reproduction ~, copy - federation of american scientists · ?itbx.typevii may give accuracyto order...
Post on 02-Jul-2019
212 Views
Preview:
TRANSCRIPT
i
i
1
631--.LA-3407’
—_———_—:-—
CIC-14REPORT Collection” ;:..:-
REPRODUCTION ~,copy ~. _
LOS ALAMOS SCIENTIFIC LABORATORYof the
University of CaliforniaLOS ALAMOS ● NEW MEXICO
A Study of Truncation Errors
in Eulerian Hydrodymics
UNITED STATESATOMIC ENERGY COMMISSION
_CONTRACT W-7405 -ENG. 36
..
I—LEGAL NOTICE—1This report was prepared as an account of Government sponsored work. Neither the UnitedStates, nor the Commission, nor any person acting on behalf of the Commission:
A. Makes any warranty or representation, expressed or implied, with respect to the accu-racy, completeness, or usefulness of the Mformatlon contained in this report, or that the useof any information, apparatus, method, or process disclosed in thts report may not infringeprivately owned rights; or
B. Assumes any liabilities with respect to the use of, or for damages reaulttng from theuse of any information, apparatus, method, or process disclosed in thts re~rt.
As used in the above, “person acttng on bebalf of tbe Commission” includes any em-ployee or contractor of the Commission, or employee of such contractor, to the extent thatsuch employee or contractor of the Commission, or employee of such contractor preparea,disseminates, or provides acceas to, any information pursuant to MS employment or contractwith the Commission, or MS employment wltb such contractor.
This report expresses the opinions of the author orauthors and does not necessarily reflect the opinionsor views of the Los Alamos Scientific Laboratory.
Printed in USA. Price $2.00. Available from the Clearinghouse forFederal Scientific and Technical Information, National Bureau of StandardS,
United &Z3kS Department of Commerce, Springfield, Virginia
LA-3407UC-32, MATHEMATICSAND COMPUTERS
TID-4500 (46th Ed.)
LOS ALAMOS SCIENTIFIC LABORATORYof the
University of California
LOS ALAMOS c NEW MEXICO
Report written: September 1965
Report distributed: January 3, 1966
A Study of Truncation Errors
in Eulerian Hydrodynamics
by
H. J. Longle y
——
. . .
,. ,
-1-
ABSTRACT
A
cation
numerical and analytical study is given of the effects of trun-
errors for vuious differencing schemes in Eulerim hydrodynamics.
Space truncation errors are studied for a conventional second
order time scheme. Correlations can be made between the analytical
theory and the numerical calculations.
-3-
●
.
.
CONTENTSPage
ABSTRACT
INTRODUCTION
TRUNCATION ERRORS AND DIJWERENCING SCHEMES
NUMERICAL RESUITS
PROBIJNV!RESUITS
CONCLUSIONS
REFERENCES
TABLES
Table
1 Type Differencing
~ Truncation Errors
3 Key to Problems
3
7
9
18
20
30
31
13
14
19
-5-
.
.
.
We desire to give some
INTRODUCTION
idea of the effects of truncation errors re-
sulting from the finite differencing of the hydrodynamic equations in a
fixed Eulerian mesh of one space dimension. The Eulerian form of the
1hydrodynamic equations wilJ be used, as in a previous report$ in conser-
vative differential and difference form. A fixed mesh in plane coordi-
nates is assumed. Each cell.of the mesh has a length of 5x. The cells
sre numbered from left (i = 1) to right (i = iM).
some time and with proper boundary conditions, the
carry the values of the quantities stored for each
in time explicitly to a small time bt later. This
Given the mesh at
usual problem is to
mesh point forward
numerical method is
for compressible flow in
Data Processing Machinej
tations.
the presence of shocks.
Type 7090,was used for
The IBM Electronic
the numerical compu-
The conservative differential form of the hydrodynamic equations in
one dimension is given by:
ap-#a(pv.---.=-
:=-&a(w)
(1)
(2)
-7-
where P
a(pE) a(pv + PVE)7F-=- dx
is the density, v is the
(3)
material veloclty} P iS the pres$~e~ Snd
E is the total energy per unit mass. E is the - of the inter~ energy
~ ~2per unit mass (&) and the kinetic energy per unit mass, i.e.j E = & ‘2 “
The extension to two dimensions is indicated elsewhere.’ A polytropic gas
is assumed (with Y = 5/3) in the numerical calculations. Thus p = (Y-1)<
2and C = 7’(7’ - lx, where c is the sound speed.
