Localized Adjoint Methods in Water Resources Problems

Localized Resources

Adjoint Methods Problems

in Water

I. Herrera

Instituto

de Geofisica, UNAM, Apdo. Postal

22-582, 14000 Mexico DF, Mexico ABSTRACT Lacallzed

Adjoint

applicabilIty, discontinuous transport

Method

based fields.

Here

diffusion

Localized addition

to

It

Method

(M.A. the

Is for

R.E.

author).

of

Green's

for

In

which

Eulerlan-Laqranqlan

the

has

been and

connection

with

formulated T.F.

by

the

Russell,

ELLAM development

treats

yleldlnq

wide

formulas

presented

Ewlnq

The

methods,

systematically,

new methodoloqy

(ELLAM)

Cella,

characteristic

a

author's

problems

Adjoint

LAM qroup

Is

on the

boundary

In

unifies

conditions

conservative

schemes.

1. INTRODUCTION The numerical equation

is

science

and

numerical

solution

a problem

of

of

engineering

treatment

the

great

advective-diffusive

importance

involve

of

such

advection

The

approaches:

standard

semidiscretization

distinguishing

feature

main

characteristics

to

formulas

that

semidiscretization techniques,

out

Herrera

and very

[1-5J

referred

as

technical

point

procedures

and

"Optimal

"Localized

method

would

Hence,

this

Test

of view,

it

Adjoint

be more clearly is

the

the

the

terminology

promising

main

in

time.

of Most

standard weighting

ad-hoc. has

been

work,

has

However,

be more appropriate Also,

distinguished

introduced

this

Method".

Methods".

use

a

on up-stream

past

Functions

the

using

approach In

that

quite

two

is

discretization

is essentially

would

is

from

latter

developed

coworkers.

The

and Eulerian-Lagrangian.

have been based

whose development

in

model.

processes

derive

of

been

approach

An alternative by

available

carry have

transport many problems

mathematical

dominated

difficult.

The

procedures

because

in

from

to call this

from

been

has been adopted

such

manner

other

a

the

procedures. more recently

[6,7). The simple to

starting idea.

functions

point

Let

of

localized

~ be a differential

defined

in

a

region

adjoint

methods

operator Q and

let

.

that

will

~

be

is

a

very

be applied its

formal

434 Computational Methods in Surface Hydrology adjoint.

Then,

when u and v satisfy

Green's

formula

.

Savfudx

= Sau~ vdx

Equation

(1.1)

the

method of

weighted

residuals.

the

equation

subjected

= fa '

formula

(1.1)

usually

considers

a

Then,

solution

of

In

this

Consider

a

the

system

in

the

order

to

customary

obtain

problem

of

of

for

of

solving

weighted

weighting

u'

is

which

(or

said

to

Green's

residuals,

one

test)

functions

an

approximate

be

a=l,...,N.

a

of

when

system of

to

the

conditions

Sa'Pa(~u'-fa)dx=O, Generally,

interpretation

(1.2)

method

a function

problem

~onditions,

.

a convenient

boundary

applies.

{'P1,...,'PN}.

allows

in

to homogeneous

boundary

(1.1)

is satisfied.

~u

suitable

N equations

system

introduce

(1.3)

possessing

a

(1.3)

has many solutions, a unique

representation

u'

but

solution, =

it

LA ~

of

is the

a a approximate

trial)

solution

functions.

bears

little

The

following that

one

derive

which that

is

of

the

this

with

the

between

the

~

permit

(1.3)

and

(1.4)

SQ~«fu'dx

or Iou'of

of Green's

of

integrable

square

equations

allows

A function

u'

projection

on the

{of.lpl""'!!~}' As a matter exact

solution

is

v,

exact

From (1.2),

exact

solution

it

is

clear

Ipa.dx

the

and given

in which

the

by IOuvdx.

spanned ~ith

in

inner

Then,

solution

space

is

space £!, product

the

of

system

of

interpretation:

coincides

contained

(1.6) the Hilbert

following

this

(1.5)

Consider

an approximate

of fact,

,N.

a. = l,...,N.

(1.1).

is

(1.4)

imply ~ = I,...

functions

u and

(1.6)

together

formula

two functions,

the

a=l,...,N.

/pa.dX = Iouof

by virtue

actual

and the

satisfies

= SQ~«fudx

..

the

solution

one.

