High-Order

Accurate

Schemes

for

Incompressible

Viscous

Flow

John

C.

Strikwerday

Computer

Sciences

Department

University

of

Wisconsin-Madison

Abstract

We

present

new

finite

difference

schemes

for

the

incompressible

Navier-Stokes

equations

The

schemes

are

based

on

two

spatial

differencing

methods

one

is

fourth-order

accurate

and

the

other

is

sixth-order

accurate

The

temporal

differencing

is

based

on

backward

differencing

formulas

The

schemes

use

non-staggered

grids

and

satisfy

regularity

estimates

guaranteeing

smoothness

of

the

solutions

The

schemes

are

computationally

efficient

Computational

results

demonstrating

the

accuracy

are

presented

AMS

-LRB-

MOS

-RRB-

classifications

Introduction

High-order

accurate

finite

difference

schemes

are

important

in

scientific

computation

because

they

offer

a

means

to

obtain

accurate

solutions

with

less

work

than

may

be

required

for

methods

of

lower

accuracy

Finite

difference

methods

are

attractive

because

of

the

relative

ease

of

implementation

and

flexibility

In

this

paper

we

present

new

finite

difference

schemes

for

the

incompressible

NavierStokes

equations

The

schemes

are

based

on

two

spatial

differencing

methods

one

a

fourthorder

accurate

method

and

one

a

sixth-order

accurate

method

There

are

several

temporal

differencing

methods

presented

in

section

These

temporal

schemes

can

be

used

with

either

of

the

spatial

differencing

methods

The

temporal

differencing

is

based

on

backward

differencing

formulas

-LRB-

BDF

-RRB-

that

are

used

for

stiff

ordinary

differential

equations

The

schemes

are

implicit

and

appear

to

be

unconditionally

stable

for

the

Stokes

equations

-LRB-

A

rigorous

stability

analysis

is

the

subject

of

further

research

-RRB-

High-order

methods

have

been

presented

by

Rai

and

Moin

-LSB-

-RSB-

and

Lele

-LSB-

-RSB-

for

the

fractional

step

method

proposed

by

Kim

and

Moin

-LSB-

-RSB-

There

is

an

excellent

study

of

these

methods

in

the

paper

by

Tafti

-LSB-

-RSB-

A

disadvantage

of

these

methods

is

that

because

they

are

explicit

there

is

a

severe

stability

limit

on

the

time

step

Moreover

as

pointed

out

by

Perot

-LSB-

-RSB-

the

pressure

for

fractional-step

methods

can

be

no

better

than

first-order

accurate

in

time

Projection

methods

also

have

difficulty

with

higher

order

accuracy

in

time

see

Shen

-LSB-

-RSB-

This

is

not

so

for

the

methods

presented

here

where

the

pressure

can

be

determined

to

a

high

order

of

accuracy

For

steady

flows

the

method

of

Aubert

and

Deville

-LSB-

-RSB-

can

be

applied

to

yield

fourth-order

accuracy

at

the

expense

of

increasing

the

number

of

unknowns

and

computational

complexity

of

the

system

All

of

these

methods

use

staggered

meshes

y

This

work

was

supported

in

part

by

the

U.S.

Army

Research

Office

under

grant

and

the

National

Science

Foundation

under

grant

The

schemes

presented

in

this

paper

are

for

orthogonal

Cartesian

grids

on

nonstaggered

grids

that

is

the

velocity

components

and

pressure

unknowns

are

assigned

to

a

common

grid

The

schemes

are

for

the

two-dimensional

Navier-Stokes

equations

however

the

methods

for

obtaining

the

equations

extend

easily

to

three

dimensions

and

to

generalizations

of

the

Navier-Stokes

equations

A

second-order

scheme

similar

to

the

ones

presented

here

was

presented

in

-LSB-

-RSB-

although

it

used

a

less

efficient

solution

procedure

The

methods

presented

here

have

been

incorporated

into

a

domain

decomposition

method

The

schemes

presented

here

have

been

tested

on

sample

problems

with

low

Reynolds

numbers

The

tests

show

that

the

schemes

are

very

accurate

and

efficient

for

these

low

Reynolds

number

computations

The

extension

of

these

methods

to

high

Reynolds

number

is

the

subject

of

further

research

The

nondimensional

time-dependent

incompressible

Navier-Stokes

system

of

equations

is

ut

u

r

-LRB-

u

uT

-RRB-

rp

f

-LRB-

-RRB-

r

u

g

-LRB-

-RRB-

The

vector

function

u

is

the

velocity

and

the

scalar

function

p

is

the

pressure

The

Reynolds

number

R

measures

the

strength

of

the

inertial

effects

relative

to

the

viscous

effects

Notice

that

the

pressure

appears

only

in

-LRB-

-RRB-

and

only

in

terms

of

its

spatial

derivatives

We

refer

to

equations

-LRB-

-RRB-

as

the

momentum

equations

and

equation

-LRB-

-RRB-

as

the

divergence

equation

The

functions

f

and

g

are

considered

to

be

given

data

In

most

problems

the

function

g

in

-LRB-

-RRB-

is

identically

zero

but

we

include

the

general

case

because

it

fits

in

naturally

with

our

methods

and

it

is

useful

in

checking

the

accuracy

of

the

computer

implementation

of

the

methods

In

particular

the

accuracy

can

be

checked

by

choosing

the

velocity

and

pressure

to

be

arbitrary

polynomials

of

the

proper

degree

-LRB-

see

section

-RRB-

In

the

limit

as

R

tends

to

zero

with

a

rescaling

of

t

p

and

f

the

Navier-Stokes

system

can

be

replaced

by

the

time-dependent

Stokes

system

which

is

ut

u

rp

f

-LRB-

-RRB-

r

u

g

-LRB-

-RRB-

We

consider

the

Navier-Stokes

system

holding

in

a

domain

and

to

specify

a

unique

solution

boundary

conditions

must

be

given

The

simplest

conditions

are

to

specify

the

velocity

u

on

the

boundary

i.e.

