functions I),.( xj, t) can be removed from the integrand by repeated application of (18). This systematic procedure reduces the evaluation of I to the evaluation of integrals of the following kind:
I’
= f/,@G3(
x’, t; j?‘, t”)&(
x”, t”)&(
y”, t”)&,(
z’, t’)dx’“dt”,
(19) where k, I, m, and n denote arbitrary non-negative integers. of I’ is readily accomplished using the integral relation
which follows easily from the following
composition
2, t; j?“‘, t”)G3( x”‘, t”, ii?‘, t’)&”
Hence,
I’ can be readily evaluated
spatial coordinates
The evaluation
rule for GJZ,
= G3( F’, t; x”, t’).
by performing the integration using the result of (20) f o 11owed by an integration
t; 2, t’):
(21)
over the over the
temporal coordinate. As a final note, since the components of the concentration field can be analytically evaluated using the present solution methodology, the problems associated with the presence of the singularity (Dirac-delta point source> at the origin have been effectively resolved. This should be contrasted with finite-difference or Gale&in schemes, where the specification of the appropriate procedure for the approximation of the singularity at the source point and the concomitant rapidly varying concentration field in its immediate vicinity, still remains a topic of contention [15]. Furthermore, the analytic evaluation of the components of the decomposition can be easily performed using computer algebra.
13
Decomposition Method 4.
SOLUTION
FOR
REACTION-CONVECTION
DIFFUSION
EQUATION 4.1. Scalar case To show how the present methodology can be applied to the solution of the convection-diffusion equation for a chemically reactive material, we assume that the rate of formation or dissociation of the material by chemical reaction can be described by a reaction term with the following form: R(C)
= yC”,(a=
1,2,3
,...
),
(22)
where y and a are the reaction rate coefficient and the reaction order of the material, respectively. In particular, the reaction term of (22) represents the effect of a chemically reactive material dissociating (7 < 0) or recombining (7 > 0) through an a-order reaction. Observe that except for a material which undergoes a first-order chemical reaction ((Y = 11,the incorporation of the reaction term of (22) into the convection-diffusion equation results in a nonlinear processes chemically
partial differential equation. Furthermore, it should be noted that undergoing chemical reactions typically involve a number N of reactive
constituents,
with
the
concentration
Cci)
of the
ith
constituent satisfying a convection-diffusion equation of the form exhibited in Eq. (2b). This system of partial differential equations is coupled through reaction terms R(C(‘), Cc’), . . , Cc”)) that typically involve products of various powers of the component concentrations. However, for ease of exposition, we focus at nresent on a svstem comoosed of one constituent whose reaction is describkd by the form’exhibited in (22). In view of (71, the incorporation of the reaction term of (22) leads to the following result: +~;L.,,;S(~,t). (23) As before, the solution C(x’, t) is decomposed as per (8). To deal with the nonlinearity in the reaction term, it is useful to expand this term in the following form: R(C)
=
f A”A,, n=O
(24)
where A,, are the Adomian polynomials whose evaluation has been discussed in detail elsewhere [2, I] for various classes of nonlinearities. In the represen-
E. YEE
14
tation
of the
nonlinear
reaction
term
embodied
in (24),
it is implicitly
assumed that R(C(h)) is analytic in h. Again, substitution of (8) and (24) in (23) followed by the collection of all terms with equal powers of h on both sides of the equation,
leads to the following sequence
of equations:
(25a)
or, in general,
An examination of (25) indicates that the components C, of the solution C can be computed in terms of the preceding components provided the coefficients A,, associated with the nonlinear reaction term R(C), can be evaluated explicitly. A systematic computational procedure for the calculation of A,, that is readily amenable to algebraic or symbolic processing has been described by Rach [13]. T o illustrate the application of the recurrence relations of (25), consider a second-order reaction (i.e., LY= 2) with R(C) = y C ‘. For this particular nonlinearity, the Adomian polynomials are given as follows:
