Numerical solution of differential equations by predictor-corrector process which selects optimum corrector at each ste

Wesson, J. R.

English

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*, QUARTERLY TECHNICAL REPORT 
for 
October, November, December, 1965 
Contract NAS 8-2559 ( M I )  
During t h e  current quarter ,  t h e  contract  work has been done ch ief ly  by 
Professor Wesson and M r .  John Edwards. As  usual,  Professor Shanks' par t ic ipa-  
t i o n  has been reduced s u b s t a n t i a l l y  because of administraf,ive r e s p o n s i b i l i t i e s .  
However, work has continued on "error s t a b i l i z e r s "  under t h e  supervision 
of Professor Shanks. A grea t  par t  o f t h i s  work has been of a computational 
nature  i n  order t o  get  addi t ional  information on these s t a b i l i z e r s .  
Professor Wesson has continued his  inves t iga t ion  of predictor-corrector 
processes i n  t h e  solut ion of d i f f e r e n t i a l  equa.tions. A method i n  which a new 
cor rec tor  i s  chosen each s t e p  has been developed for several  orders,  and has 
been t e s t e d  for orders  3 and 5 .  The invest igat ion included a c lose  look  a t  
t h e  r o o t s  of t h e  c h a r a c t e r i s t i c  equation. 
Professor Wesson read a paper out l ining h i s  r e s u l t s  a.t the  November 1 2  
meeting of t h e  American Mathematical. Society. The paper i s  submitted as p a r t  
of t h i s  re-aort . 
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Re spec t f u l l y  submitted , 
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Audie Anderson 
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A REPORT ON A PREDICTOR-CORRECTOR 
PROCESB IN WHICH THE SEUECTION OF 
AN OPTIWY CORRECTOR IS MADE EACH STEP 
By James R. we88on, Vanderbllt Wvereity 
1. 
use ef a predictorccorrector process In which a new corrects. I 8  sebcted 
a t  each step. 
between truncation error and propagated error. 
be applicable to  the numemical solution of a wide variety of different ia l  
equations. 
Introduction. It I s  ths purpose here t o  report t h b  developmmt and 
The purpose of the program is t o  make a good compronise 
The new procecls aeems t o  
We present in section 2 a me-parameter familp of correctors, along 
with an examination of the characteristic roots. In section 3 m explain 
the method for choosing the optimum corrector from the fifth-order family. 
Ths program l a  skotcbd in section 4, and a comparison of the 1 1 8 ~  mthed 
with othsr methods l a  givun in section 5. 
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2. A f a d 4  of stable correctors. Th6 A b 6  corractor 
is noted for 
Nenton-Cotea 
its faverabb s tab i l i ty  prepertiies. On ths othsr hand t& 
corrector 
TMm work raa dona under NASA contract ItAS8-2ss9, 
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has a smaller step error  but is on the boundary of the set of stable 
correctora. For h = 0, the characteristic mots of (1) are l,O,...,O 
and the  characteristic root8 of (2) a m  the (n+l)-th root6 of unity. 
We now consider th, me-parameter famlly of correctors obtained by 
ml81pWng rqurblonr (1) ur8 (0) by r urd 1 - r ~0109()8biYOlY, and 
then adding ths re8ult8. As r varies from 1 to 0, ths correotor ChaIAg06 
from (1) t o  (2). 
Far h = 0 and 0 2 r 5 1 it is ea8y to show that no characteris- 
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t i c  root is outside tb tanit circle. 
x f o give 1x1 2 r 4 (1-r) 1x1". Then Ixlsi  i a  impossible. m n  
0 r 
For any o t b r  root m ~ l d  satis0 (l+a)* + xn-a + Pa + *.. + I + 1 = 0, 
where r/(Lr) = a > 0. ~ h s n  multipIying by IC - 1 g i m s  
For -**' + r$ + (1-r) = 0 and 
1, every root except tbb single root  1 i r  within th unit c i rc le .  
= 1 + a. The point 
with center -a and 
x" 1 g i n 8  
pa + x*' + + 1 * 0, (l+o)fl = 0, and x = 0, a contradiction. 
3. 
best corrector for oolving 7' f(x,y) w i t h  stepsiee h depend8 on 
K = h fy [2, p.8; 3, p. 197; 4, p. 218). Recall t h a t  each corrector of 
t h s  family of section 2 is determined by r, and euppose m use K to 
Choice of a corrector. It is well knm that the selection of the  
determim r - r(K). Then m may change tb corrector aa often ab we 
take tip6 to  computa K and r(K). 
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I n  this report, the function r(K) has been determined empirically 