u

b

on

-LRB-

-RRB-

This

is

called

the

Dirichlet

boundary

condition

To

limit

our

discussion

we

only

consider

Dirichlet

boundary

conditions

in

this

paper

The

modifications

needed

for

other

boundary

conditions

should

not

be

difficult

to

implement

The

system

-LRB-

-RRB-

and

-LRB-

-RRB-

has

a

solution

only

if

the

integrability

condition

Z

n

b

Z

g

-LRB-

-RRB-

is

satisfied

This

condition

is

a

constraint

relating

function

g

in

-LRB-

-RRB-

and

-LRB-

-RRB-

and

the

boundary

data

b

in

-LRB-

-RRB-

The

schemes

we

develop

are

derived

using

the

difference

calculus

By

considering

the

total

system

in

the

derivation

we

obtain

schemes

that

are

compact

that

is

the

stencil

of

the

scheme

is

about

as

small

as

possible

In

particular

the

schemes

presented

here

have

smaller

stencils

than

those

of

Lele

-LSB-

-RSB-

and

Rai

and

Moin

-LSB-

-RSB-

However

to

obtain

usable

schemes

two

other

aspects

must

be

taken

into

account

These

are

the

regularity

of

the

scheme

and

the

behavior

at

boundaries

The

regularity

of

the

scheme

is

important

to

assure

that

the

solutions

are

smooth

that

is

the

high

frequency

modes

are

prevented

from

dominating

the

error

The

schemes

have

parameters

that

remove

these

high

frequency

modes

often

referred

to

as

checkerboard

pressure

oscillations

The

difficulty

at

the

boundaries

is

related

to

the

size

of

the

stencil

Since

the

stencil

increases

in

size

as

the

order

increases

the

amount

of

modification

required

at

the

boundary

also

increases

These

topics

are

addressed

in

sections

and

The

schemes

we

derived

can

be

used

for

both

the

steady-state

and

time-dependent

equations

We

consider

only

schemes

for

which

the

temporal

differencing

and

spatial

differencing

are

independent

of

each

other

The

spatial

differencing

is

discussed

in

sections

and

and

the

temporal

differencing

is

discussed

in

section

We

avoid

modifying

the

Navier-Stokes

equations

such

as

is

done

with

the

Poisson

pressure

equation

method

One

difficulty

with

such

methods

is

the

need

to

decide

on

additional

boundary

conditions

especially

on

the

pressure

This

is

also

true

for

projection

methods

see

Gresho

-LSB-

-RSB-

In

our

approach

the

linear

systems

that

must

be

solved

to

determine

the

solution

at

each

time

step

involve

both

the

momentum

and

divergence

equations

These

large

systems

are

solved

by

preconditioned

GMRES

methods

-LSB-

-RSB-

One

advantage

of

our

approach

is

that

the

pressure

can

be

obtained

with

the

same

order

of

accuracy

in

space

as

the

velocity

and

better

than

first-order

in

time

which

is

the

limit

with

fractional-step

methods

-LSB-

-RSB-

We

do

not

use

the

finite

volume

approach

relying

on

the

power

of

the

symbolic

difference

calculus

to

obtain

high

accuracy

with

compact

stencils

Our

schemes

do

not

satisfy

exact

conservation

laws

for

mass

or

momentum

The

accuracy

of

the

solutions

implies

that

the

conservation

laws

should

be

satisfied

to

a

high

degree

of

accuracy

The

schemes

are

based

on

the

conservation

form

of

the

differential

equations

The

structure

of

the

paper

is

as

follows

In

section

we

present

the

notation

for

the

basic

difference

operators

In

sections

and

we

present

the

spatial

differencing

methods

for

orders

four

and

six

respectively

In

section

we

discuss

the

numerical

boundary

conditions

needed

for

both

schemes

Section

discusses

the

regularity

of

the

two

schemes

for

steady-state

computations

The

multistep

schemes

used

for

the

temporal

differencing

are

discussed

in

section

and

in

section

the

numerical

tests

of

the

methods

are

discussed

Conclusions

are

presented

in

section

Notation

We

develop

our

schemes

for

regular

two-dimensional

Cartesian

grids

with

grid

spacing

x

and

y

respectively

We

use

the

notation

to

denote

the

first-order

central

difference

with

respect

to

x

and

is

defined

by

fi

fi

x

The

forward

and

backward

operators

are

denoted

by

ffix

and

ffix

respectively

and

are

defined

by

ffix

fi

fi

fi

x

and

ffix?fi

fi

fi

x

The

standard

second-order

central

difference

is

denoted

and

is

defined

by

ffix

ffix?fi

fi

fi

Difference

operators

ffiy

etc.