4 = syG$,, A, =
r(Cl” + 2C,C,),
A, = 2y(C,C, + C&J,), A, = r(C;
+ 2C,C,
A, = 2y( C,C,
+ 2C,C,),
+ C&z, + C&),
43 - y(C3” + c,c, +
c&T, +
c,c,),
Decomposition Method
15
and so forth. An important point should be noted from this example. For a given reaction term, the coefficient A, depends only on components C, for k < 12. Now, observe from (25) that the component C, of the solution depends only on C,_ r and A,, _ i, However, A,, _ 1 depends only on components C, for k < n - 1 and, as a consequence, the component C, is computable since it only depends on the preceding components of the solution. It should be noted that the terms Ki A,, generally cannot be evalu‘f;” terms of ‘t is form can be evaluated with ated analytically. Nevertheless, standard quadrature methods such as the commonly used Monte Carlo procedure [El. In connection with the latter point, it should be pointed out that very efficient numerical methods for multi-dimensional integration have been developed based on concepts form number theory 151. In any event, the quadratures involved in the evaluation of the terms qi, y, z A, can be efficiently executed with available numerical methods. 4.2. Vector case The decomposition method for the solution of the scalar reaction-convection-diffusion equation can be easily generalized to the vector reaction-convection-diffusion equation describing the dispersion of a number of chemically reactive species under the action of some prescribed flow field. As a simple illustration, consider a reacting mixture of two chemically reactive material substances A and B whose reaction processes can be summarized by the following kinetics:
A+A+I?%‘A+A+A, k;.k;
B+B+A
*
B+B+B,
where kl , ki and k, , k, denote th e forward and reverse reaction coeffcients for constituents A and B, respectively. Assume the reaction terms R, and R, characterizing the rates of formation or depletion of chemical constituents A and B, respectively, are given by RA(CA,CB)
= k,+CA2CB
i- k,cB3
- k,cA3
- k,+CACB2,
RB(CA,CB)
= ke+cAcB2
+ k,CA3
- k,CB3
- k,+CA2CB.
and
16
E. YEE
Here C * and C B denote the concentrations In view of (23), corresponding
the
parametrized
of species A and B, respectively.
reaction-convection-diffusion
equations
to species A and B assume the following form:
+
+=%;x’,,~~,(x’,t), (=a)
q::,y,zR~(CA>CB)
and
+w-,’. y R,(CA,CB)+%;;,,,.SB(~& I
where before,
S, and S,
,
are source
let us decompose
z
functions
for species
CA, CB, R,(C*,
RA(CA,CB)
ll=O
As
CB> as follows:
a
c A”Ct n=O
= 2
A and B, respectively.
CB), and R,(C*,
cc
CA =
(26b)
and CB =
c h”Ct ; n=O
h”A, and RB(CA,CB)
= t
h”B,,
n=O
to the where A,, and B, are the Adomian polynomials corresponding nonlinear reactions terms R, and R,, respectively. Now, substituting these parametrized decompositions into (26a) and (26b) and equating terms in equal powers of A yields the following systems of equations:
17
Decomposition Method or, in general,
The calculation of the Adomian polynomials A,, associated with the nonlinear reaction term R, is easily effected using the computational scheme developed by Rach [13] and leads to the following form:
A, = k;C;‘C;
+ k&‘,B3 - kiCff3
A, = k,+(2c,BC,$;
+ C;C;‘)
-k,(3C,f2C;)
+2c;c;c3B
+ CJ$;~)
+ 3c,B2c,B ) - k,(3C,f2c,A + C;Cf2
+ 2c($;c3A
+ 2c;C;c; + c,“cff”
+2C;C;C;4) - k;(C;’
+ C+Z;~),
+ 2C;C;C;
+ C;Ct2
+k,(3C,B2c,B
A, = kA+(C;Cf’
+ k,(3C,B2C,B)
- k;(2C,AC,fc;
A, = k,+(zC,BC,$,A
-k;(eC;C;C;
- k;c,$OB2,
+ c;c;2
+ c$C;~), + 2c$$c;
+ k;(C,B3
+ 3C,A2C3A + 6C;C~C;) + 2c;c($;
+ 3Cf2c;)
+ 2c;c$;
+ 3C,B2C,B + 6C;C;C9
- k,f(C,Acf= + 2c;c;c;
+ 2c;c;c9
and so forth. Analogously, the Adomian polynomials B, associated with the nonlinear reaction term R, can be calculated. However, from symmetry considerations, it is apparent that B, can be obtained from A,, by exchanging CA with CB, k: 5.
ILLUSTRATIVE
with ki,
and kA with k,.