fo r  fifth-order correctors. Note that  equations of the form y' - ay 
present 8 c o n e t a t  K. For such an equation variow correctors (that 
Is, various values of r) were tested and compared, The beet one was 
uelcd %a d * W M  P(K)~ ~ M P  ~ & W I  a? K knee oBBMllkB W v-8 
a In y' - ay, or by varying h. Each of these -8 used to determine 
another best value of r. A least  squares moothing gave ths equation 
r - 0.9K' - 1.18K + 0.18 for IKlr 0.5. Then as K v a r l a ~  from 
-0.5 to  0 t o  0.5, r variee from 0.91 to -0.27. T b  decision 
t o  permit r to  get outside the interval [0,11 i r  risky. However, 
the best resul ts  fo r  the equations solved were obtained U s i n g  r 88 
defined above. 
4. A sketch of t he  program. 
irr used t o  produce three starting points. 
The following process by Shanks [6, p, 193 
T b n  the predictor . 
gi res  y4, and y: 'is predicted as f(x4,y4). Ths correator (take r - 1 
i n i t i a l l y )  
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5 s L 
gives a corrected value of y4, which lead8 t o  a corrected value of 
y: f(+,y4), etc. (In ths examples described here, t h e  corrector is 
applied twice- each step,) 
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Now ue compute a new value of r f o r  the next step. Take - - 
7 -  K - h(y: - y:) / (y4 - y4), w b r s  y4, y4 are predicted values, and 
y:, y4 are ths l a s t  corrected values. 
r = 0.57Ka - L18K + 0.18 if K 5 0.5. In  case - y4 0, use 
r m 1, Far K 5 -0.5 take r constant. S i m i l a r l y  fo r  k f 0.5 take 
Thsn compute 
r aonstant. 
5. Conparison with other methods. The follaning table indicates how 
th method formulated above competes w i t h  methods i n  which a fixe$ 
corrector (value of r )  is used a t  each step, 
a t  (0,l) w i t h  h = 0,Os and continued to  x = 10. Th6 programs a m  
Each solution is started 
identical  in every respect, except in the choice of r. 
In the first s ix  columns of the table r is fixed. The first 
colunm are the resul ts  f o r  the Adam corrector ( r  = 1). IVhn E 3 0, 
values of 1: close to 0 are used, and this gives a mall truncation 
error  with no apparent damage t o  the numerical stability of tho process. 
For K 0, r must be nearer t o  1, or e lm the process becomes un- 
stable, For all values of K tb new process competes very well with 
any of the 0ther8. For k 0 it ia on a par  with the best of the 
others, and for k * 0 it 8utpa888s the others. It seem t o  be depend- 
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able f o r  a broad class of equations. 
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6, Conclueion. 
matically achieving a balance between numrical a tab i l i t y  and truncation 
error. 
f inctional evaluations and with a relatively small  increase in canputing 
time. It 58 to be expected that the empirical formula fur r (aee 
section 3) can be improved, Indeed, the author has experinrented with a 
number of different functions r = r(K), but none ham been appreciably 
superior t o  the one used. 
Changing t h 6  corrector each step is one way of auto- 
A new corrector each step can be erlected without additional 
APPENDIX 
The derivation and treatmsnt of faprmulas (1) and (2) of section 2 
term fo r  the f ifth-order corrector 
I - 0, the corrector i 8  of  order r ix .  
r ' .  . , 
REFERENCES 
. 
1, Collatz, L. Tim n u r i c a l  treatment of different ia l  equationa, 3rd 
edition, Springer, Berlin, 1960, 
2, Dahlquiat, Germund, Stabil i ty and error bounds in ths numerical 
integration of ordinary different ia l  equations, Almqubt and 
Wiksells BOktr$%bfi AB, Uppsala, 1958. 
3. Hamming, R. W. Numerical methods fo r  scientieta and engineera, 
McGraw-Hill, New York, 1962. 
b. Henrici, Peter. Discrete variable methods i n  ordinary d i f fe ren t ia l  
equations, Wlley, New York, 1962. 
5. Milne, W. E. Numerical Solution of differential  equatione, wiley, 
New York, 1953. 
6, Shanka, E. B. Highbr order approximation8 of Runge-Kutta tyfm, 
National Aeronautics and @ace Administration Technical Note D-2920, 
Washington, D. C., 1965, 
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Predictor-corrector process in which the selection of an optimum corrector is made each step Quarterly technical report, oct. - Dec. 1965

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