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dissipative processes in the gas, that is, viscous friction and heat conduction. ... conservation laws, the equations of ideal compressible flow (ICF). Even in ... Mathematically, this leads to the theory of weak solutions of nonlinar conservation equations. ...... for gasdynamical simulations", Lawrence Berkeley Lab. Report LBL-.
N AS.A Contractor Report

172180

leASE

NASA-CR-172180 19830026406

COMPUTATIONAL METHODS FOR IDEAL COMPRESSIBLE FLOW

Bram van Leer

Contract No. NASl-l58l0 July 1983

INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING NASA Langley Research Center~ Hampton, Virginia 23665 Operated by the Universities Space Research Association

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NF02533

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National Aeronautics and Space Administration

Langley Research Center Hampton, Virginia 23665

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COMPUTATIONAL METHODS FOR IDEAL COMPRESSIBLE FLOW

Bram van Leer* Leiden Observatory, P.O. Box 9513, 2300 RA Leiden, The Netherlands

Lectures presented at the Von Karman Institute for Fluid Dynamics, Rhode-StGenese, Belgium, in Lecture Series 1983-04 on Computational Fluid Dynamics.

I

1.

Conservative dissipative difference schemes

p.

2.

The recognition and representation of discontinuities

p. 13

3.

Multi-dimensional methods

p. 18

The research reported here was partially supported

under N 0,

b

> O.

(43)

For convenlence the points (xi' Yj)' (xi-I' Yj)' (xi' Yj-l) and (xi-I' Yj-l)' and

the volumes

centered at these

points are indicated by A,

B,

C and

D,

respectively; the center (xi-t' Y j-t) of the staggered volume is indicated by M. This is shown in Figure 3a.

Af ter moving

the

piecewise-uniform distribution with

a

velocity

(a, b)

over a time interval La we find~ by weighting q according to area (see Figure 3b):

o x

For comparison with Eq. (10) we ~lY rewrite this as

,0

Y

~

t .

(44.1)

20

n - a [{( t-t a )q n + O+t a ) qn }- {( t-t a )q ne + (t+ta )qD }] y x A x B x x

n+! q W)

n+t

n+!)

ay ( q N - q S

(44.2)



n+! Here q E is the time-average of the space average of q on the East side (Ae) of volume M;

the other averages are defined similarly. This formula differ$

from the version given orignally by Lax (1954), where the fluxes through the sides of volume M are not centered in time. This removes the cross-difference n n n n term Ox ay(q A - qB - qe + qD) from

the

scheme

and

reduces

the

scheme's

stability domain. Scheme

(44)

can

be

written as

the product

of

two

one-dimensional LF

schemes:

[t(l+T -1) _ a (l-T -l))[t(l+T -1) y y y x

-

n. LLF(At) u L LF (b t) qi Y x J

a(1-T x x

-1

n )]qi' J

( 45)

where Tx and Ty denote a forward translation by one volume in the x- and ydirection.

Hence,

the method of fractional

time-steps or time-splitting is

exact for the linear LF scheme. Godunov's scheme, illustrated in Figure 4, can be written as

n+1 q A

or

n (1- a ) 0- a ) qA x y

n n n + a (1- a ) q B + a (1- a ) q e + a a qD x y y x xy

21

n+t) q W

whE!re E,

W,

n+t)

q

(46.2)

s '

Nand S now refer to the sides of volume A. Again the proper

cross-difference term is included and exact factorization is possible:

n { (1-0 (I-T -I ) } { 1-0 (I-T -1 ) }qi. Y Y x x J

(46)

3.2

From convection to IeF

The

LF

scheme

can

be

exactly

implemented

for

a

nonlinear

system

of

conservation laws

o" The flux

aq~

n+t FE

(47)

through the East side of M is changed into

1

= /).t/).y fn

n 1 t + A

t

f

c

F~,

n f(wAC(t,y)) dy dt ,

with

(48)

n where wAC (t,y) is the time-dependent solution of the one-dimensional Riemann

problem

on

the

line

CA with

initial

values

at

tn.

The

other

obtained similarly. An lllseful approximation of (48) results i f we first average w:

fluxes

are

22 1

n-H

WE

2

1

= ~t~y tItn

n+1

.A

f1"1 n ( ) C WAC t,y dy dt

(49)

and then adopt

(50)

which

differs 0((~x)2) from

(48).

