At present the authors are developing a procedure, based on Herrera's algebraic theory of boundary value problem, which systematically uses localized adjoint.
s t , , American Institute 01 Hydrology i-p 321-327
Localized Adjoint Method as a New Approach to Advection Dominated Flows ISMAEL HERRERA
MICHAEL A. CELIA
Insl de Geofisica. UNAM ApdO Pola122-582
Mass Insl of Technology Depl of Civil Eng
14000 Mexico. D.F.
Cambridge, Massachusetts 02139
JUAN D. MARTINEZ Insl deGeofiscia, UNAM 04510 Mexico. D.F.
ABSTRACT
Solution of advection-dominated transport problems by discrete interior methods is usually accomplished by employing some type of upstream weighting. Upwinded finite element formulations have also been developed. At present the authors are developing a procedure, based on Herrera's algebraic theory of boundary value problem, which systematically uses localized adjoint formulas. Here, the basic ideas of the method are explained. Also, the present state of development, as it applies to one and multi-dimensional steady-state and transient problem is revised.
INTRODOCTION
\ The numerical
solution of the advectlve-dlfuslve transport equation is a problem of great importance because many problems is science and engineering Involve such mathematical model. The numerical treatment of advection dominated proceses is quite difficult. The procedures available derive from two main approaches: standard semidiscretlzation and Eulerian-Lagrangian. The lain distinguishing feature of the latter is the use of characteristics to c~ry out the discretization in time. Most formulas that have been developed ~lng a standard semidiscretization approach have been based on upstrea, ~Ighting techniques, whose development is essentially ad-hoc. ~ alternative and very promising approach has been introduced by Herrera (1-4]. In past work, this has been referred as "Optimal Test Function ~thod". However, from a technical point of view, it would be more appropriate to call such procedures "Localized Adjoint Methods". Also, in this manner, the method would be more clearly distlnguised from other Focedures. Thus, this is the terminology that will be adopted in what follows. 321
(1.1)
(1.2)
SO GenerallY. obtain
the a
system
representation trial)
that
u'
= La
exact
(l.u'
-fO)dX
N equations a f
IX IX However.
to the information exact be established
the
"'IX
of
possessing
functions.
The relation actual one. can
clear
System
unique
of
u
solution about making
solution
terms
IX
has it of
the
representation
the u.
u
the
= 1.
many
solution. in
this
= O.
(1.3)
is
solutions.
System is
exact solution following
but
customary
an
to
{fl
in
f
artifice
.,..,
contained observations.
order
introduce }of
base
K that
in
an From
bears
approximate (1.2) it
satisfies
u = 1
NI.
(1.4)
N
N
322
to the
(1.5)
(or no
is
.LOCALIZED
ADJOINT METHOD AS A NEW APPROACH
TO ADVECTION
DOMINATED
FLOWS
necessary to have available Green's formulas similar to (1.1), but which are applicable even when the functions considered are not smooth. This is because in most applications weighting functions are localized (i.e., they have local support) and they usually violate the smoothness assumptions at the boundary of their support. Even more, the development of a theory spplicable to carry out the analysis when both base and test functions are fully discontinuous, is most desirable since standard theory of distributions is not applicable to that case. Herrera [1-4), recently developed an "algebraic theory of boundary value problems", in which the analysis of the information contained in an approximate solution can be carried out, when both trial and test functions are fully discontinuous. Such setting is Ideal for localizing the adjoint equation (1.6).
LOCALIZED
ADJOINT
METHODS
"Localized adjoint methods" consist in making systematic use of the resulting equations to analyse the information contained in approximate solutions. SInce the quality of the results obtained with a numerical method depends, in an important manner, on the weighting functions used, one of the main goals of localized adjoint methods, thus far, has consisted in developing improved weighting functions. In recent years the authors have applied localized adjoint methods to advection diffusion equations. The numerical results indicate that this kind of methods possesses definite advantages over other procedures (5-8). Further improvements are expected to come from work presently under way, in which Herrera's theory is applied in space-time. The localized adjoint methods derived in this manner, clarify and perfect Eulerian Lagrangian procedures (9,10). In what follows Herrera's (1-4) algebraic theory of boundary value problems is briefly explained. For a more detailed presentation and proofs of the results, the reader is referred to the original papers (1,2,4). The general abstract boundary ~ problem to be considered corresponds to one with prescribed jumps and it is defined by the system of equations:
=
=
=
V c D
(2.1)
Where D is the linear space of admissible functions, P, B and J are bilinear functionals, while f, g and j are prescribed functionals (the data of the problem). Generally, the bilinear forms P, Band J are associated with the differential operator, the boundary values and the jumps accross discontinuities. When the sought solution is smooth, the prescribed jump j=O. Under the assumptions of the general theory, the three equations (2.1) are equivalent to the single variational formulation «P-B~'J)u.
v> = vcD
(2.2)
This variational formulation is said to be "the variational formulation in terms of the data of the problem", because Pu, Bu and Ju are prescribed. Making use of the General Green's formula (Q.!:, ooerators in discontinuous llilQ§.
