Internat. J. Math. & Math. Sci.
(1987) 173-179
No.
173
Vo]. i0
AN ENERGY CONSERVING MODIFICATION OF
NUMERICAL METHODS FOR THE INTEGRATION OF
EQUATIONS OF MOTION
ROBERT A. LABUDDE
821 Hialeah Dr.
Virginia Beach, VA 23462
and
DONALD GREENSPAN
Department of Mathematics
University of Texas
Arlington, Texas 76019
(Received July 2, 1982)
ABSTRACT.
In the integration of the equations of motion of a system of particles, con-
ventional numerical methods generate an error in the total energy of the same order as
the truncation error.
A simple modification of these methods is described, which re-
sults in exact conservation of the energy.
KEY WORDS AND PHRASES.
Conventional numerical methods, ordinary differential equations,
multiplicative parameters, truncation error.
1980 AMS SUBJECT CLASSIFICATION CODE.
i.
70F34.
INTRODUCTION.
When applied to the motion of a system of particles, conventional numerical methods
for the integration of ordinary differential equations only approximately conserve the
total energy of the system.
The error in the calculated value of the energy is of the
same order as the truncation error in the velocities.
In previous work [1]-[5], a new
class of methods was described, which maximally conserve the constants of motion.
These methods exactly conserve the total energy and linear momentum, and conserve the
total angular momentum to at least one higher order than the corresponding conventional
methods.
In what follows, our purpose is to show how conventional numerical methods--exemplified by the third-order Taylor series and Adams’ formulae--can be modified so
that exact conservation of energy occurs.
This modification simply involves the in-
troduction of adjustable, multiplicative parameters, whose values are unity for the
conventional case.
2.
EQUATIONS OF MOTION.
The following is a brief description of the equations of motion of a system of
n
particles, interacting according to a pairwise-additive potential.
For more details,
see [i] or [5].
Suppose particle
r.
l
velocity vector
i
(xi,Yi,Zi)
has mass
mi,
position vector
(2.1)
-
R.A. LABUDDE and D. GREENSPAN
174
.dY i dzil
i
[dx
dt
[
vi
and acceleration
d2xi
a.
i
dt
Newton’s
d
2
2
dt
Yi
2
2
zi
dt
law of motion
m.a.
I i
d
(2.2)
(2.3)
2
=.
(2.4)
I
relates the acceleration
to the force
a.1
Fi,
given by
(2.5)
r.
where
It will be assumed that
is the potential of interaction.
has the pair-
wise-additive form
rn)
(ri,r2,
where
r
Z
(2.6)
ij(rij)
i<j
is the magnitude of the vector distance
13
ro
between particles
13
i
and
j
r.. =r.
(2.7)
-r.1
J
1J
As a consequence of equation (2.6),
n
(2.8)
IF..
F.
j=l 31
where
31
and
0
Fji
if
The introduction of equation (2.8) into equation (2.4) gives
i.
j
(2.9)
drij rij
drji rji
13
the equation of motion
Z F..
j=l 31
m.a.i
For
particles, equation (2.10) yields a system of second-order ordinary differ-
n
ential equations for the
t’
at any later time
.
This system may be used to solve for the
r..1
t
+ At,
E
and
v.1
at time
and
n
occurs because of the existence of the poten-
n
Z
mi(v i- vi)
i=l
a-b
+
Z
i=l
mi(v i" vi)
denotes the scalar product of two vectors
a
+
and
Z
i<j
b.
ij
Conservation of
energy is expressed by the equation
E(t’)
for any two times
E(t)
t
and
t’,
with
E
i’
t.
Here,
E
where
r.1
given the
Conservation of the total energy
tial
(2.i0)
drij rij
j=l
evaluated along the trajectory.
NUMERICAL METHODS FOR THE INTEGRATION OF EQUATIONS OF MOTION
175
3. CONVENTIONAL NUMERICAL METHODS
A simple example of a conventional approximation method for the numerical solu-
(2.10) is provided by the truncated Taylor-series formulae
tion of equations
r’
r
c,i
+ v.z A
i
t
i
+
mi
Z
j=l
I
i
2
n
(At)2 + ji
2
ji
(At)31
(3.1)
6
and
v’ ,i
=v.z +--
where the
t’
r’c,i
+ At,
t
Z
j=l
m.
z
V’c,i
and
At+
(3.2)
are the calculated values for the
r
ol
and
v_
at time
and
dq
Goo
31
i
d0ji vji
drj i rJ i
dt
0ji
dr..
2
i
d0 jirji
ro
rJ i drj i
(3.3)
3i
where
rji
drji
rji’vji
dt
r..
and
v
vji
vj
i
The method of equations (3.1) and (3.2) is of third-order, since
7c,i
’r.
i
+ 0[(At) 4]
(3.4)
0[(At)3]
(3.5)
and
-v.’