are

defined

similarly

We

obtain

most

of

our

difference

formulas

from

basic

identities

relating

derivatives

to

the

difference

operators

and

For

the

first-derivative

we

use

the

identity

@x

xffix

sinh

xffix

xffix

-LRB-

-RRB-

see

-LSB-

-RSB-

The

basic

identity

we

use

that

relates

the

second

derivative

to

is

see

-LSB-

-RSB-

sinh

xffix

x

-LRB-

-RRB-

By

expanding

these

expressions

as

Taylor

series

in

x

to

appropriate

powers

we

may

obtain

difference

approximations

of

any

order

To

handle

the

modifications

at

the

boundaries

we

use

two

formulas

that

relate

forward

and

backward

differences

These

are

ffix

ffix

xffix

-LRB-

-RRB-

and

ffix

ffix

xffix

These

two

relations

both

arise

from

the

identity

ffix

ffix

ffix

ffix

x

The

Fourth-Order

Scheme

Our

fourth-order

scheme

for

the

Navier-Stokes

equations

is

based

on

the

approximations

@x

O

-LRB-

x

-RRB-

-LRB-

-RRB-

and

@x

O

-LRB-

x

-RRB-

-LRB-

-RRB-

for

the

first

derivatives

from

-LRB-

-RRB-

and

O

-LRB-

x

-RRB-

-LRB-

-RRB-

for

the

second

derivatives

from

-LRB-

-RRB-

Using

the

approximations

-LRB-

-RRB-

and

-LRB-

-RRB-

we

have

that

the

first

two

equations

of

-LRB-

-RRB-

are

approximated

as

ut

-LRB-

-RRB-

-LRB-

uv

-RRB-

R

R

u

O

-LRB-

-RRB-

and

vt

-LRB-

uv

-RRB-

-LRB-

-RRB-

R

R

v

O

-LRB-

-RRB-

We

have

used

the

symbol

O

-LRB-

-RRB-

for

O

-LRB-

x

-RRB-

O

-LRB-

y

-RRB-

The

discretization

of

the

derivative

in

time

is

discussed

in

section

Operating

through

with

the

product

-LRB-

-RRB-

we

obtain

ut

-LRB-

-RRB-

-LRB-

uv

-RRB-

R

R

u

O

-LRB-

-RRB-

for

the

first

component

of

the

velocity

and

similarly

for

the

other

component

The

stencil

for

the

second-order

difference

terms

from

the

Laplacian

has

the

shape

ffl

ffl

ffl

fflfflffl

ffl

ffl

ffl

where

the

scheme

is

centered

about

the

center

of

the

stencil

The

coefficients

for

the

difference

in

the

x-direction

are

The

terms

for

the

first

difference

in

x

for

the

convection

terms

and

pressure

gradient

become

O

-LRB-

-RRB-

O

-LRB-

-RRB-

The

stencil

for

the

terms

in

this

last

expression

-LRB-

other

than

the

O

-LRB-

-RRB-

terms

-RRB-

has

the

shape

ffl

ffl

fflffl

fflffl

ffl

ffl

where

the

marks

the

center

point

The

difference

approximations

in

the

y?direction

are

similar

Notice

that

the

stencil

must

be

modified

for

points

one

grid

spacing

from

the

boundary

The

modifications

are

discussed

in

section

To

insure

the

regularity

we

take

the

pressure

gradient

expression

to

be

p

As

shown

in

section

the

term

with

positive

will

insure

the

regularity

of

the

solution

Notice

that

this

term

does

not

degrade

the

accuracy

of

the

difference

formula

Thus

the

scheme

for

the

first

momentum

equation

with

the

exception

of

the

time

differencing

is

ut

-LRB-

-RRB-

-LRB-

uv

-RRB-

p

R

R

u

-LRB-

-RRB-

and

similarly

for

the

other

momentum

equation

Notice

that

the

terms

on

the

right-hand

side

of

-LRB-

-RRB-

are

essentially

the

standard

fourth-order

accurate

scheme

for

the

Poisson

equation

derived

by

Rosser

-LSB-

-RSB-

We

next

consider

the

approximation

of

the

divergence

equation

-LRB-

-RRB-

or

-LRB-

-RRB-

Using

the

approximation

-LRB-

-RRB-

on

-LRB-

-RRB-

we

have

g

O

-LRB-

-RRB-

Operating

with

the

product

we

obtain

g

O

-LRB-

-RRB-

-LRB-

-RRB-

The

stencil

for

the

terms

for

the

differencing

in

u

has

the

shape

ffl

ffl

ffl

ffl

ffl

ffl

where

the

marks

the

center

point

The

stencil

for

the

differencing

of

v

is

similar

but

rotated

a

quarter

turn

To

insure

the

regularity

of

the

scheme

we

modify

-LRB-

-RRB-

to

give

the

scheme

ffiy?v

g

-LRB-

-RRB-

Notice

that

the

divergence

operator

and

the

gradient

operator

are

not

adjoints

of

each

other

The

Sixth-Order

Scheme

The

sixth-order

accurate

scheme

for

the

Navier-Stokes

equation

is

based

on

expanding

-LRB-

-RRB-

to

terms

that

are

O

-LRB-

x

-RRB-

The

approximation

for

the

first

derivative

is

@x

O

-LRB-

x

-RRB-

-LRB-

-RRB-

This

equation

is

used

to

approximate

first

derivatives

in

the

convection

terms

and

the

pressure

gradient

It

has

a

stencil

involving

seven

points

and

for

the

divergence

equation

it

is

desirable

to

find

a

formula

of

sixth-order

accuracy

with

a

smaller

stencil

An

approximation

giving

a

smaller

stencil

is

@x

O

-LRB-

x

-RRB-

-LRB-

-RRB-

Similarly

we

obtain

from

-LRB-

-RRB-

O

-LRB-

x

-RRB-

-LRB-

-RRB-

for

the

second

derivatives

Using

the

approximations

-LRB-

-RRB-

and

-LRB-

-RRB-

we

have

that

the

first

of

the

two

components

of

-LRB-

-RRB-

may

be

approximated

as

ut

-LRB-

-RRB-

-LRB-

uv

-RRB-

R

u

R

u

O

-LRB-

-RRB-

and

similarly

for

the

second

momentum

equation

Operating

through

with

the

product

-LRB-

-RRB-

we

obtain

ut

-LRB-

-RRB-

-LRB-

uv

-RRB-

R

R

u

O

-LRB-

-RRB-

The

stencil

for

the

second-order

difference

terms

in

the

x

direction

has

the

shape

ffl

ffl

ffl

ffl

ffl

fflfflfflfflffl

ffl

ffl

ffl

ffl

ffl

centered

around

the

center

point

Notice

that

this

difference

approximation

requires

modification

for

two

points

away

from

the

boundary

The

terms

for

the

first-order

difference

in

x

become

O

-LRB-

-RRB-

O

-LRB-

-RRB-

O

-LRB-

-RRB-

The

stencil

for

these

terms

-LRB-

other

than

the

O

-LRB-

-RRB-

terms

-RRB-

has

the

shape

ffl

ffl

ffl

ffl

fflfflffl

fflfflffl

ffl

ffl

ffl

ffl

where

the

marks

the

center

point

To

insure

the

regularity

we

take

the

pressure

gradient

expression

for

the

first

momentum

equation

to

be

ffix

p

As

shown

in

section

the

term

with

positive

will

insure

the

regularity

of

the

solution

and

again

the

additional

regularity

term

does

not

degrade

the

order

of

accuracy

Thus

the

scheme

with

the

exception