EXAMPLE
In the present section, an example of the application of the decomposition method to the convection-diffusion equation will be given, mainly in order to
18
E.YEE
clarify the features of the method. In particular, the efficacy and accuracy of the method will be evaluated with reference to the example. Consider the gravitational settling of particles in a simple velocity field described by 3 = CU,*z*, 0, O), where IJ,* is a prescribed constant which characterizes the horizontal velocity gradient. In this case, the dimensional convection-diffusion equation reduces to dC*
d2C*
-_KK*-
_K*-=
dXS2
a*
d2C*
dC*
-u,*z*--
cw2
Ax*
dC*
+
v,*-
dz*
+ S*(x*,z*,t*),
where V, * is the gravitational settling velocity that is assumed to be directed along the z*-axis (i.e., along the vertical direction) and K * is the diffusivity of the particles. With reference to the previous equation, the source term is to be an instantaneous line source S*(x*, z*, t*) = where Q* denotes the mass of material released per unit length of the source. Now, it is convenient to introduce a characteristic implicitly
assumed
Q*s(x*)s(z*)13(t*),
length L = V,*/U,* and a characteristic velocity U = V,*. Introducing dimensionless coordinates with the aid of (2a) and rendering C” dimensionless by means of the reference concentration c = Q* U:/4rrK*, we may write the convection-diffusion
equation
for this example
in dimensionless
form as
follows: dC
--
dt
d2C
d2C K-
-K-=
c?x2
c?X2
-zE +g +S(x,
z,t),
diffusivity and S( x, z, t > where K = K * U,* /V, * 2 is the nondimensionalized = 47r K S( x)6( z)6(t). It is assumed that the material is released into an initially clean (i.e. uncontaminated) flow field, SO C(X, Z, t = 0) = 0. For this example, it is clear that the nth component of the Adomian decomposition
of the solution possesses
the form
Here, it is convenient to identify K (cf. (3)) with K. For simplicity in the following calculations, we assume that K = 1. In the present example, the integrals in (27) can be integrated analytically, as described previously. The Mathematics language for symbolic manipulation developed by Wolfram [18]
Decomposition
19
Method
was used to calculate ten components C, of the solution. The symbolic form of the components generated by Mathematics were output as expressions in the programming language C and these C-form expressions rated directly into a C control program for computation printing of the numerical presented below:
results. For reference,
were incorpoand formatted
the first five components
are
E.YEE
The exact solution for this example has been derived by Neuringer [lo]. In particular, the solution for the dimensionless concentration C is given by
1
C(x,z,t)
= t(1 + P/12)
(X/2 1/2=P
-
i
t(1
- t( z/4)y + P/12)
_
(z + ty 4t
I
.
Figure 1 compares the exact solution with the three-term approximation at z = 0 for the nondimensional time t = 0.01. It should be noted that just using three terms, the maximum relative percentage error between the exact
21
Decomposition Method -
True C(x,z=O,t=O.Ol) Series (3 terms)
80.00
0.00
FIG. 1. A comparison of the exact solution C(x, .z, t) at .z = 0 and t = 0.01 with a three-term approximation obtained by the decomposition method.
and approximate Figure
2 contrasts
solutions
is only 0.000314%
the exact
solution
for the cross-section
with the five-term
shown.
approximation
z = 0 for the nondimensional time t = 0.1. In this case, the f or the cross-section relative percentage error is 0.0000719%
at
maximum indicated.
Similarly, Figures 3 and 4 illustrate the comparison between the exact solution and seven-term and eight-term approximations at z = 0 and z = 1, respectively, for the nondimensional time t = 1. Again, it should be noted that the truncated solutions provide excellent approximations for the exact solution. Indeed, the maximum relative percentage errors between the exact and approximate solutions were 0.0510% and 1.26% for the cross-sections displayed in Figures 3 and 4, respectively. As an indication of the convergence of the N-term approximation to the exact solution. Figure 5 exhibits a comparison of truncated solutions for N = 3, 5, 7 with the analytical solution for the cross-section at z = 1 and t = 1. A more informative comparison is displayed in Figure 6, which shows the maximum relative percentage error between the exact solution and the N-term approximations for N = 2, 3, 4, 5,
E. YEE
22
s
True C(x,z=O,t=O. Series (5 terms)
1)
X FIG. 2. approximation
A comparison obtained
of the exact solution
by the decomposition
C(x,
7 t) at z = 0 and t = 0.1 with a five-term _,
method.