Values of p<, v<, and L ~ are stored and will be considered as the
values
mesh.
is the
J. J. -L
of p, v, and L, respectively, at the center of the ith cell of the
Strictly speaking, the product of pi and the volume of the ith cell
mass in the ith cell. The prOdUCtS Of V. and&4 tith this mass ueJ. J.
the momentum and energy, respectively, in the ith cell. The conservative
difference forms of equations (l), (2), and (3) am given by
tit(4)
n+lVn+l nnPi i - pivi
M.
n+l~n+lPi - p>;
-%r----
- (p + Pv2)i-A
.. (5)VA
(PV + pvE)ie - (PV + PvE)i 1
=-5X
(6)
where if n indicates some time t, and n + 1 indicates a time t + bt (one
time step later). The method of calculating the barred quantities on the
right side of these eqwtions and the resulting truncation errors will be
the subject of this report. For conservation they must be unique, i.e.,
-8-
.
.
.
.
for example, the number calculated for ~ 1 in the ith cell is to bei+=
used in the (i + I)th cell for ~(i+l)-*”
Or, if the left side of the
first cell is the left boundary and the right side of the iMth cell is the
left boundary of the region being studied, then byEq. (4),
n%%=~1-+ --w+i= 1
i.e., the change in mass of the region each time step
flow in and out of the boundaries in that time step.
and (6) must be made to conserve momentum and energyj
is given by the mass
Similarly, Eqs. (5)
respectively. ThiS
report will not discuss the stability of the various differencing schemes}
which may be found elsewhere.2
TRUNCATION ERRORS AND DSFFERENCING SCHEMES
Since the equations sre all similar, we will show only the expansion
of Eqs. (1) and (4), i.e., for
procedure is used for Eqs. (2)
advancing the density in time. The same
and (3). Thus,
For an explicit calculation we must evaluate the coefficients of M, btz,
M3, etc. from the values of p, v, and L which exist at the various mesh
points at time step n. We wilJ truncate (7) after second order (no bt3
ap a(Pv)term). The coefficient of M is given by Eq. (1)} i.e.} ~ = - ~ ●
The coefficient of btz is given by Eqs. (1) and (2), i.e.,
-9-
2=-&P*l=-&P+l=+$(p+ .v2)Thus, we must calculate the coefficients of M ~ bt2 by the use of
space derivatives. These space derivatives tilJ be calculated by finite
differencing schemes frcm the values of p, v, snd & at time step n
which are given at the various mesh centers. Any
scheme for the space derivatives till.be accurate
in 5x, i.e., wild have truncation errors.
finite difference
only to some order
We will use the following methods of space differencing. Let
be a function of x which we desire to approximate by some scheme of
finite differencing, using the values of f at various mesh points;
we want
fi+ - fi4
6X
(2)to approach ~ within some order (of truncation) in 5x. In thisi
port, of course, f may be p, v, ~, p,etc. or some combination of these
dependent variables. For differencing we will assume a three point
scheme such that
[
= ‘Ifi + ‘2fi+l + ~3fi-1go
ifvT>o
f =0i.+ ifvT=o
(8)
\
= ‘Ifi+l + “fi + ‘5fi+2E.
where 50 = E, + E’ + ~., VT =>
stants. The only exception to
ifvT<o
‘i + ‘i+l’and the set (~ .~ E ) are con-
1’ ‘~ 3
Eq. (8) is that for VT = O and f = p, then
-10-
.
.
.
.
‘i + ‘i+lf l=—i+~ 2
.
VT is a number to test the direction of flow. Since, in a three
point scheme, one must have two mesh values
between two cells and one mesh value on the
it is evident that VT is used to select two
on one side of the boundary
other side of this interface,
mesh values on the side of
the interface from which the lqfdrodyrugnicflow is approaching this inter-
face.
Let
fl ~~f’.,
() ()a2f
i ~t9f:= ~i>
and, for VT > 0, expand from Eq.
etc.
(8) as
or, doing the same for fi I(VT > O), we have7
f l=fi+i+=
fi&=fi+-2
For VT < 0,
“f: C&-?)+$f~t2;“)‘ ?+’i’’(.-)‘ “o”
we have
-11-
Equations (9) and (10) give
+ 5X30
+ ● *9 VT>O
+5X3” +.** V*<O()Thus, for our three point scheme for the calculation of f on an inter-
face (fi~) ~ the selection of the three const~ts (~1J~2~~~) till deter-
mine a differencing scheme and determine the truncation errors. The
terms in Eq. (11) containing bx, bx’, 5X3, etc. may all be considered to
be errors in
as an
given
approximation to f‘.i
We will consider in this report the types of differencing schemes
in Table 1.
(lo) ‘
(11) I
.
.