(1. 4 )
«P-B-J)u.v>
= «Q-C-K)v,U>
323
(2.31
HERRERA,
the
variational
formulation
(2.2)
CELIA AND MARTINEZ
is
«Q-C-K)
transformed v,u>
Into
=
(2.4)
This is said to be "tIle v.ariation~l formulation in terms of the souglJt information", because Qu Cu and Ku are not prescribed. Generally, Qu supplies \nformation at the interior of the region of definition of the p~oblem, C u supplies information About the complementary boundary values, and K u yields the average of the solution (and derivatives) accross surfaces of discontinuity. Regarding the general Green's formula (2.3), it is assumed that the bilinear forms P and Q are !:Q!:!!!li adloints, in the sense of the theory. The boundary bilinear forms Band C, can be constructed using arguments which are essentially standard. However, the construction of the operators J and K is not standard. The systematic manner supplied by Herrera's algebraic theory will be succintly explained now. Let the region of definition of the problem a, be decomposed into two subregions a and a and assume the discontinuities of the trial and test I
functions Then, are
let
II
can
only
the
space
linear
spaces
having
constructed
occur of
on the
common boundary
admissible
whose
functions
elements
are
r,
be D=O
I
functions
separating + D
II
,
01
where
defined
in
and
0
I
01
all
and
0 11 all,
and
respectively. After
I the
operators
Band
C,
two
classes
of
functions,
50
and 5"0, are considered. Elementes belonging to 51 will be said to smooth while those belonging to 5" will be said to be right-smooth. I " classes 5 and 5 , are taken to be con.1u~ate subspaces; i.e., «P-B)u,v> Given 1
any
U = {u I' t U II c
DI
element U II }
I
is
ucD, t
where
the
u
I
write c D
I
left-smooth
u = {u ,u I
is
lu] (2.6)
together,
}
with
of
u c 1
extension 0
I I
of
the
1
= uI
and
to
0
1
I
I
I
the
(see
left The
(2.5
D
u c D
theory
(u 1 + u) /2
u of
cD.
II
u
I I
cD,
Define
II
[1).
while
I I
existence
of
Then,
such it
.1
(2.6a) (2.6b)
-[u)
1
/2
1
Observe that u c D and [u] c D. These functions will be called the (abstract) average and jump, respectively, based on the left-smoothness criterium. Correspondingly, given any function vc D, one considers the functions v v r
or
is
lmply U = U
1
to
hypothesis
.1 U =
Equations
II
left-smooth
extension
extensions is a fundamental possible to define
UE S I & v E S
= «Q-C)u,v>,
be
I II
r
,
v, v and [v], which are defined replacing the left-smooth class of functions, by the right smooth class of functions. ( 1 r By as~umption, equation 2.5 holds whenever ucS and vc S. However, P-B (Q-C) .in general. Thus, one defines R = P -B-
324
(Q
C)
(2.8)
LOCALIZED
ADJOINT
In addition, every
u:
{u
METHOD AS A NEW APPROACH TO ADVECTION
the
operators
,u
} £ D,
by
R u,v>
=
Dand
I
y.>;
I
v={v
.v
I
II
II
Using
->
=
validity
of
equations
(2.10)
is
pre
(lJ.