Note

that,

owing

to

conservation,

the

n+t does not require knowledge of the detailed

computation of the average wE Riemann solution.

Eq. (49) is, of course, the

one-dimensio~al

LF scheme for flow along the

line CA. This inspires the following implementation of the LF scheme for Eq. (47).

""Il.+t

LLF(~) wn.. 1J Y 2

(51.1)

~

LLF(~t) Wn.. X 2 1J

(51. 2)

wi'J-"21

w.1-"2l J'

~t

~x

(fU+t ij-t

(51.3)

This version is symmetric in x and y, just as the time-splitting suggested by Strang (1968) for two-dimensional operators:

t {L x (~t)L Y(~t) + LY(~t)L x (~t)} ,

(52.1)

23

It

must

be

said,

however,

that

computational

exclusively use pure product operators like betwen

x

and

y, the sequence

LyLx

fluid

(45).

dynamists

almost

To reduce the asymmetry

is alternated with

LyLx:

L (At)L (At)L (At)L (At) • x y y x

(52.2)

The exact formulation of Godu!lov's scheme for Eq. (47) requires knowledge of the solution of Riemann's problem in two dimensions, with initial values uniform

in

each

quadrant.

Th:ls

solution

is

only

known

for

a

single

conservation law and was obtained by Wagner (1980). In the same way as (51) is derived from (44b), we may derive from (46.2)

an approximation to Godunov's

scheme based entirely on the solution of one-dimensional Riemann problems:

"1:1+t wi'J

At n LG (Z)w y ij

(53.1)

w ij

~t

At n LG (Z)w x i;

(53.2)

n+1 wi .

n w.. 1J

J

At t.x

(in+ t

i+1 j

:fD+: .) i-2]

