Accurate Finite Difference Algorithms - NASA Technical Reports ...

NASA

Technical

Accurate

Memorandum

107377

Finite Difference

Algorithms

John W. Goodrich Lewis Research Center Cleveland,

Prepared Barriers

Ohio

for the and Challenges

in Computational

Fluid

Dynamics

sponsored by the Institute for Computer Applications and Engineering and NASA Langley Research Center Hampton,

Virginia,

National Aeronautics and Space Administration

August

5-7,

1996

Workshop

in Science

ACCURATE

FINITE

JOHN

W. GOODRICH

NASA

Leud8 Research

Cle_ela,d,

1.

DIFFERENCE

ALGORITHMS

Center

OH 4_185,

USA

Introduction

Clear

examples

of the

parently

simple

tational

electromagnetics.

over long numerical

diitlculties

problems

associated

are provided These

with

applying

by computational

applications

CFD

techniques

aeroacoustics

require

accurate

distances for a wide range of frequencies, placing algorithms, and raising issues related to efficiency,

to ap-

and compu-

wave

propagation

a severe accuracy,

demand on compatible

space and time treatments, high frequency istic surfaces, isotropy, stable and accurate nonrestrictive stability bounds. This paper

data, propngatica along characterartificial boundary treatments, and briefly presents two methods for the

development

which

issues. up

of finite

High

order

to eleventh

sense per

difference

single

order

step

explicit

accuracy

step

which

with

are virtually

in [0, 1].

If our

most

accurate

periodic

sine wave,

then

after

five

periods

and

after

grid points

error is O[10-e], per wavelength,

are relatively the

error

dimension in the

more

bound

increases.

number

five periods

time

Our

boundary

for the source

with

Applied Euler

equations.

information

or an expanding

has

Algorithms

As a first dimension.

for the

wave

example

Linearized

we will

The equations

consider

This

about front.

similar here.

Euler the

for the isentropic

The

linearized case

condition or direction and

properties

Euler

difference at

conditions. [5] will is local

in

of either

an

propagation

[4]. Additional

in

as

spatial

of O[10 -4]

by Hagstrom

boundary

Equations

as the

boundary

boundary

the distance

with eight

to increase

and bound

developed

initial

algorithms

of magnitude

require

been

order

tends

increases,

CFL

wavelength,

periods

High

an error

change

an

per

thousand

orders

computations that

gorithms have a consistent derivation and are raised by shocks, but are not addressed

2.

several to meet

phase

to propagate

efficiency

time

show

required

condition

linearized

relative

in the

in [0, lr] and

grid points

error is O[10-4].

their

algorithms

and relative

is used four

with

algorithms

frequencies

five hundred

simulation

of multiplications

and does not require

assumed

[7], and as the

of propagation.

A new artifical be shown

efficient

decreases,

factor

these

and examples

resolution

mode

algorithm

the maximum

to address

are possible, High

amplification 1 for normal

numbers

the maximum

are intended

algorithms

will be shown.

of [9] are also possible, time

algorithms

alissues

1D

equations

can be written

in one

space

in nondimension-

alized form as the system _

+

ax +

+_r where

Mr is the

pressure, solved

and

constant

u is the

by the

method

mean

ccmvection

disturbance

velocity.

=2(ui(z

O)

+_=o,

of characteristics

.(_, t)

= O,

Mach

number,

System

to produce

(1) the

p is the

disturbance

can be cliagonalized general

and

solution

(M + 1)t)+ #(_ - (M + 1)t))

-

+_(.i(_ - (M - s)t) - #(_ - (M - 1)t)), l_Z,t)

with

intial The

polate

data first

1 --_(p/(z

ui(z) step

-

(M

-4- 1)t)

+ ui(z

1 . +_0_(_ .....(M

1)t)

ui(z

= u(z,O),

and

in producing

u and p at t.

with

ivi(z)

(M

1)t)),

= iv(z,O).

algorithms

order

for solutions

D polynomials

of (1) is to locally

D

,,(_, + _,t.) _. ,,,,(_1= _,,o:,

_,

+ _,t.) _ r_(_) = _p°:.