(or

that

u.

about

Sa'Pa(~u-fa)dx=O, Equations

base

an artifice

establishing

information

u,

is

solution

an approximate

solution

{~l""'~N}of

an approximate

actual in

system

representation

observations

exists

contained

the exact

terms

However,

relation

relation and

in

by that

~

~

if and

the

system

of

the

exact

information

an approximate

one.

only

if,

its

of functions solution

u.

about

the

Computational Methods in Surface Hydrology In this

light,

the

a procedure the

for

The very

For

simple

this

it,

are

the

and

to

that

boundarv which

lear

analysis

adjoint

developed

by the

the

the

of

Green's

numerical

(i.e.,

satisfy

the

the

more,

the

analysIs

discontinuous,

distributions

they

smoothness

Even

out

when the

most

localized

carry

me~hods.

even

support.

fully

of

the

is

corresponds

on u.

system

out

is

when

is

not

most

applicable

diffusion

problems.

which R.E.

formulated one

with

a

use of

that

a

for

problem space jumps.

thus

Ju = j

used, far, In

connection

has this with

EQUATIONS

considered

by the

functions

D

and

by

the

of It

to

Since

functions

in

and

theory

functions.

explained

~

Herrera

method

methods,

weighting

being

solutions.

weighting

adjoint

linear

prescribed

Bu = g;

ideal

numerical

is

defined

of equations: Pu = f;

in test

presently I.

with

on the

is

boundary in

Q[

is,

and

is

Ewing,

S FORMULAS FOR TRANSPORT-DIFFUSION

abstract

theory that

trial

are

in approximate

obtained

improved

methodology

both

setting

systematic

localized

LAM

property;

(1.~).

manner,

of

"alRebraic

when

(LAM)",

(H.A Celia,

results

an that

Such

contained

the

developing

to

precisely

in making

main goals in

tions

.but e

to are

methods

information

The general

t the

their

clarifies

available

in

in

.

be desirable

discrete

since are

carried

LAM group

GREEN-HERRERA'

theory

of

equation

in an important

transport

its

have

do not

discontinuous.

consist

quality

consisted

2.

be

adjoint

Russell),

paper

to

developed

with

can

~e

"Localized

depends,

of

recently

localizing

one of

analyze

would

can be applied

usually

theory

problems"

fully

the

:tl-.

[1-5],

~

the

analyze

contained

presented

to

functions

functions

are

T.F.

Just and it

smooth,

applicable

standard

functions

Itlon

as

case.

Herrera

that

test

that

not

boundary

a theory

since

but

and they the

of

base

result

manner,

weighting

at

development

can be in~erpreted IX

information

solutions

necessary

consIdered

support)

desirable

is (1.1),

requirements

both

it to

applications have local

IX

actual

precise

in a systematic

similar

functions

onelions

and

of approximate

purpose

formulas

the

= rA ~

solution.

much the nature apply

u'

extrapolating

approximate

to

representation

435

(2.11

436 Computational Methods in Surface Hydrology where and

P, Band

J are

.. are jcD

..

Here,

D

linear

prescribed

is

the

functionnals procedure C and K,

as Q,

depend

on the

smoothness

the

are

conditions

boundary

this

case

single

on D).

considered.

system

of

operators

J are

equations

Band

of a

J,

as

definitions

conditions

functionals

and

f,

g and

in particular,

when

jump j=O.

constructed

are

space supplies

Their

boundary

gcD

problem).

the

P,

on.

the problem;

which

the

is

.

fcD,

theory

the prescribed

P,

of

D

later the

Band

for

data

The linear of

functionals operators

the

the

operator,

data

while

The general

deriving

is smooth,

The bilinear

of

be introduced

by the

solution

are

D (i.e.

differential

determined

sought

dual

for to

operators, (the

defined

systematic

valued

functionals

algebraic

well

j,

functional

f.!!lli

(2.1)

so that

disjoint

is

they

[2,5].

In

to

the

equivalent

equation (P -B

This

equation

be verified

This

is

data

of

-J)u

is

= f

«P

-Bto

-j

a variational

observing

said

-g

formulation

that

J)u,v> be

(2.2)

-j,v>

variational

because

of

the problem,

is equivalent

=

functional,

C u and

the

-g

...

formula

(2.4)

(2.3)

variational

~

[2,5]

-K

=