’.+
C,l
l
due to the neglect of the succeeding Taylor-series terms.
error of
V
0[(At)
3]
in the value of the energy
E’C
These errors generate an
calculated using the
.:
Cl
AE c
E
0[--(At)
E
c
’
ci
3]
and
(3.6)
The third-order Adams’ method arises via equations (3.1) and (3.2) and the ap-
proximation
G..
zj
/a
(3.7)
G.. + 0[At]
j
where
/a
F
G..
c ,ij
-ij
In equation (3.8)
using the
(3.8)
At
zj
r’c,zj+o..
used for the
Gij.
’
denotes the value of
c,ij
Equations
(3.4), (3.5),
and
Fij
obtained from equation
(3.6) also hold when the
ij
(2.9)
are
R.A. LABUDDE and D. GREENSPAN
176
.
4.
ENERGY CONSERVING MODIFICATION OF CONVENTIONAL METHODS.
3, with
Consider the third-order methods of Section
G.*.
replacing either the
or
r’c,i
r
v’
v
i
+
+m
v At
i
(t)2
l
ji
j=l
i
+
2
3
(At)
G
(4 i)
6
i
and
c,i
i
+-- E
mi
2
(At)
+
2
j=l
(4 2)
When equations (4.1) and (4.2) are used to obtain estimates for
AE c
r.l
and
v
i,
an error
is made in the total energy, which is given by
n
E’c
AE c
E
7.
I m
i= 1
At
i
,
c
n
Z
Z
j=l
I.+
At
+-m.
vi
l
+
Y.
i<j
+
($ij
+
aij
-)
n
Z
k=l
Fki).
+i
z
i+m.
k=l
l
At
i
1
F-
n
i=l
+--(v
(v
i
+
aijAt
Fij
b
F
ji
G*.
3
+ A
G.*lj
i3
+
At
(4.3)
where
aij
A
ij
n
l
Fki
k=l Lmj
’
mi
ij (r’c ,ij
ij
c,ij
ij(rij ),
and
.
Suppose now, instead of using
AE
to an error
G..
e..
c
of
or
*
Gij
+a
e
ij
(4.4)
c,i
c,j
c,lJ
Gij
0[(At)3]--that
Gii
Gij
adjustable
or
Gij
*..lj
in equation (4.3)--which leads
given by
(4.5)
NUMERICAL METHODS FOR THE INTEGRATION OF EQUATIONS OF MOTION
is used.
The
g..
are to be chosen so that
lj
+ 0[At]
1
l]
177
(4.6)
(preserving the order of the method) and so that exact conservation of energy occurs.
Solving
AE
0
C
e..
for the
ij
/+
+
(Vij
(At)
4
ij
+
aijAt
+
2
t
aij--)
}
At
ij
+
Fij
Aij
At
s
ij
(4.7)
0
particles, equation (4.7) yields a set of implicit, coupled equations in the
For
n
ij’
since the
and
bi3
At
For small
depend upon the values of the
c,ij
e
occur through the
ij
In both these cases, the terms involving the
r’c, ..).
13
proportional to
(At)3.
(vij
+
ij
aijAt)
0[At].
are of
-
bij
e.o
lJ
and
...
c,ij’
ij’
The
(through
occur with coefficients
(Compare equations (4.1), (4.2), (4.4), and (4.7).)
the coefficients of the linear terms in
trast
ij"
the equations of (4.7) are strongly linear in the
only nonlinear dependences on the
the
-
gives, for example, for (4.5), the equation (cf. [i] and [5])
In con-
namely
At
.
Because the equations of (4.7) are linear except for terms of
0[(At)3],
they
may be easily solved via the iteration formula
..
&ij
2
At
ij
At
I]
For small
At
+ (V.o+a.
13
lj
At
--)
z
(.)
(vij +-"
aij At+bij 4
At
(4.8)
the equations of (4.8) are solved via successive substitutions,
starting with
..
(4.9)
i
Iteration to convergence of the
e..
guarantees exact conservation of energy in the
me tho d.
Higher-order formulae may be obtained directly in the same way as equation (4.7).
If the highest-order terms involve
13
dt
m
then these are replaced by
/(m)*
F..
13
+(m)
e.. F..
1J
R.A. LABUDDE and D. GREENSPAN
178
where the
satisfy equation (4.6).
S.o
lj
The formulae for the
v’
C,l
(4.3), the
in equation
sum transformed to
and the
i <j,
ij
are substituted
terms set
individually
e..
methods, with the first approximations given by equations (4.9).
For very high order methods, the extra algebra needed to obtain the
to zero.