-12-
Table 1. Type Differencing
I 1 1 0 2
II 1 0 0 1
III 6 3 -1 8
Xv 4 1 -1 4
v 7 4 1 12
VI 3 0 -1 2
VII 5 2 -1 6
Type I gives a linear differencing scheme. Type II indicates that the
value on the boundary is to be taken as the value from the near celJ.
frm which the flow is approaching the interface. Type III results
from a quadratic fit of the three selected mesh values which is eval-
uated at the interface. Type IV is the result of an extrapolation f%om
the nearest mesh value behind the flow to the interface, using the slope
given by the two outside cells. Type V is
least squsres fit with three data points.
the result given by a linear
TypeVI uses the two mesh
values behind the flow for a linear extrapolation to the interface.
TypeVII does not seem to have a simple geometrical interpolation. How-
ever, its value as a differencing scheme to reduce the truncation errors
will be shown below. Though there does not seem to be any profitable
--13-
reason for doing so, Type VII may be considered
of Type I and Type VI, i.e., for V > 0,T
as a linear combination
is the same as Type VII.
We can now return to the coefficients in Eq. (11) which determine
the truncation errors. Table 2
truncation errors.
Table 2.
illuminates the effect of the g’s on the
Truncation Errors
I (1,1,0) o 0 1
II (1,0,0) -1 +1 1
III (6,3,-1) o 0 1/4
Iv (4,1,-1) o 0 -1/2
v (7,4,1 ) -1 /2 +1/2 . 3/2
VI (3,0,-1) o 0 -.2
VII (5,2,-1) o 0 ,0
.
.
The fact that Types II and V do not give an accuracy even to order 5X may
be noted. In f%ct, both give the effect of a diffusion te~ being added
to the differential equations, with T3@e II giving twice as much diffusion
-14-
.
.
as that given by Type V. This fact and the effect of the errors in the 5X2
terms will be evident in the numerical.exsmples given later in this paper.
Let us again returnto Eq. (7). The coefficient of 5t2 will have a
term involving
rivative in an
the second space derivative. Differencing this space de-
ordinary
Pi+l-Pi Pi-pi,
5X - ?lX
will give accuracy to
conserving manner, i.e., like
P.1+1 ‘%-l - *pi= p,,+ 2 ,,15.2+ ● emi ~pi
(12)5X2
order 5x. This term is then accurate to order
bt2bx, i.e., third order. In this report, all.the numerical exsmples
used a linear (first order) differencing scheme for the calculation of the
coefficient of the term which is second order in time.
of differencing schemes listed above were used for the
coefficient of the bt term in Eq. (’j’).Thus, Types II
The various types
calculation of the
and V give accur-
acy only to order bt,while Types I, III, IV, and VI give accuracy to order
?itbx. Type VII may give accuracy to order as high as ?Ytbx2for this term,
depending on the considerations discussed in the next psragraph.
Because the right side of our hydrodynamic equations are nonlinear
in the dependent variables, there are several.ways these terms may be
grouped for differencing. We will discuss two methods of grouping and
designate them as Group A and Group B. By Group A we will mean that
f of Eq. (8) is to be replaced by pv when used to calculate the coeffi-
cient of M in Eq. (7), i.e.,
‘i+= (Pv)~+>= (glp~vi + ‘*p-j+lv.j+l + ‘~p~-,vi..l)/E()
-15-
for VT > 0, etc. Group A wi12 thus result in the truncation errors in-
dicated in Table II being valid for
as an approximation to
[1
a(pv)74
A
Similarly, Group A
is replaced by p +
for the first order time terms means that f of Eq. (8)
PV2 and fisreplaced bypv+pvE inEqs. (5) and(6),
respectively. In other words, Group Ameans that the terms on the right
hand side of the hydrodynamic equations are fitted as a group. By Group
B we will mean that each dependent variable on the right hand side of the
hydrodynamic equations (for coefficient of M terms) will use Eq. (8) in-
dividually, i.e.,
for VT > 0, etc. Thus, by Group B one can expand and obtain (for VT > O)
N+?. -?;2?1}+i-2p Vt
2
{[“ + v p“
%‘ivi i i 1 (13)
,
.