of

the

time

differencing

is

ut

-LRB-

-RRB-

-LRB-

uv

-RRB-

p

R

u

-LRB-

-RRB-

and

similarly

for

the

other

momentum

equation

We

next

consider

the

approximation

of

the

divergence

equation

-LRB-

-RRB-

or

-LRB-

-RRB-

Using

the

approximation

-LRB-

-RRB-

on

-LRB-

-RRB-

we

have

g

O

-LRB-

-RRB-

Operating

with

the

product

we

obtain

g

O

-LRB-

-RRB-

-LRB-

-RRB-

To

insure

the

regularity

of

the

scheme

we

modify

-LRB-

-RRB-

as

ffiy?v

g

-LRB-

-RRB-

The

stencil

for

the

difference

operator

on

u

other

than

the

term

multiplied

by

is

ffl

ffl

ffl

ffl

fflffl

fflffl

ffl

ffl

ffl

ffl

where

the

marks

the

center

point

One

disadvantage

of

the

sixth-order

scheme

over

the

fourth-order

scheme

is

that

because

of

the

necessary

boundary

modifications

the

stencil

for

the

Laplacian

is

not

symmetric

for

the

sixth-order

scheme

Some

implications

of

this

are

described

in

the

section

Boundaries

and

Extrapolation

of

Pressure

For

the

higher-order

methods

the

stencils

are

so

wide

that

some

modification

of

the

schemes

is

required

at

boundaries

Also

for

all

of

the

schemes

the

pressure

values

on

the

boundary

must

be

determined

by

extrapolation

The

regularity

terms

those

multiplied

by

are

removed

whenever

they

conflict

with

boundaries

We

index

the

grid

points

by

nonnegative

integers

starting

from

For

a

rectangle

the

grid

points

are

indexed

by

-LRB-

i

j

-RRB-

for

i

M

and

j

N

for

some

integers

M

and

N

We

consider

only

the

boundary

points

with

i

the

other

boundaries

are

handled

similarly

We

use

the

identity

-LRB-

-RRB-

to

replace

backward

differences

with

forward

differences

In

particular

we

use

ffix

ffix

-LRB-

xffix

-RRB-

O

-LRB-

-RRB-

-LRB-

-RRB-

We

also

use

the

relation

-LRB-

ffix

ffix

-RRB-

ffix

ffix

xffix

ffix

O

-LRB-

x

-RRB-

-LRB-

-RRB-

For

the

fourth-order

scheme

the

differencing

for

the

convection

and

gradient

terms

need

modification

near

the

boundary

For

i

the

terms

are

replaced

by

ffix

For

the

sixth-order

scheme

both

the

differences

for

the

Laplacian

and

the

convection

terms

need

modification

near

the

boundary

From

the

Laplacian

the

terms

j

for

i

are

replaced

by

-LRB-

xffix

-RRB-

The

expression

-LRB-

xffix

-RRB-

OE

at

grid

point

i

is

OEi

Similar

modifications

are

made

at

the

other

boundaries

For

the

pressure

gradient

and

convection

terms

we

also

use

-LRB-

-RRB-

At

i

the

expression

-LRB-

-RRB-

from

the

sixth-order

scheme

-LRB-

-RRB-

is

replaced

by

-LRB-

xffix

-RRB-

-LRB-

xffix

-RRB-

ffix?p

-LRB-

-RRB-

In

this

last

expression

the

fourth-order

divided

difference

was

modified

using

the

relations

-LRB-

xffix

-RRB-

O

-LRB-

-RRB-

and

the

central

difference

was

modified

using

-LRB-

-RRB-

The

expression

-LRB-

xffix

-RRB-

in

-LRB-

-RRB-

expands

to

x

and

xffix

j

expands

to

j

j

j

j

j

j

j

x

At

i

the

expression

-LRB-

-RRB-

is

replaced

by

-LRB-

xffix

-RRB-

ffix

p

-LRB-

-RRB-

where

we

used

-LRB-

-RRB-

on

the

central

first-order

difference

The

values

of

the

pressure

on

the

boundaries

must

be

set

by

extrapolation

from

values

in

the

interior

For

a

scheme

of

order

r

the

extrapolation

should

have

order

r

to

insure

that

the

scheme

is

exact

for

polynomials

of

degree

r

In

this

work

the

formula

used

to

determine

j

was

ffir

x

j

-LRB-

-RRB-

For

the

fourth-order

scheme

the

extrapolation

is

the

fifth-order

formula

j

j

j

j

j

j

Similarly

for

the

sixth-order

scheme

the

extrapolation

is

the

seventh

order

formula

j

j

j

j

j

j

j

j

Values

of

the

pressure

must

also

be

set

in

the

corners

of

the

Cartesian

grids

In

this

work

the

formula

used

to

determine

was

for

the

method

with

r

ffitx

ffit

y

ffit

x

ffity

where

the

other

boundary

values

of

p

have

been

determined

by

the

boundary

extrapolation

-LRB-

-RRB-

For

the

fourth-order

scheme

the

corner

extrapolation

is

-LRB-

-RRB-

Similarly

for

the

sixth-order

scheme

the

corner

extrapolation

is

-LRB-

-RRB-

The

Regularity

of

the

Schemes

In

this

section

we

check

the

regularity

of

the

schemes

As

shown

in

-LSB-

-RSB-

a

scheme

must

be

regular

in

order

to

insure

that

the

solution

be

smooth

We

consider

only

the

steady

equations

since

the

theory

has

only

been

developed

for

the

steady

state

case

The

importance

of

regularity

is

shown

in

the

examples

in

section

To

test

the

regularity

of

the

fourth-order

scheme

consisting

of

-LRB-

-RRB-

the

similar

formula

for

v

and

-LRB-

-RRB-

we

examine

the

symbol

of

the

principal

part

of

the

scheme

We

use

the

Fourier

transform

to

determine

the

symbol

by

replacing

u

m

by

u

-LRB-

x

y

-RRB-

and

similarly

for

v

and

p

Because

of

the

factor

of

in

the

exponential

we

are

only

concerned

with

and

in

the

range

ss

to

ss

We

may

write

the

symbol

as

B

L

-LRB-

-RRB-

iG

-LRB-

-RRB-

x

L

-LRB-

-RRB-

iG

-LRB-

-RRB-

y

iD

-LRB-

-RRB-

x

iD

-LRB-

-RRB-

y

CA

-LRB-

-RRB-

where

L

-LRB-

-RRB-

-LRB-

-RRB-

G

-LRB-

-RRB-

sin

-LRB-

-RRB-

-LRB-

-RRB-

and

D

-LRB-

-RRB-

sin

-LRB-

-RRB-

i

-LRB-

-RRB-

The

determinant

of

this

system

is

L

-LRB-

-RRB-

G

-LRB-

-RRB-

D

-LRB-

-RRB-

G

-LRB-

-RRB-

D

-LRB-

-RRB-

-LRB-

-RRB-

The

scheme

is

regular

precisely

when

this

determinant

vanishes

only

for

with

j

and

j

less

than

or

equal

to

ss

The

factor

L

-LRB-

-RRB-

does

not

vanish

for

nonzero

and

so

we

need

consider

only

the

other

factor

in

-LRB-

-RRB-

We

first

evaluate

the

product

G

-LRB-

-RRB-

D

-LRB-

-RRB-

We

have

G

-LRB-

-RRB-

D

-LRB-

-RRB-

sin

An

examination

of

this

expression

easily

shows

that

the

real

part

is

a

sum

of

non-negative

terms

and

for

j

and

j

less

than

or

equal

to

ss

it

vanishes

only

for

Thus