t.t :::n+t t.y (G i i_ +!.2

~~~t 1) 1J-"2

(53.3)

Here

-

. . ("'n+! W •• 'if

1J

~+!

Gij+t

with the Riemann flux tl'1(w ,w ) defined previously. Thus, L R Riemann solver is used twice in ea.ch space dimension. ExpreBsing (53) in terms of

L~D only leads to

(53.4)

the one-dimensional

24

L~O(f1t)

= I

+

G (L (f1t) - I) LG(f1t) y 2 x

+

(LG(f1 t) - 1) LG(~) y x 2

(54.1)

G Since LIO(T) is linear in T, this may also be written as

G L (M) 20

G G (L (f1t) - I) t(L (f1t) y x

+

I

G (L (f1t) - I) y

+

I)

HL xG(f1t) +

.

I)

(54.2)

Using the homogeneity property of the Euler equations and,

therefore, of the

Riemann solution, we may further work this out:

L

HL~ (f1 t)

G (f1t) 20

+

This

would

(LG(M) y

G L (f1t) (LG(f1 t) y x

reduce

to

+

+

1) - LG(f1 t).

x

I) - LG(f1 t) ] y

Strang's

splitting

,

(54.3)

(52.1)

if

L~D

would

have

the

distributive property. This, of course, is only true in the linear case.

3.3. Second-order schemes

The mimic

examples

of the previous section teach us that

two-dimensional

wave interactions. order accuracy.

wave-interactions

by

combinations

it

is possible to

of

one-dimensional

These schemes contain terms" (f1t)2, but only have first-

It is possible to apply the ideas of Section 3.1 and 3.2 to

the second-order schemes of Section 2. This clearly involves more algebra and will not be carried out here. One of the reasons is that none of these twodimensional

schemes

has

yet

been

programmed.

It

remains

to

be

shown

in

25

practice if the more subtle ways of splitting like (53) are indeed preferable to a pure product operator like (52.2).

3.4. Related subjects

In two dimensions the grid plays an even more important role than in one dimension. Various ingredients of the schemes discussed in these lectures may be used more than once, moving the grid or

in order to make decisions about refining the grid,

just rotating the frame in which the solution of a one-

dimensional Riemann problem is considered. For the latter decision we may use the direction of the gradient of the solution -

readily available from 0 wand 0 w in a second-order scheme x y

or

the projection angle at which the initial values in a Riemann problem could be best connected ("best" in some least-squares sense) by a single running wave. The latter idea is due to Harten (private communication). Moving a grid interface at the speed of the best-fitting single wave was reallzed in one dimension by Harten and Hyman (1982); two-dimensional results will soon follow. It appears that the improvement due to the grid movement is dramatic

for

a

first-order

scheme and not nearly so much for second-order

schemes. A two-dimensional extension of Roe"s matrix theory is being considered (Baines,

M.

J.,

communicati()n).

(1982),

University

of

Reading,

United

Kingdom,

private

26

REFERENCES 1.

J. P. Boris and D. L. Book (1973), J. Computational

2.

P. Colella and P. R. Woodward (1982), "The piecewise-parabolic method

Phys.~,

38.

for gasdynamical simulations", Lawrence Berkeley Lab. Report LBL14661. 3.

P.

Colella (1982),

"Approximate solution of the Riemann problem for

real gases", Lawrence Berkeley Lab. Report LBL-14442. 4.

J. E. Fromm (1967), J. Computational Phys.

5.

S. K. Godunov (1959), Matern. Sb •

6.

A. Harten (1977), Comm. Pure Appl. Math.

7.

A. Harten (1978), Math. Compo

8.

A. Harten (1981),

~,

.i,

~,

176.

271. ~,

611.

363.

"On the symmetric form of systems of conservation

laws with entropy", ICASE Report 81-34. 9.

A. Harten (1982), "High-resolution schemes for hyperbolic systems of conservation laws", Report DOE/ER/03077-67, New York Univ.

10.

A. Harten and J. M. Hyman (1981), "Self-adjusting grid methods for onedimensional

hyperbolic conservation laws",

Los Alamos

Sci.

Lab.

Report LA 9105. ~,

11.

A. Harten and P. D. Lax (1981), SIAM J. Numer. Anal.

289.

12.

A. Harten, P. D. Lax and B. van Leer (1983), "On upstream differencing and Godunov-type schemes for hyperbolic conservation laws", Rev.

32,

35.

13.

A. Jameson, W. Schmidt and E. Turkel (1981), AlAA Paper 81-1259.

14.

P. D. Lax (1954), Comm. Pure Appl. Math.

15.

P. D. Lax and B. Wendroff (1960), Comm. Pure Appl. Math.

16.

R. W. MacGormack (1969), AlAA Paper 69-354.

17.

G. Maretti (1979), Computers and

~,

Fluids~,

159.

191.

~,

217.

SIAM

27

18.

K.

W. Morton (1982), "Shock capturing, fitting and recovery", Lecture Notes in Physics 170, 77.

19.

W. A. Mulder and B. van Leer (1983), "Implicit upwind methods for the Euler

equations",

Proc.

AlAA

6th

Computational

Fluid

Dynamics

Conference, Danvers, Mass., July 1983, Paper 83-1930, pp. 303-310. 20.

S. J. Osher (1981), North-Holland Math. Studies!!2., 179.