or=O

dents

expansion using

coefficients the

of a particular considered.

known mesh

The

obtained

type,

of single

(3)

a=O

by the

on a given

or data

use

to simultaneously

are

data

inter-

in x:

D

The

(2)

-(M + 1)_))

and

stencil. that

interpolation

apprc0dmating

separate

of Undetermined

that

and

up

to the

ivo =

there

derivative

polynomials

derivatives

1 c3aua 10au u° = a! Oz °. _ at Oz °'

Method Note

terms

of order D t_ order,

I _iva a! Oz °

CoeflL

is no specification are

not even

D is equivalent with

10_iv _. _ a! Ox °"

(4)

The km.al inter/_/aticms(3) to u mad ivat time t. axe used as initialdata for the exact

solution

(2),

producing

u(zi + z,t.

an approximate

1 +t)_(ua(z-(M +_(ua(x

p(zi

+ x,t.

1 + t) _,_(pa(x +_(pa(z

local

+ 1)t)+ -(M-

solution pa(z - (M

+ 1)t))

1)t)- pa(z --(M" - 1)t)),

(5) -

(M

- (M 2

+ 1)t) + ua(z

- (M

- 1)t)- ua(z --(M

+ 1)t)) - l)t)).

The

approximate

local

solution to (1) with exact local propagator fundamental instead seen

viewpoint

in the

use

Fourier

for a wide obtain

(5) is a function

the local interpolants for (1) and correctly

of particular

of local

solution

of this method terms

of Riemann

variety

This

with

separation

[1].

at the

exact

stencil

[2], and

to develop

local

center,

of the system

fundamental

shocks

of variables

The approximate

algorithm

it is an

the solution

equations.

for problems

and

of problems

a computational

is to approximate

solvers

t, and

(3) as initial data, so that (5) is an incorporates the dynamics of (1). The

in the govern_

decompositions

of z and

solution

idea

is

in the

use

algorithms

(5) is used

to

with

u(zi,t, + k) _ u_+1= l(ua(-(M + 1)k)+ pa(-(M + 1)k)) .6 l(ua(-(M - 1)k)-pa(-(M - 1)k)),

(6)

p(z,, t,, .6k)_ p_+1__l(pa(_( M .61)k) + ua(-(M-k1)k)) + I(_(-(M

- 1)k)- _(-(M - 1)k)).

Algorithm (6) uses the exact local propagator (5) with approximate local data (3), so that the time evolution introduces no new error, but merely propagates what has

been

introduced

not

speci_ed,

interpolant

and (3).

by the that

interpolation.

(6) represents

If order

a family

D interpolant8

exact

local

coefficients

of Characteristics,

since

propagator

is obtaind

are

as spatial

viewed

that

of algorithms

in this case,

the general

by this method. derivatives

(4),

then

size

k, with

- k_((1 =_ where include

.6 Mul)

ks((1

.6 k2(2Mp2

.6 3MZ)pa

- k(ul

forms

are

dependant

upon

the

realization

+ MFI)

"6 M(3

+ M(3

the

local

of (6),

form

for the

expansion

solution

expansion. expansion

in spac_

The algorithm in the time step

.6 (M z Jr 1)u2)

"6 M2)u3)

+ k2(2Mu2

+ 3M2)u3

solution

If the interpolation

time (5) can be viewed as a Cauchy-Kowaleskaya also be reformulated as a truncated Taylor series

- k(z_

di_erence

of order D in both space and time. There (6). It may be viewed as an application of

and can

u_ +1 =u0

finite

are used for a partimlar

then the algorithm will have accuracy are several interpretations of algorithm the Method

Note

"6 ...,

(7)

+ (M 2 + 1)/_)

+ M2)I_)

+...,

the grid ratio _ = _ is implicit in (7), since the coefficients ua the factor h -° for space step size h. A truncated Taylor expansion

and p., in time

be viewedas a gener_t_edL_-Wendro_method {8].Algorithm(0) _ be reformulated spatial

interpolations

as a conventional use local

finite

difference

polynomials. 3

Further

method, details

since

the

are in [3].