G

-LRB-

-RRB-

D

-LRB-

-RRB-

G

-LRB-

-RRB-

D

-LRB-

-RRB-

does

not

vanish

except

for

and

both

zero

The

regularity

of

the

sixth-order

scheme

is

analyzed

similarly

The

symbol

has

the

same

form

as

-LRB-

-RRB-

with

L

-LRB-

-RRB-

-LRB-

-RRB-

G

-LRB-

-RRB-

sin

-LRB-

-RRB-

-LRB-

-RRB-

and

D

-LRB-

-RRB-

sin

-LRB-

-RRB-

i

-LRB-

-RRB-

The

symbol

L

as

defined

in

-LRB-

-RRB-

vanishes

only

when

with

j

and

j

less

than

or

equal

to

ss

For

the

sixth-order

scheme

the

product

G

-LRB-

-RRB-

D

-LRB-

-RRB-

is

G

-LRB-

-RRB-

D

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

sin

As

with

the

fourth-order

scheme

with

positive

this

expression

vanishes

only

for

equal

to

Therefore

the

expression

-LRB-

-RRB-

for

the

sixth-order

scheme

does

not

vanish

for

non-zero

values

of

and

and

thus

the

scheme

is

regular

The

Temporal

Differencing

In

this

section

we

discuss

the

temporal

differencing

of

the

time-dependent

NavierStokes

equations

We

consider

only

schemes

based

on

multistep

methods

from

ordinary

differential

equations

To

simplify

the

discussion

we

consider

the

Navier-Stokes

and

Stokes

equations

in

the

form

ut

L

u

rp

f

r

u

g

-LRB-

-RRB-

where

u

denotes

the

velocity

and

p

denotes

the

pressure

The

operator

L

represents

the

terms

involving

spatial

derivatives

of

the

velocity

Our

approach

is

motivated

by

the

similarity

of

the

system

-LRB-

-RRB-

to

differential-algebraic

systems

for

ordinary

differential

equations

The

incompressible

Navier-Stokes

equations

are

similar

to

differential-algebraic

systems

of

index

see

-LSB-

-RSB-

For

the

system

-LRB-

-RRB-

the

temporal

differencing

using

a

general

multistep

method

is

defined

by

t

K

X

k

k

un?k

K

X

k

kL

un?k

K

X

k

k

rpn?k

K

X

k

k

f

n?k

r

un

gn

-LRB-

-RRB-

The

two

arrays

of

coefficients

k

and

k

are

normalized

by

K

X

k

k

and

K

X

k

k

Any

of

the

second-order

fourth-order

or

sixth-order

spatial

differencing

can

be

used

with

any

of

these

time-dependent

schemes

We

consider

the

temporal

differencing

-LRB-

-RRB-

to

be

applied

before

the

spatial

differencing

such

as

applying

the

operations

-LRB-

-RRB-

or

-LRB-

-RRB-

The

stability

of

the

scheme

depends

on

the

two

polynomials

A

-LRB-

z

-RRB-

K

X

k

kzK?k

and

B

-LRB-

z

-RRB-

K

X

k

kzK?k

A

necessary

condition

for

the

stability

of

the

overall

scheme

is

that

the

polynomial

A

satisfy

the

standard

root

condition

for

stability

in

the

sense

of

ordinary

differential

equations

That

is

the

roots

of

A

-LRB-

z

-RRB-

must

be

inside

the

unit

circle

or

simple

on

the

unit

circle

-LSB-

-RSB-

-LSB-

-RSB-

As

the

following

theorem

shows

for

the

standard

multistep

methods

-LRB-

-RRB-

stability

also

requires

that

the

polynomial

B

-LRB-

z

-RRB-

satisfy

the

root

condition

Theorem

A

necessary

condition

for

stability

of

the

multistep

method

-LRB-

-RRB-

is

that

the

roots

of

B

-LRB-

z

-RRB-

be

inside

the

unit

circle

or

simple

on

the

unit

circle

Proof

Let

z

be

a

root

of

B

-LRB-

z

-RRB-

and

let

q

be

any

non-constant

function

of

the

spatial

variables

A

solution

of

-LRB-

-RRB-

is

constructed

by

setting

u

and

p

z?q

If

the

magnitude

of

z

is

larger

than

then

this

solution

will

be

unbounded

in

norm

If

z

is

a

multiple

root

with

magnitude

then

take

p

z?q

Thus

it

is

necessary

for

B

-LRB-

z

-RRB-

to

satisfy

the

root

condition

As

is

seen

in

the

proof

the

roots

of

the

polynomial

B

-LRB-

z

-RRB-

govern

the

growth

of

the

pressure

errors

The

restriction

on

the

roots

of

B

-LRB-

z

-RRB-

is

a

result

of

the

pressure

appearing

only

in

the

spatial

differencing

portion

of

the

equations

Similar

situations

occur

with

semi-explicit

differential-algebraic

equations

of

index

However

defining

p

by

pn

K

X

k

kpn?k

we

can

replace

-LRB-

-RRB-

with

t

K

X

k

k

un?k

K

X

k

kL

un?k

r?pn

K

X

k

k

f

n?k

r

un

gn

-LRB-

-RRB-

The

function

pn

is

at

least

a

first-order

accurate

approximation

to

p

-LRB-

tn

t

-RRB-

where

K

X

k

k

k

Note

that

a

disadvantage

of

this

modified

scheme

is

that

the

pressure

is

not

be

obtained

to

the

same

accuracy

as

the

velocity

without

some

post-processing

Theorem

is

a

severe

limit

on

multistep

schemes

however

the

modified

multistep

scheme

-LRB-

-RRB-

allows

for

many

schemes

to

be

used

An

adequate

theory

for

the

stability

of

schemes

for

the

Navier-Stokes

and

Stokes

equations

has

not

been

developed

Here

we

rely

on

the

experience

and

theory

of

differential-algebraic

equations

see

-LSB-

-RSB-

to

guide

our

choice

Primarily

we

have

used

schemes

based

on

backward

time-differences

These

schemes

called

backward

differencing

formula

-LRB-

BDF

-RRB-

schemes

are

widely

used

for

solving

stiff

ordinary

differential

equations

and

in

differential-algebraic

equations

see

-LSB-

-RSB-

For

these

schemes

is

the

other

k

are

and

the

k

are

chosen

from

the

formula

@t

ln

-LRB-

t

ffit

-RRB-

t

t

ffit

-LRB-

t

ffit

-RRB-

-LRB-

t

ffit

-RRB-

ffit

truncated

after

K

terms

for

a

scheme

with

order

of

accuracy

K

The

coefficients

k

for

a

scheme

of

order

K

are

given

by

K

X

j

j

and

k

-LRB-

-RRB-

k

K

X

j

k

j

k

j

for

k

K

Our

finite

difference

schemes

are

obtained

by

replacing

the

temporal

derivative

in

equations

-LRB-

-RRB-

and

-LRB-

-RRB-

with

the

BDF

operator

of

order

K

In

this

paper

we

consider

only

K

equal

to

and

These

schemes

are

stable

as

multistep

schemes

for

ordinary

differential

equations

for

K

unstable

for

K

and

possibly

all

larger

values

of

K

see

-LSB-

-RSB-

In

the

numerical

tests

the

BDF

schemes

out

performed

the

non-BDF

schemes

The

backward

differencing

schemes

we

have

used

are