6, 7, 8, 9, 10 for the previous cross-section. Observe that the approximate solutions converge uniformly to the exact solution rather rapidly. Additional components of the solution can be readily added to provide an approximation to the exact solution to any prescribed degree of accuracy. The calculation of additional comoonents is lengthy but straightforward and, as already mentioned, can be easily determined with a symbohc manipulation program such as Mathematics. &
6.
CONCLUSION
The decomposition method has been applied to construct solutions for the fully three-dimensional nonlinear reaction-convection-diffusion equation. The solution is determined by computing the components of a particular decomposition without the need to discretize first the partial differential equation. As a consequence, the solution is obtained continuously in contrast to
Decomposition
23
Method
-
True C(x,z=O,t=l) Series (7 terms)
0.80 3
0.60
-
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
X FIG. 3. A comparisonof the exact solution C(x, .z, t) at z = 0 and t = 1 with a seven-term approximation obtained by the decomposition method.
finite-difference methods, which only provide a discrete approximation. Furthermore, each component of the decomposition depends only on the preceding component. In turn, the first component depends only on the source term and the boundary and initial conditions, both of which are known. Consequently, the components of the solution can be obtained through a sequence of easily programmable recurrence relations. Indeed, it is shown that the components obtained by application of the decomposition method to the convection-diffusion equation can be evaluated analytically for certain functional forms of the mean velocity field and eddy diffusivity tensor. Unfortunately, it is not possible to evaluate analytically the components in the solution of the nonlinear reaction-convection-diffusion equation although, in this case, these components can be calculated using standard quadrature methods. Even in the latter case, the new method for solving the parabolic partial differential equation seems somewhat better than the conventional finite-difference methods since discrete values are assumed only for the quadrature and not for the original partial differential equation. In other
E. YEE
24
M+H
True C(x,z=l ,t=l) Series (8 terms)
FIG. 4. A comparison of the exact solution C(w, z, t) at z = 1 and t = 1 with an eight-term approximation obtained by the decomposition method.
words, in the present
methodology,
discretization
is the last step rather than
the first so that this source of error does not propagate through the analysis. It has been demonstrated that the decomposition method performs well in the context of a simple example for which the exact solution is known and can be used for comparison. Indeed, this example indicates that the decomposition method provides an excellent approximation for the exact solution with the use of only a small number of components. Accuracy can be improved by simply adding additional terms in the decomposition, which involves routine but cumbersome calculations. Naturally, these calculations can be facilitated with some form of computer algebra system. Further studies are required in order to see how well the decomposition method may be adapted to the solution of the convection-diffusion equation for more complex flow configurations. In connection with the latter point, it should be mentioned that most of the work involves the analytical evaluation of the various integrals composing the components of the solution. While this may be -tedious and timeconsuming, it needs to be done only once for each type of flow geometry.
Decomposition
Method
0.40
25
7
FIG. 5. A comparison of the exact solution C(x, .z, t) at z = 1 and t = 1 with approximations obtained by the decomposition method for N = 3, 5, 7.
N-term
Indeed, this paper has provided an algorithm for the evaluation of these integrals and, although for complex flows, this may lead to long and tedious algebraic calculations, it is important to emphasize that these calculations can be readily performed using computer algebra processing systems such as Macsyma, Maple, or Mathematics. Work is presently underway to develop a general computer algebra program to automatically generate the components of the decomposition solution for any prescribed mean velocity field and diffusivity tensor and to combine these results with numerical algorithms in order to automatically generate computer programs for numerical computation. Some preliminary experience with this solution-scheme suggests that the combination of symbolic algebra with fast numerical algorithms for multidimensional integration leads to computer codes that are relatively simple and versatile when compared with the elaborate formulations and difficult implementations that generally characterize the finite-difference and finiteelement methods have been used to solve unsteady reaction-convectiondiffusion equations.
26
E. YEE
0
10
5
Number
of Components
FIG. 6. Number of Components. The maximum relative percentage error between the exact solution and N-term approximations for the cross-section described by C(x, z = 1, t = 1).
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