2l.

P. L. Roe (1980), Lecture Notes in Physics 141, 354.

22.

P. L. Roe (1981), J. Computational Phys. 43_, 357.

23.

G. Strang (1968), SIAM J. Numer. Anal.

24.

G. D. van Albada, B. van Leer and W. W. Roberts, Jr. (1982), Astron.

.2.,

506.

Astrophys. 108, 76.

.!!'

25.

B. van Leer (1972), Lecture Notes in Physics

26.

B. van Leer (1974), J. Computational Phys. ~, 36l.

27.

B. van Leer (1977), J. Computational Phys. ~, 276.

28.

B. van Leer (1979), J. Computational Phys.

29.

B. van Leer (1980) , in "Numerical Methods for Engineering", Eds. E. Absi,

R.

Glowinski,

P. Lascaux and H.

.E.,

163.

10l.

Veysseyre, Vol. 1, p.

137,

Durand, Paris. 30"

B.

van Leer (1981),

"On the relation between the upwind-differencing

schemes of Godunov, Enquist-Osher and Roe,

ICASE Report 81-11,

to

appear in SIAM J. Sci. Stat. Compo v~m

31..

B.

Leer (1982), Lecture Notes in Physics 170, 507.

32.

D. Wagner (1980),

"The Riemann problem in two space dimensions for a

sllngle conservation law", Report TSR 112363, University of Wisconsin. 33.

P. R. Woodward (1980), sehemes

for

Vector

"Trade-offs in designing explicit hydrodynamic Computers",

Rodrigue, Academic Press, New York.

in

Parallel

Computing,

Ed.

G.

28

34.

P. R. Woodward and P. Colella (1982), "The numerical simulation of two-

dimensional fluid flow with strong shocks", Lawrence Livermore Lab. Report UCRL 86952.

29

(b) 5

5

F:lg.

1.

Stability

5

domain of

four-step

Runge-Kutta

methods

in

the

complex

plane. a) With ex Fourier

(1)

= ex

(2)

= ex

transform of

(3)

the

= ex

ef)

= 1. The inner contour is the locus of the

second-order upwind spatial-differencing operator,

forcr=1.3. b) With

( 1) (X

.86,

.44;

0

= 1.9.

2.5 2.0 1.5 1.0

.Sll \\ll0~ 0.0

(a)

GOOUNOV c:;;:,

Ln

~ELOCITY

.

.

- CS)

IJ.)

In

~ X N

C'-J

CS)

IJ.)

~ElOCITY

-.

IJ.) •

-

CS)

(b)

MUSCL

.

IJ.)

C'-I

w

0

2.5 Fig. 2. Propagation of a one-dimensional

2.0

shock, as represented by first-order (a),

1.5

second-order (b) and third-order (c) up-

~

..."

1.3

" "

,

\~,

~\\~~

wind methods. Note that the overall dis-

.5 tribution is always monotone. Reproduced from Woodward (1980).

.

CS)

C'-I

(c)

PPl'l

~.0

.

CS)

.

IJ.)

~ELOCITY

CS)

Ln

.

.

cs)

IJ.)

C'-I

C'-I

31

(a)

yt

-=+= -=

B

y.1

A

M

y.1- 1

2

~-1

0

C

x

x·1

x·1- 1

(b)

r ---- - - - - -

I - - - - -, I

I

y.

I

1

--1

b~t

I

I

f

I

L___ _

I

--'

,

a~t

r

-L

X·1- 1

X·1-1

X

X·1

2

...

F:lg. 3. Two-dimensional convection by the Lax-Friedrichs scheme. (a) Staggered grid of finite volumes. time

step

wc~ighted

(b) Contours of the convected distribution after one

(broken lines). average

d:lstinctively.

of

the

The updated average values

in

A,

B,

value

C and

D;

in zone M is an areathe

areas

are

shaded

32

(a)

yt y

I

B

A

0

C

X.1-1

X.

Y.1- 12 Y.1- 1

x

I

.. (b)

r-Yj

bAt

----I-N---'

I

I

/

WE/

L-__

_./

/

I L __ _

I I

I I

I

- -L---

_..J

QAt X.1- 1

X.

I

x

Fig. 4. Two-dimensional convection by Godunov's scheme. Same as Fig. 3, except updating is done in zone A.

1. RepeIrt No.

I

NA.,SA CR-172180 4. Title and Subtitle

2. Government Accession No.

3 . Recipient's Catalog No.

5. Report Date

July 1983

Co mputational Methods for Ideal Compressible Flow

6. Performing Organization Code

7. Auth orIs)

8. Performing Organization Report No.

83-38

Br am van Leer

10. Work Unit No.

9. PerfeIrming Organization Name and Address

In.stitute for Computer Applications in Science and Engineering Ma.il Stop l32C, NASA Langley Research Center Ha.mpton, VA 23665

11. Contract or Grant No.

NASl-158l0 13. Type of Report and Period Covered

12. Spon soring Agency Name and Address

Contractor report

Na.tional Aeronautics and Space Administration 20546 Wa.shington, D.C.

14. Sponsoring Agency Code

15. Suppdementary Notes

Langley Technical Monitor: Fj.nal Report

Robert H. Tolson

16. Abst ract

LE!ctures presented at the Von Karman Institute for Fluid Dynamics, Rhode-St--Genese, !lgium, in Lecture Series 1983-04 on Computational Fluid Dynamics.

B~

p.l

1.

Conservative dissipative difference schemes

2.

The recognition and representation of discontinuities p.13

3.

Mul ti·-dimensional methods

17. Key Words (Suggested by Author(s))

p.18

18. Distribution Statement

co nservative difference schemes Euler equations sh ock waves

19. SecUI' ity Classif. (of this report)

Un classified

20. Security Classif. (of this page)

Unclassified

64 Numerical Analysis Unclassified-Unlimited

21. No. of Pages

34

22. Price

A03

For sale by the National Technical Information Service, Springfield, Virginia 22161

NASA-Lang1 ey, 1983

End of Document