_so

underlying

We will introduce four relatively conventional realizations of algorithm (6), with central stencils for the interpolauts (3), and with values for u and p at each grid point. These al_thms c/Is, they are second, fourth, time,

and they

methods,

are

refered

respectively.

to as the "c3o0ex,"

These

truncation algorithms

of each

four methods.

of the

four

This

from an additional below.

point. boundary

at the

boundary

modified

treatments expansions.

for u and

in this

interval,

point.

data

at the

with

data

u(z,0)

inital

and

M -

u-p are

The and

additional

imposed,

with

eighth

accuracy

order TABLE

1: Data

p(1,t)

and

boundary

We

Table

in both

time

From

fee the

and

Explicit

c3dOex

nlo

8

50

1.871)-01

16

100

4.57D-02

32

200

1.32D-02

64

400

3.50D-03

128

800

8.94D-04

cgo0ex

point

The

space.

data are

that

method

results

are are on a

is not

must

stencil

be with

for a simulation

_< z _< 1, with

_ = 0.8

at the z = I boundary, CFD

boundary

in Table stable details

with

Boundary

conditions

1 shows

that

from

second

with

Further

the to

are in [3]. Treatments

in u or p at t = 10 c,SdOex

c7dOex

cgdOex

1.17D-02

1.01D-02

8.65D-05

9.98D-04

5.26D-05

8.50D-07

8.11D-05

8.79D-07

4.69D-09

5.54D-06

1.29D-08

2.09D-II

3.58D-07

1.89D-10

3.75D-13 4.91D-13

256

1600

2.25D-04

2.27D-08

3.35D-12

512

3200

5.66D-05

1.43D-09

1.89D-12

1024

6400

1.42D-05

9.12D-11

Algorithms

will introduce

cal propagator

detail

that

are computed

boundary

for -1

typical

Algorithms

Error

grid point

is computed

and

treatments

_"

Hermltlan

in more

methods

variables

1 presents

u+p given.

at each

eight

= Sin(_rz),

boundary,

Maximum

3.

"c9o0ex"

if these algorithms from the data

Riemann

treatment

treatment, and

can be derived The interpolant

a truncated point.

z = -1

is discussed Hermitian

computed

= 0 and p(z,0)

,(--1,t)

algorithms

data

outgoing

boundary

boundary

at the

propagation

p are and

it uses

0. For the boundary

is computed

and

is used as initial data for the approximation over the interval from the stencil center to the

Values

for stability,

"c7o0ex,"

all aiagle

grid refinement

single stencil next to a boundary of the evolution of the solution on the

are

class of high resolution

Stable high order boundary viewed as Cauchy-Kowaleskaya

boundary

"cSo0ex,"

algorithms

point central stenin both space and

step explicit methods 1 Grid r__,_-_t errors, and each is stable for _ _ 1+--'fir" is presented in Figure 1, co_ the order of accuracy

with dispersive data for these with data introduced

have three, five, seven, and nine sixth, and eighth order accurate

a second (5),

but

family

which

of algorithms

are distinguished 4

for (1), from

the

which

use the

relatively

exact

conventional

lo-

algorithms introduced above by the use of Hermitian interpolants for (3). This particular family of algorithms uses and computes values for u and p, and for their

spatial

at each

derivatives.

grid point,

are viewed

Various

depending

as approximate

expansions

that

differentiated

are

upon local

locally

with

the

by differentiating

the

that

for u and with

p. These

for u and

are analogous forms

uz_ +1 =ul

particular

p.

to and

with

for the

similar

forms

Hermitian

and p, using

local

Coett;cients

with

Hermitian

note u and

four a first

grids. exact

a third

time,

stable

and time,

These

seven

_ive step

size

step

ing from

propagator

algorithm

referred

is _.

algorithms

staggered

or second

are

to as the

"c5ols2,"

are all single

step

errors, Note

and that

with two point

each two

time

half

step

grids

time

size

if just

which

u and

which

p are

we will de-

fifth,

or seventh

"c3o2s2,"

and

if just

u and

to

order

"c3o3s2"

p are

used,

which

we will denote

in addition

to u and

p,

accurate

in

or eleventh

"c5o2s2"

steps

can

respectively.

on staggered

for _