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.

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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-