Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / DAAB-07-67-C-0199Office of Naval Research / ONR N00014-67-A-0305-001

Watanabe, D.S.

Illinois Digital Environment for Access to Learning and Scholarship Repository

REPORT R-573 JUNE ,1972 UILU-ENG 72-2234COORDINATED SCIENCE LABORATORYBLOCK IMPLICIT ONE-STEP METHODSD. S. WATANABEUNIVERSITY OF ILLINOIS -  URBANA, ILLINOIS' THIS DOCUMENT HAS BEEN APPROVED FOR PUBUC RELEASE AND SALE; ITS DISTRIBUTION IS UNLIMITED.”UILU-ENG 72-2234BLOCK IMPLICIT ONE-STEP METHODS byD. S. WatanabeThis work was su p p o r te d  i n  p a r t  by th e  J o i n t  S e r v i c e s  E l e c t r o n i c s  Program (U. S. Army, U. S. Navy and U. S. A i r  F orce )  under C on tra ct  DAAB-O7 - 6 7 -C -0 1 9 9 , and in  p a r t  by the  O f f i c e  o f  N aval Research  under C on tra c t  ONR N0001 4 -6 7-A-O305-0019 .R ep rod u ct ion  in  w hole  o r  i n  p a r t  i s  p e r m it te d  f o r  any purpose  o f  the U nited  S ta te s  Government.This document has been  approved f o r  p u b l i c  r e l e a s e  and s a l e ;  i t s  d i s t r i b u t i o n  i s  u n l im ite d .BLOCK IMPLICIT ONE-STEP METHODSD. S . WatanabeC o ord in a ted  S c ie n c e  Laboratory- U n iv e r s i t y  o f  I l l i n o i s  a t  Urbana-Champaignlo I n t r o d u c t i o nMany p h y s i c a l  systems are d e s c r ib e d  by o r d in a r y  d i f f e r e n t i a l  eq u a t io n s  whose s o l u t i o n s  c o n ta in  time c o n s ta n ts  d i f f e r i n g  g r e a t l y  in  magnitude,,Such e q u a t io n s  are c a l l e d  s t i f f .  When a c l a s s i c a l  n u m er ica l  in t e g r a t i o n  p roced u re  i s  a p p l ie d  to  a s t i f f  system  o f  e q u a t io n s ,  the s t e p  s i z e  is  g e n e r a l l y  determ ined  by the component o f  the s o l u t i o n  w ith  the l a r g e s t  decay  r a t e ,  w h ile  the r e g io n  o f  in t e g r a t i o n  i s  determ ined  by the component w ith  the s m a l le s t  r a t e .  A f t e r  the i n i t i a l  t r a n s i e n t ,  the r a p id ly  d eca y in g  components are i n s i g n i f i c a n t ,  but the s te p  s i z e  must remain sm all  to  p rev en t  nu m erica l  i n s t a b i l i t y .  As a r e s u l t ,  the time r e q u ir e d  to  in t e g r a t e  a h ig h ly  s t i f f  system  can become e x c e s s i v e .A - s t a b le  methods are  o f t e n  used to  overcome t h i s  problem  because  the s t e p  s i z e  o f  an A - s t a b le  method i s  governed  o n ly  by the a l lo w a b le  d i s ­c r e t i z a t i o n  e r r o r .  The s t e p  s i z e  f o r  A - s t a b le  l in e a r  m u l t i s t e p  methods must remain sm a l l ,  how ever ,  becau se  the o r d e r  o f  such methods cannot e x c e e d  two. I m p l i c i t  o n e - s t e p  methods are f r e e  from t h i s  r e s t r i c t i o n  on o r d e r ,  and s e v e r a l  c l a s s e s  o f  A - s t a b le  i m p l i c i t  o n e - s t e p  methods o f  a rb i t r a r y  o r d e r  e x i s t .  