These resulting implicit equations in the
are then solved by standard
siderable, and substantially reduces the relative efficiency of the method.
e.. is conHowever,
it should be noted that conservation of energy guarantees stability in the usual sense
(bounded motion), which is always a desirable computational property.
5. NUMERICAL EXAMPLE.
As an illustration of the affect of the modification described in Section 4, the
modified and unmodified forms of the third-order Adams’ method are compared numerically
on a sample two-dimensional problem involving two particles.
Here
m
I
2,
n
m
(5.1)
2
2
and the gravitational interaction
i
12(r12)
is used.
(5.2)
r12
The initial conditions are chosen so that the center-of-mass of the system
is at rest with
r12(0)
(,0)
(5.3)
v12(0)
(0,1.63).
(5.4)
The value of the energy is then
E
0.6715500000...
Because of the form of
(5.5)
12
in
(5.2), the
exact motion that occurs traces out
a closed ellipse with major-axis
2a
1.48909 23855
corresponding to upper and lower bounds on
0.98909 23855
r>
r<
and
(5.6)
r12
of
0.50000 00000.
The motion repeats itself with period
equal to
4.0366 15087,
The implicit equations of the third-order methods were iterated to a relative con-8
A constant step-size of
ergence of I0
t
/80
In order to focus attention on the errors made in the methods, results were
obtained at times t which were multiples of the period
where the exact solu-
was used.
NUMERICAL METHODS FOR THE INTEGRATION OF EQUATIONS OF MOTION
tion returns to the initial conditions.
E,
error in the calculated value of
the deviation of
from
r12
,
Measures of the errors at these points are the
the deviations from zero of
dX/dt and
unmodified Adams’ method makes an error in
E
t
mr.
It can be seen that the
as well as larger errors in
and compares unfavorably with the modified method.
dX/dt and
Another simple measure of the
error for this problem is the number of steps over which a phase error of 180
i.e., the time at which
r12
0.985
this was about 2800 steps (35T).
phase error of less than
180
and
1/2.
Table I gives these quatities for several times
Y,
179
instead of
0.5.
is made:
For the unmodified methods,
For the modified methods, at 20000 steps (250T), a
had been made.
Programs for the methods are given in the Appendix of [6].
m
0
i
E
Method
a
Exact
b
U
M
c
dX
dt
r
0.00000
Y
0.00000
0.67155
0.50000
0.67140
0.50221
0.20630
-0.08704
0.67155
0.49997
0.02164
-0.00462
2
U
M
0.67099
0.67155
0.50873
0.49997
0.40254
0.04328
-0.17213
-0.00923
3
U
M
0.67040
0.67155
0.51924
0.50001
0.58036
0.06492
-0.25351
-0.01385
5
U
M
0.66905
0.67155
0.55019
0.50017
0.86162
0.10818
-0.39996
-0.02311
i0
U
M
0.66679
0.67155
0.65934
0.50116
1.15127
0.21592
-0.64976
-0.04639
I00
U
M
0.66561
0.67155
0.97998
0.62554
0.82003
1.35684
-0.97598
-0.57888
At times t
alnitial
m
conditions
bunmodified
CThird-order
third-order Adams’ method
Adams’ method modified to give exact energy conservation.
TABLE I.
Comparison of Modified and Unmodified liethods
on a Simple Gravitation Problem
REFERENCES
i.
LaBUDDE, R. A.
and GREENSPAN, D., "Discrete Mechanics--A General
Treatment",
to be
published in J. Computational Phys.
2.
LaBUDDE,
3.
LaBUDDE, R. A. and GREENSPAN, D., "Energy and Momentum Conserving Methods of Arbi-
R. A. and GREENSPAN, D., "Discrete Mechanics for Anisotropic Potentials",
Univ. of Wis. Computer Sciences Dept. Report WIS-CS-203 (1974).
trary Order for the Numerical Integration of Equations of Motion. I. Motion
of a Single Particle", Univ. of Wis. Computer Sciences Dept. Report WIS-CS-
208 (1974).
4.
LaBUDDE, R. A. and GREENSPAN, D., "Discrete
with Application to the LEPS
port WIS-CS-210 (1974).
Form",
Mechanics for Nonseparable Potentials
Univ. of Wis. Computer Sciences Dept. Re-
GREENSPAN, D., "Energy and Momentum Conserving Methods of Arbitrary Order for the Numerical Integration of Equations of Motion. II. Motion
of a System of Particles," Univ. of Wis. Computer Sciences Dept. Report WIS-
5.
LaBUDDE, R. A. and
6.
LaBUDDE, R. A., and GREENSP/%, D., "An Energy Conserving Modification of Numerical
Methods for the Integration of Equations of Motion," Univ. of Wis. Computer
CS-215 (1974).
Sciences Dept. Report WIS-CS-217
(1974).