Second-order

backward

in

time

-LRB-

K

-RRB-

Third-order

backward

in

time

-LRB-

K

-RRB-

Fourth-order

backward

in

time

-LRB-

K

-RRB-

In

addition

three

schemes

were

used

in

the

first

time

steps

when

the

above

schemes

could

not

be

used

since

they

require

several

past

time

steps

Crank-Nicolson

-LRB-

second-order

accurate

-RRB-

Fourth-order

in

time

using

three

time

levels

Fourth-order

in

time

using

four

time

levels

These

schemes

are

obtained

by

factoring

the

backward

operators

as

done

in

sections

and

for

spatial

difference

operators

-LRB-

Scheme

is

equivalent

to

-LRB-

-RRB-

applied

in

time

rather

than

space

-RRB-

As

multistep

schemes

as

in

-LRB-

-RRB-

schemes

and

are

unstable

by

Theorem

but

appear

stable

when

used

as

in

-LRB-

-RRB-

Runs

using

the

second-order

scheme

used

scheme

to

compute

the

first

time

step

Runs

using

the

third-order

accurate

scheme

scheme

the

first

time

step

and

scheme

the

second

time

step

Subsequent

steps

then

used

scheme

Runs

using

the

fourth-order

scheme

used

scheme

to

compute

the

first

time

step

scheme

to

compute

the

second

time

step

and

scheme

to

compute

the

third

time

step

Subsequent

steps

then

used

scheme

The

use

of

a

second-order

accurate

scheme

to

initialize

a

fourth-order

accurate

appears

not

to

reduce

the

overall

order

of

accuracy

There

was

no

reason

to

use

a

first-order

accurate

scheme

The

BDF

schemes

are

dissipative

of

order

when

applied

to

parabolic

equations

For

dissipative

schemes

with

order

of

accuracy

r

initializing

schemes

may

be

accurate

of

order

r

and

still

have

the

overall

order

of

accuracy

be

r

For

the

Stokes

equations

because

they

are

linear

the

determination

of

the

solution

at

the

next

time

step

requires

the

solution

of

a

linear

system

The

system

can

be

written

as

un

t

L

un

r?pn

K

X

k

k

un?k

t

K

X

k

kL

un?k

K

X

k

k

f

n?k

r

un

gn

-LRB-

-RRB-

This

system

determines

the

solution

un

and

pn

for

the

new

time

level

Note

that

the

spatial

operators

such

as

-LRB-

-RRB-

or

-LRB-

-RRB-

must

be

applied

to

-LRB-

-RRB-

We

have

used

a

preconditioned

GMRES

method

-LSB-

-RSB-

for

the

solution

of

this

linear

system

The

usual

method

was

GMRES

-LRB-

-RRB-

with

a

restart

The

preconditioner

was

an

inversion

of

the

operator

t

-LRB-

-RRB-

on

the

first

two

equations

in

the

system

Other

methods

of

solving

the

linear

system

could

also

be

used

An

advantage

of

GMRES

is

that

it

does

not

require

the

system

to

be

symmetric

As

mentioned

the

preconditioner

for

the

system

-LRB-

-RRB-

was

the

inversion

of

the

operator

-LRB-

-RRB-

For

the

second-order

and

fourth-order

accurate

schemes

this

was

done

using

the

preconditioned

conjugate

gradient

method

with

SSOR

as

the

preconditioner

For

the

sixth-order

scheme

because

the

operator

is

not

symmetric

the

GMRES

method

was

used

Although

accurate

solutions

were

obtained

this

method

was

not

particularly

efficient

More

experience

is

needed

with

these

methods

to

improve

the

overall

efficiency

Because

the

pressure

can

only

be

determined

to

within

an

additive

constant

the

system

-LRB-

-RRB-

is

singular

Moreover

the

existence

of

the

solution

is

dependent

on

satisfying

the

integrability

condition

An

important

issue

is

the

choice

of

norms

to

determine

convergence

of

the

system

-LRB-

-RRB-

By

requiring

only

that

the

quantity

r

un

gn

be

constant

as

described

in

-LSB-

-RSB-

we

effectively

have

a

nonsingular

system

The

average

value

of

r

un

gn

over

the

grid

is

a

measure

of

the

consistency

of

the

data

In

all

the

results

shown

in

section

this

value

is

less

than

the

errors

themselves

by

several

orders

of

magnitude

For

the

nonlinear

Navier-Stokes

equations

we

modify

the

equations

to

obtain

a

linear

system

for

the

solution

that

does

not

degrade

the

accuracy

We

linearize

the

quadratic

expressions

uv

and

at

time

level

n

using

the

following

idea

Consider

a

two

functions

A

-LRB-

t

-RRB-

and

B

-LRB-

t

-RRB-

depending

on

the

independent

variable

t

Using

the

relations

ffirt?A

-LRB-

t

-RRB-

and

ffirt?B

-LRB-

t

-RRB-

we

obtain

approximations

A

and

B

to

order

r

for

A

-LRB-

t

-RRB-

and

B

-LRB-

t

-RRB-

The

formula

for

An

is

An

rX

k

-LRB-

-RRB-

k

r

k

An?k

and

similarly

for

Bn

The

product

AB

involving

past

values

of

the

variable

t

is

approximated

using

the

relation

-LRB-

A

A

-RRB-

-LRB-

B

B

-RRB-

O

-LRB-

t

-RRB-

In

particular

at

tn

n?t

with

An

A

-LRB-

tn

-RRB-

we

can

write

AnBn

AnBn

AnBn

AnBn

O

-LRB-

t

-RRB-

where

An

and

Bn

depend

on

values

of

t

less

than

tn

We

use

these

formulas

for

t

n?t

and

with

A

and

B

being

velocity

components

We

take

r

equal

to

the

number

of

time

levels

available

in

the

scheme

The

expression

-LRB-

uv

-RRB-

@x

at

time

step

n

is

approximated

by

-LRB-

unvn

unvn

unvn

-RRB-

@x

and

the

spatial

derivative

is

approximated

using

either

the

fourth-order

or

sixth-order

methods

given

previously

In

this

way

the

equation

being

solved

for

the

solution

at

each

time

step

is

a

linear

system

and

the

accuracy

of

the

solution

is

not

affected

For

schemes

and

the

above

approximations

were

used

only

at

the

time

step

being

solved

for

Numerical

Experiments

Several

tests

are

described

in

this

section

that

illustrate

the

accuracy

of

the

methods

The

first

series

of

tests

check

the

formal

order

of

accuracy

and

the

second

tests

examined

the

accuracy

for

an

analytically

known

solution

for

which

there

is

no

analytic

solution

The

finite

difference

methods

were

implemented

using

the

C

programming

language

with

double

precision

variables

The

finite

difference

schemes

were