-16-
,
.
using Type VII (5, 2, -~) differencing, one sees that the last term in
the coefficient of 5X2 does not vaish. Another instructive example of
Group 11is also given by considering spherica geomet~. Considering
the divergence in one dimension
with r. 1 = r + br/2 and VT > 0, we find1*
~-
2 2‘i+$fi++ - ‘i-~fi-+= f, + 2fi
~&
r? br i—‘i 2
.L
The last term in the coefficient of
(f; +
3f:+—
‘i
2f ‘ -$ + E2 - 3E3~
-)(i
‘i 50 )(14)
c,+E2+553
~o )
5r2 cannot be made to
~ ‘;1‘2T+”””
‘i
vanish by
any selection of ~,) E2J and ~5. The advantage to using the Group B
method is that, since p 1, vi+
1, andi~ & 1 are calculated once for
i~
each ceil and used to calculate all the other terms which are combina-
tions of them, calculation time is saved. However, the numerical ex-
smples given later in
truncation errors are
For a stagnation
●this report will indicate that the resulting
not desirable.
region one may note that, if the flow is into a
cell on both interfaces (VT < 0 for‘i@
and VT > 0 for fi+)j
+ ● . . (15)
and, if the flow is out of a cell.on both interfaces (VT > 0 for f 1i*
-17-
and VT < 0 for fi~),
(16)
The result is that the truncation errors are essentially increased by one
order for alJ the schemes (except Type I) considered in this report.
To summarize the methods discussed above and used in this report, we
note that, for Eq. (4)}
Hi+= (PVQ[
-~ (P+ PV2):+1 -( P+ PV2); 1(17)
The first term on the right is calculated by Group A and Group B for the
various types of differencing. The second term on the right is the
second order time term which is used in this report.
NUMERICAL RESUID!S
The numerical solution to a steady shock was used to study the
effects of truncation errors. In front of the shock all the cells con-
-tainedinitial values of P. = 1.0, Vo= O, and&o= O. Withy= 5/3,
the theoretical values behind the shock are PS = 4.0, v~ = 1.0, and
L5 = 0.5. The shock was assumed to enter the mesh on the left boundary,
and its progress to the right was calculated as a flmction of time.
The left interface of the first cell and all mesh values to the left of
this cell, i.e., the left boundary conditions, were assumed to have the
&values of ps, vs} and s. The theoretical shock velocity is 4/3 vs...
The theoretical position of the shock and values of p, v, and ~ are
-18-
plotted as straight lines (connecting dots) in the numerical examples
below.
Table 3 gives the “key” to the problems.
‘Table 3. Key to I%oblems
Problem TypeNumber Differencing 5X Group
1
101
2
102
3
103
4
104
5
105
6
106
7
107
1)1(3 Plots)
(3’;lots)
I (1,1,0)
I (1,1,0)
II (1,0,0)
11 (1,0,0)
III (6,3,-1)
III (6,3,-1)
Iv (4,1,-1)
Iv (4,1,-1)
v (7,4,1)
v (7,4,1)
VI (3,0,-1)
VI (3,0,-1)
VII (5,2,-1)
VII (5,2,-1)
I (1,1,0)
VII (5,2,-1)
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1/4
1/4
A
B
A
B
A
B
A
B
A
B
A
B
A
B
A
A
-19-
A constant value
Values of N from 0.4
ence in the numerical
However, Problem 6 is
of w = 0.3 bx/vs was used for each problem.
bx/vs to 0.05 5x/vs made essentially no differ-
results for the better differencing schemes.
given as an example to ilAudn?ate the difficulty
of using a constant 5X for all time, i.e., one should use a different
bt=0“3*
for each cycle, where Ivmu I isthe velocity of maximummagnitude which
exists in the mesh at the existing time. IYoblem 6 is shown at a time
when it has just become unstable. If bt had been calculated for each
cycle in Problem 6 as indicated above, then the problem would have re-
mained stable, but large oscillations would be evident because of the
inaccuracies of the differencing scheme, i.e., due to truncation errors.
Space runs from x = O to x = 20.0 for all problems shown.
PROBLEMI?Fs.mrs
The density versus x plots for
sity, velocity, and internal energy
are shown on the following pages.
Problems
versus x
1-7 and 101-107 and the den-
plots for Problems 14 and 17
-20-
I
I
,
.i
I
I
i
. . . .. ... . ..-. —— ---— . . . . . . . .. . . M18DENSITYVSX
FRC6W2= ! ox= foooE 00 Im= to Tllcz i.otooooE 01
tooo-o!b4000-01
+000 -a ,woo-a
. ... ——-— .——---●2O8.- —.-.—— ,.——
DENSITY w x
FR@ M@ = !01 OX= I.00E0O w to TfKs loOtOOOOE01
-21-
.
A
. .
4000-01,4000 -oi
I,I1
.. . .. . . . . ..●ct
OENSITY W X
FRC6 K! = 2 Dx= f.oof Go 1* 20 TIlm 1.020000E 01
(
. .- . .. —-- ---
●ZO8------ .—.. ... .—
0EN51TY W X
FRCU N@: 102 Dx= 1 .00E 00 M= to TI14V 1. OKOOOOE 01
,
.