ADVECTION-DIFFUSION EQUATIONS
The methodology Is applicable to steady state and time dependent problems In one and several dimensions. A.~ Dimensional Problems. The steady state leads to ordinary differential equations. for which four algorithms were developed by Herrera et al. [3] They were derived by seeking nodal Information only. Algorithm Aigorithm Algorithm Algorithm
For steady
state
1.2.3.4.-
the
The value of The value of The value of The function others.
advection
the function and Its derivative; the function only; the derivative only; at some nodes and the derivative
diffusion
~u =d (Ddu/dx)/dx
equation
-Vdu/dx+Ru
with
= fa(x),
linear
at
sources
o~ x ~ l
some
is
(3.1)
and i"v
E d(Ddv/dx)/dx
The domain [O,l) is divided equal, [Xo,Xl) ,... [x ,X). [-1
nodal
points,
test
[
functions
-d(Vv)/dx
into E subintervals or elements, not necessaryly In order to concentrate all the information at are
required
to
algorithms 2 to 4, three-diagonal matrices are yields tetra-diagonal matrices [3). Thus far, ewployed for constructing the wieghting functions l "«;0 In [5) the (variable) coefficients approximated approximate 1.°9>«=0 at
by piece-wise polynomials, the test functions, which collocation
points
(3.2)
+ Rv
only.
satisfy
while in [6] were required Using
325
to"
«;0
,locally.
For
contained, while algorithm 1 two procedures have been which satisfy the equation of the equation were polynomials to satisfy
a semi-discretization
were used to the equation approach,
HERRERA,
the the
extension to time advection-diffusion
dependent equation .fu=
CELIA AND MARTINEZ
For
problems is straightforward can be written as c3 u/c3
case
(3.3)
fa (x,t)
t
this
where f is differential operator in x, given by equation (3.1). Thus, in the term au/at was incorporated in the right-hand side of equation (3.1), each time step, and the solution procedure for steady state problems applied to the resulting equation.
[7] at was
B.- Problems in Several Dimensions. The results for advection dominated transport-diffusion problems, in several dimensions, have been presented in [8). The procedure differs from that for one-dimensional problems, mainly, due to the fact that T-complete systems [11) are infinite in this case. Of course, only a finite number of test functions are used in actual applications, introducing a truncation error in this manner.
C.Numerical Results. For one dimensional steady state problems the procedure yielded [5,6] very efficient and highly accurate algorithms, which advantageously compare with other algorithms. For time dependent one-dimensional situations, only algorithms 1 and 2 were 0 applied. These correspond to the use of fully discontinuous and C weighting functions, respectively. Two test problems were treated to demonstrate the performance of the new method: the propagation of an initial step discontinuity and of a Gauss hill. The results and the efficiency of the procedure were compared with other methods, testing the effect of changing several parameters such as e (in time), element Peclet number, as well as Courant number. The conclusion was reached that solutions obtained with this method are as least as good as those from the best available interior methods
[7]. Similar conclusions were drawn in the case of two-dimensional problems. For details see [8]. In general, localized adjoint methods allow very effective treatment of boundary conditions, but this is specially noticeable for two-dimensional problems, due to its greater difficulty.
REFERENCES
Herrera, Methods
I., for
Herrera, Equations
I., Ibid, Part 1(3), 159-186,
Herrera, I., for Partial [4]
Herrera, Symp.,
[5]
"Unified approach Partial Differential
Chargoy, Differential
I., Laramie,
"Unifying
L.
to
numerical methods", Equations 1 (1), 25-44.
2, Num. 1985.
Methods
for
and Alduncin, G. Ibid, Equations (4), 241-258,
concepts
in Applied
Partial Part 3, 1985.
Mathematics",
Part 1985
I,
Num.
Differential Num.
Proc.
Methods
G.
Gail
1985.
Celia, M. A. and Herrera, I., "Solution of ordinary equations by a unified theory approach", Num. Methods Differential Equations, 3(2), 117-129, 1987.
326
differential for Partial
[6)
Herrera, I., "The algebraic theory approach for ordinary equations: highly accurate finite differences", Num. Partial Differential Equations, 3(3), 199-218, 1987.
(7)
Celia, M.A.; Herrera, I; Bouloutas, E. and Kindred, J.S. "A New Numerical approach for the advective-diffusive transport equation", Num. Methods for Partial Differential Equations. 1988 (in press).
(8)
Celia, M.A.; Bouloutas, E. and Herrera, Test Funtions Methods to Multi-dimensional 7th. International Conference, 1988. Herrera, published).
I.,
[ 10]
Celia, M.A., published).
[11)
Herrera,
I.
Ewing,
R. E.;
Russell,
Boundary
T.;
Methods:
Cella,
1.
M. A.
Herrera, An algebraic
1984.
327
"Application Flow Problems.
of Optimal Proceedings
and Russell. and
Ewing,
theory
differential Methods for
T. R.E
Pitman,
(to
be
(to
be
London,