tested

extensively

to

assure

that

when

the

velocity

and

pressure

were

polynomials

of

appropriate

degree

the

solutions

satisfied

the

schemes

to

within

machine

precision

This

served

as

both

a

test

of

the

methods

and

a

means

to

detect

programming

errors

in

the

implementation

of

the

methods

For

example

the

solution

u

-LRB-

-RRB-

v

-LRB-

-RRB-

p

g

was

used

with

the

steady-state

Stokes

equations

-LRB-

-RRB-

and

-LRB-

-RRB-

for

the

sixth-order

method

Other

sixth-degree

polynomials

were

also

used

By

considering

these

solutions

after

translations

and

rotations

in

the

plane

a

large

class

of

solutions

could

be

obtained

Similar

tests

were

made

for

the

fourth-order

scheme

For

positive

Reynolds

numbers

because

of

the

quadratic

convection

terms

the

fourthorder

scheme

is

exact

for

all

polynomials

of

degree

two

for

the

velocity

and

of

degree

four

for

pressure

Similarly

the

sixth-order

method

is

exact

for

third-degree

polynomials

for

the

velocity

and

sixth-degree

polynomials

for

the

pressure

In

all

these

tests

with

polynomial

solutions

the

solutions

were

computed

to

within

machine

precision

Similar

tests

were

used

to

check

the

temporal

differencing

For

a

test

of

the

method

on

a

less

trivial

application

we

take

the

solution

used

by

Pearson

-LSB-

-RSB-

Chorin

-LSB-

-RSB-

and

others

to

test

their

methods

The

solution

is

given

by

u

e

R

cos

x

sin

y

v

e

R

sin

x

cos

y

p

R

-LRB-

cos

sin

-RRB-

on

the

square

x

ss

and

y

ss

for

t

We

took

the

Reynolds

number

to

be

Computational

results

are

displayed

in

Tables

and

The

order

of

accuracy

of

the

method

is

given

as

an

ordered

pair

-LRB-

p

q

-RRB-

where

p

is

the

temporal

order

of

accuracy

and

q

is

the

spatial

order

of

accuracy

The

runs

marked

with

an

asterisk

are

those

for

which

the

parameter

was

nonzero

These

are

discussed

later

in

this

section

Table

displays

the

error

for

runs

in

which

the

initial

steps

were

given

by

the

exact

solution

For

the

runs

given

in

Table

the

solutions

on

the

initial

steps

were

computed

from

schemes

with

fewer

time

levels

The

pressure

errors

were

computed

using

the

standard

deviation

as

a

norm

That

is

the

norm

is

the

mean

square

of

the

error

minus

the

average

error

see

-LSB-

-RSB-

There

are

two

principles

illustrated

by

the

data

in

the

tables

The

first

is

that

for

a

given

choice

of

method

the

smaller

values

of

the

time

step

and

grid

spacing

gives

better

case

order

t

n

error

u

error

v

error

p

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Table

accuracy

The

second

principle

is

that

higher

order

methods

give

better

order

accuracy

than

lower

order

methods

Moreover

for

fewer

grid

points

and

fewer

time

steps

a

higher

order

method

can

give

better

accuracy

than

lower

order

accurate

methods

A

difficulty

with

interpreting

the

numerical

results

is

that

the

errors

due

to

the

time

discretization

and

the

spatial

discretization

are

combined

Thus

in

few

cases

do

we

observe

a

nice

reduction

in

the

error

that

corresponds

to

the

order

of

accuracy

in

time

or

space

Notice

however

that

because

of

the

unconditional

stability

of

the

implicit

method

the

time

steps

are

much

larger

than

that

required

by

the

fractional-step

method

The

errors

in

the

runs

in

Table

with

order

-LRB-

-RRB-

are

principally

due

to

the

time

discretization

This

is

due

in

part

to

the

smoothness

of

the

solution

in

space

The

error

for

case

is

not

that

much

smaller

than

that

for

case

Also

the

ratio

of

the

error

between

cases

and

is

approximately

as

would

be

expected

if

the

error

where

predominately

due

to

the

time

discretization

Notice

that

the

reduction

in

the

pressure

error

is

consistent

with

the

second-order

accuracy

in

time

For

the

-LRB-

-RRB-

runs

in

Table

the

error

is

affected

primarily

by

the

spatial

step

size

There

is

a

reduction

in

the

error

of

about

as

the

mesh

parameter

n

is

increased

from

to

Also

the

run

with

n

and

t

case

produces

smaller

errors

than

does

the

run

in

case

for

less

effort

-LRB-

The

work

for

a

given

run

is

proportional

to

-LRB-

t

-RRB-

:-RRB-

The

first

two

-LRB-

-RRB-

runs

in

Table

cases

and

illustrate

that

for

this

choice

of

scheme

the

spatial

error

for

the

grid

with

n

dominates

the

temporal

discretization

error

The

smaller

error

for

the

pressure

for

case

may

be

due

to

the

initial

data

being

exact

and

there

being

only

time

steps

of

actual

computation

in

case

and

such

steps

in

case

and

is

also

influenced

by

the

oscillatory

modes

which

are

allowed

by

taking

equal

to

Note

however

that

the

computation

in

case

is

much

more

efficient

than

is

case

with

better

accuracy

Cases

and

illustrate

the

effect

of

the

regularizing

parameter

on

the

accuracy

Because

the

exact

solution

is

so

smooth

the

oscillations

allowed

by

having

equal

to

are

small

initially

and

because

there

are

relatively

few

time

steps

in

these

tests

the

error

due

to

pressure

oscillations

remain

small

and

are

dominated

by

the

usual

discretization

error

However

for

case

the

dominant

error

for

the

pressure

is

due

to

this

oscillatory

mode

The

errors

are

reduced

in

case

by

taking

For

runs

involving

more

time

steps

the

importance

of

having

a

nonzero

value

for

is

even

more

important

Also

comparing

the

errors

for

cases

and

in

Table

we

see

that

there

is

a

reduction

in

the

velocity

errors

that

is

approximately

in

excellent

agreement

with

the

sixth-order

accuracy

in

space

If

the

error

were

due

solely

to

the

spatial

differencing

we

could

expect

a

reduction

of

The

reduction

in

the

pressure

is

only

a

factor

of

The