-22-
,
.
.
42080000-016WOO-(71
OENSf TY W X
FRc6 K s 3 0 B= 1.GGE GO lIG 20 TIK= 1,020000E 01
0000-01 ●4000-01
. . -...—— -—. - .-. +W8OENSITV VS X
FRC6 M = !63 ox= : ,00E00 Iw 20 TIME=1.OZOOOOE01
-23-
.
4000-01,woo -01
-_ —.- . . ..●ZO8
0EN3fTYW X
FRcO K z 4 Dx= 1 .Ooc 00 lM: 20 TIME: 1.OZOOOOE Ot
awo-ol ++000-0;
. . . _____ .wolf—... — .—-.—
OENUTY W X
FRCO W : 104 ox= 1 .00E 00 IM: 20 TIWS 1.OWOOOE 0!
,
-24-
I
I
i
4000-01 ●4000 -0!
OENU TY W X
FRCLI NO = 5 ox: ! .00E 00
. . -.. -...— .—— .. -—.. . . _--iii
m= 20 TIW= ! .OMOOOE 01
. ..—— .
I
I
1
0000-01++000-0!
FRc# N@ = 105
-.——— “—- —--”–--*EI8DENSfTY W X
ox= f .00E 00 m= ml TIMES ! .omoooE 01
,
-25-
4000-01+400Q-01
GENWTY V5 X
-. -.--— --———.—- -------- .... .. ...-●2O8
+oOo-o!b+000 -0;
FRC8M3Z 6 OX= 1.00E 00 Ill= 20 ThlE= 5. KWOOE 00
. .. .----- . . ..-... . . .. .--— ---- .-. .-. —●toil
5fNSf TY WI X
FW w = 106 ox: 1 .Ooc 00 w 20 TIMES 1 .omooE 01
,
.
-26-
. .+208
TIK= I o020000E 01
@Go -01,4000-01
OENSITY VS X
FRc6 K : 7 OX= I .00E 00 In: eo
...—- —. —___ .. .. ..●2O 8
C%NSITY VSX
FRC8 W = 107 ox= 1 .00E 00 I* 20 TIlc= : .OZOOOOE 01
U700-016aoo -al
-27-
i
wDO-ol&@oo-oI
FRCS142S
L:...—Ctmln Vs x
Ud
14 01s Z. fof-ot I* m lIW 1Ju$mw 01
t
I
-0X(+ ..-. .-@oa-ul
–*
WutlTv VI x
Pn@nos :4 Dxs S. fof-a IW m TIIRX 1mwcuf 01
1
* -q
w-o] . ----- .— 4QL+
-28-
t
.
I
I
40004; A4000-0:
“--”‘----”~Ofmm w x
-L Uo-or,Wloo+
FRc#!83r 17
I
I
. ..—●
wLaflv V9 x
on Z.sol?.a Ins m TIXIH 1 .oom~ Oi
A
,
+ooo-a j@oo.ol ..— ____
alNT ENERWw x
mom. :7 ox= t.$ol?.o: IXP m TIM. i AmxxoOr 01
-29-
CONCLUSIONS
The problems show that grouping by Group A definitely gives fewer
truncation errors than by Group B. The exception, of course, is shown
for Type II differencing (one point scheme) where both groupings should
be the same.
A comparison of the theoretically expected truncation errors given
in Table 2 with the results in Problems 1 to 7 (Group A) is enlighten-
ing. That Type II differencing gives about twice the diffusion as that
given by Type V is evident. The effect of a smaller magnitude coeffi-
cient in the 5X2 term for Types III, IV, and VII as compared to Type I
is evident. A simihr comparison can be made for the effect of the
second order space term between Types I and VI.
In general, the second order time terms in conjunction with accur-
ate space differencing gives a very fast and accurate differencing scheme.
A look at the density versus x plot of l?roblem17 shows the shock
front to be essentially two cells wide. It is suprising that so few
terms in the expansions sre necessary to produce this near perfect mov-
ing step function. It should be noted by observation of Eq. (11) and
Table 2 that no viscosityl (explicit or otherwise) is used to “smear”
the shock and that if any viscous type effects are present, they are
produced by the fourth space derivatives.
-30-
REFERENCES
1, Longley, H. J., Los Alamos Report LAMS-2379 (1760).
2. Lax, Peter D. and Wendroff, Burton, NYO-9759, 17 Ed. “Difference
Schemes with High Order of Accuracy for Solving Hyperbolic
Equations” July 31, 1962.
-31-
top related