Lawrence Berkeley National Laboratory
Recent Work
Title
AN ADAPTIVE FINITE DIFFERENCE SOLVER FOR NONLINEAR TWO POINT BOUNDARY
PROBLEMS WITH MILD BOUNDARY LAYERS
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https://escholarship.org/uc/item/569825w9
Author
Lentini, M.
Publication Date
1975-11-01
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SIAM
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LBL-4226
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'
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print
AN
ADAPTIVE
FINITE
DIFFERENCE
SOLVER
FOR
NONLINEAR
TWO
POINT
BOUNDARY
PROBLEMS
WITH
MILD
BOUNDARY
LAYERS
M.
Lentini
and
V.
Pereyra
November
1975
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for
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S.
Energy
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under
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This document was prepared
as
an account
of
work sponsored by the United States
Government. While this document is believed
to
contain correct information, neither the
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of
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•
'i'
..
"An .Adaptive
Finite
Difference
Solver
for
Nonlinear
Two
Point
Boundary
Problems
with
Mild
Boundary
Layers,"
M.
Lentini
and
V.
Pereyra
ABSTRACT.
A
variable
order
variable
step
finite
difference
algorithm
for
approximately
solving
m-dimensional
systems
of
the
form
y'
==
f(t,y),
t E
[a,b]
subject
to
the
nonlinear
boundary
conditions
g(y(a),y(b))
==
0
is
presented.
A
program,
PASVAR,
implementing
these
ideas
have
been
written
and
the
results
on
several
test
runs
are
presented
together
with
comparisons
with
other
methods.
The main
feautres
of
the
new
pro-
cedure
are:
a)
Its
ability
to
produce
very
precise
global
error
estimates,
which
in
turn
allow
a
very
fine
control
between
desired
tolerance
and
actual
output
precision.
b)
Non-uniform
meshes
allow
an
economical
and
accurate
treatment
of
boundary
layers
and
other
sharp
changes
in
the
solutions.
c)
The
combination
of
automatic
variable
order
(via
deferred
corrections)
and
automatic
(adaptive)
mesh
selection
produces,
as
in
the
case
of
initial
value
problem
solvers,
a
versatile,
robust,
and
efficient
algori
tbm.
I
0
n o
'
•
AN
ADAPTIVE
FINITE
DIFFERENCE
SOLVER
FOR
NONLINEAR
TWO
POINT
BOUNDARY
PROBLEMS
WITH
MILD
BOUNDARY
IAYERS
*
M.
Lentini
and
+
v.
Pereyra
1.
Introduction
We
are
interested
in
developing
usable
software
for
two-
point
boundary
problems
for
m-dimensional
systems
of
the
form
y'
=
f(t,y)
tE
[a,b]
(
1.
1 )
g(y(a),y(b))
0
In
[8,
9]
we
have
already
presented
a
finite
difference
algorithm
(SYSSOL),
based
on
deferred
corrections,
which
has
variable
order
capabilities.
SYSSOL
uses
only
uniform
meshes,
which
can
be
refined
automatically
in
order
to
reduce
the
maximum
norm
of
the
(esti-
mated)
global
error
on
the.
current
mesh
below
a
requested
tolerance.
SYSSOL
behaves
quite
adeguately
for
many
problems
(see
[
8,
9]),
but
becomes
inefficient
or
does
not
work
at
all
as
soon
as
the
(*). .
_Department
of
Applied
Mathematics,
Caltech,
Pasadena,
California.
(+)Department
of-Mathematics,
University
of
Southern
California,
Los
Angeles.
The work
of
M.
Lentini
was
partly
supported
by
Conicit,
and
that
of
V.
Pereyra
by
the
u.s.
Energy
Research
and
Development
Administration
while
visiting
Stanford
University
and Lawrence
Berkeley
Laboratory,
and
the
National
Science
Foundation
at
USC.
Both
authors
are
on
leave
of
absence
from
Universidad
Central
de
Venezuela,
Caracas.
Z
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