pressure

errors

are

still

somewhat

oscillatory

for

this

case

a

different

value

of

might

give

even

smaller

errors

case

order

t

n

error

u

error

v

error

p

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Table

The

runs

in

Table

with

order

-LRB-

-RRB-

as

in

Table

have

errors

that

are

principally

due

to

the

time

discretization

The

errors

are

also

approximately

the

same

for

similar

-LRB-

-RRB-

runs

in

Tables

and

showing

that

the

initialization

does

not

significantly

affect

the

accuracy

-LRB-

Compare

cases

and

in

Table

with

cases

and

in

Table

-RRB-

Case

a

-LRB-

-RRB-

scheme

obtains

results

comparable

to

but

better

than

case

with

less

computational

effort

There

is

a

dramatic

decrease

in

the

velocity

errors

as

the

grid

is

refined

between

cases

and

Case

in

Table

with

accuracy

-LRB-

-RRB-

has

errors

smaller

than

cases

and

with

far

fewer

grid

points

and

time

steps

This

shows

the

advantages

of

the

more

accurate

methods

Cases

and

demonstrate

the

gains

in

accuracy

for

the

higher

order

accurate

methods

Increasing

the

spatial

resolution

by

between

cases

and

results

in

a

reduction

in

the

errors

by

a

factor

of

about

for

the

velocity

and

the

pressure

a

bit

better

than

the

factor

of

expected

if

the

errors

were

due

solely

to

the

spatial

discretization

They

also

show

that

for

fourth-order

accurate

methods

the

accuracy

of

the

solution

is

not

adversely

affected

by

the

use

of

second-order

methods

as

part

of

the

initialization

This

is

a

consequence

of

the

dissipativity

of

the

BDF

methods

for

parabolic

equations

Recall

from

section

that

that

the

solution

on

the

first

time-step

was

computed

using

the

CrankNicolson

scheme

which

is

second-order

accurate

in

time

For

case

a

value

of

equal

to

reduced

the

pressure

error

by

factor

of

about

again

showing

the

importance

of

regularity

This

same

value

of

caused

a

slight

increase

in

the

errors

for

case

Case

gives

the

best

results

for

the

accuracy

of

the

velocity

of

any

of

the

cases

showing

the

high

accuracy

of

the

methods

The

pressure

errors

could

be

improved

by

choosing

to

be

positive

see

the

discussion

of

cases

and

and

cases

and

Case

when

compared

to

cases

and

shows

that

the

-LRB-

-RRB-

method

can

produce

comparable

accuracy

for

less

effort

As

mentioned

in

section

a

disadvantage

of

using

the

sixth-order

method

is

that

the

iterative

solution

method

that

was

employed

was

less

efficient

in

terms

of

iterations

required

for

convergence

than

was

the

iterative

method

for

the

fourth-order

scheme

Hopefully

this

inefficiency

can

be

removed

by

experimenting

with

preconditioners

and

with

methods

Cases

and

illustrate

the

effect

of

the

parameter

on

the

computation

For

computations

which

required

few

time

steps

notice

that

the

stopping

time

was

taking

to

be

was

adequate

and

gave

very

good

accuracy

In

fact

taking

positive

increased

the

pressure

errors

for

the

-LRB-

-RRB-

cases

However

in

the

cases

with

equal

to

the

pressure

error

was

noticeably

oscillatory

for

many

cases

and

for

runs

involving

more

time

steps

such

as

case

this

error

was

significant

and

seriously

degraded

the

total

accuracy

Notice

that

the

pressure

error

for

case

is

greater

than

that

for

case

For

case

the

value

of

was

and

for

case

it

was

Notice

the

dramatic

improvement

in

the

error

for

the

pressure

as

increases

in

these

three

cases

Further

work

is

needed

to

determine

how

to

set

the

parameter

for

different

computations

For

computations

involving

many

more

time

steps

than

the

tests

here

and

for

runs

with

initial

data

containing

more

high

Fourier

modes

than

this

solution

a

non-zero

value

of

would

be

critical

to

obtaining

good

results

As

discussed

in

section

the

method

GMRES

-LRB-

-RRB-

was

used

to

solve

for

the

solution

variables

at

each

time

step

An

average

of

about

applications

of

GMRES

-LRB-

-RRB-

were

required

per

time

step

Conclusions

In

this

paper

we

presented

several

new

finite

difference

schemes

for

the

incompressible

Navier-Stokes

equations

We

have

shown

that

these

schemes

have

a

high-order

of

accuracy

The

schemes

are

based

on

two

spatial

differencing

methods

one

a

fourth-order

accurate

method

and

one

a

sixth-order

accurate

method

The

temporal

differencing

methods

are

BDF

methods

of

order

and

These

temporal

schemes

can

be

used

with

either

of

the

spatial

differencing

methods

The

schemes

as

presented

are

for

orthogonal

Cartesian

grids

The

schemes

can

be

used

for

both

the

steady-state

and

time-dependent

equations

For

the

time-dependent

methods

the

storage

requirements

of

the

second-order

scheme

appear

to

be

greater

than

needed

by

fractional-step

methods

However

this

is

compensated

for

by

the

larger

time-steps

and

smaller

stencils

for

the

higher-order

spatial

discretization

For

the

higher-order

accuracy

in

time

the

storage

requirements

are

reasonable

increasing

by

one

level

of

storage

with

each

order

of

accuracy

The

methods

presented

in

this

paper

have

been

demonstrated

to

be

accurate

and

effective

methods

for

solving

the

time-dependent

incompressible

Navier-Stokes

and

Stokes

equations

Further

research

is

needed

to

improve

the

efficiency

of

the

solution

of

the

linear

systems

For

higher

Reynolds

number

solutions

better

iterative

methods

will

be

needed

to

solve

the

linear

systems

at

each

time

step

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Aubert

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Deville

Steady

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-LSB-

-RSB-

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The

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