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
1111111111111 1111 11111 1111111111 11111 1111 1111
NF02533
N/lS/\
National Aeronautics and Space Administration
Langley Research Center Hampton, Virginia 23665
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1~ 1...', - ~lQo", I(I.~;
LANGLEY RESEAR(:H CENTE1,(, Li8Rl\fN, N~\SA H.c.M:)T~)N, IJIW,INtA
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31176014168257
-I
r:
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