U n fo r t u n a te ly ,  these  i m p l i c i t  methods are r e l a t i v e l y  i n e f f i c i e n t .  T h e ir  e f f i c i e n c y  can be im proved, how ever , by o b t a in in g  a b l o c k  o f  new v a lu e s  s im u lta n e o u s ly .  These b l o c k  i m p l i c i t  o n e - s t e p  methods have been s tu d ie d  by R osser  [73  and Shampine and Watts [ 8 ] .  We p r e s e n t  in t h i s  paper a new c l a s s  o f  b l o c k  i m p l i c i t  methods p o s s e s s in g  d e s i r a b l e  s t a b i l i t y  p r o p e r t i e s .We s h a l l  r e s t r i c t  our d i s c u s s i o n  t o  a s i n g l e  e q u a t io n  f o r  s i m p l i c i t y  The g e n e r a l i z a t i o n  to  systems o f  eq u a t io n s  w i l l  be o b v io u s .  We f i r s t  d e s c r i b e  g e n e r a l  b l o c k  i m p l i c i t  methods and our new c l a s s  o f  methods.We then p r e s e n t  a con v erg en ce  theorem f o r  g e n e r a l  b lo c k  i m p l i c i t  methods F i n a l l y  we d is c u s s  the s t a b i l i t y  o f  ou r  m ethods.2 .  B lock  I m p l i c i t  MethodsWe wish  t o  approxim ate the s o l u t i o n  o fy '  = f ( x , y ( x ) ) , y ( a )  = a ,  ( i )on the i n t e r v a l  [ a , b 3 .  R ather than make s p e c i f i c  d i f f e r e n t i a b i l i t y  a ssu m ption s , we s h a l l  assume y has co n t in u o u s  d e r i v a t i v e s  on [a ,b 3  o f  any o r d e r  r e q u i r e d .L et  -  a + nh f o r  n = 0 , 1, . . .  and h > 0 .  We w ish  to  g en era tea sequence { yn3 which approxim ates  the sequence o f  e x a c t  va lues  { y ( x  ) } .b e t  y^ -  a .  An s - b l o c k  method g e n e ra te s  a b l o c k  o f  s a d d i t i o n a l  termss im u lta n e o u s ly  and u l t im a t e ly  prod u ces  va lues  f o r  a l l  n € I  . wherehI h = { n :  0 £ n £ ms} and m = C ( b - a ) / s h ] ,  Each b l o c k  o f  va lu esT0 ~ (y n+ l ’ * * ’ ,y n+s^ J w^ere  n = s a t i s f i e s  e q u a t io n s  o f  the formn + i = y „  +lh S cp . (x  ,3=1 r  n ’y n + s ’ h> ’ i  “  1. s . ( 2)The increm ent fu n c t io n s  cp . are determ ined  by f  and are fu n c t io n s  o f  xJyn ’ •••» yn + s > and h o n ly .  We s h a l l  assume they are d e f in e d  in  the r e g io n  R o f  (x ,^z,h) space  d e f in e d  by x € [ a , b - s h 3 ,  z € ( - co^ cd)^t v3k 0 ,  s ,  and h € [  0 , ^ 3 ,  where < ( b - a ) / s .  The l o c a l  d i s c r e t i z a ­t i o n  e r r o r s  are d e f in e d  byd i (V h> = hcpi (x n , y ( x n ) , . . . , y ( x n+s) >h) -  ( y ( x n + . )  - y i x ^ . p ) ,i ~  1» Sj (3 )and the o r d e r  o f  the method i s  d e f in e d  to  be the la r g e s t  in te g e r  r such th a tr+1d i (x >h)| = 0 (h  ) ,  i  = 1, s , (4 )in  the r e g io n  S o f  ( x ,h )  space  d e f in e d  by x € [ a , b - s h ]  and h € [ 0 , h Q] .We p rop ose  s - b l o c k  methods where each increm ent fu n c t io n  i s  ani n t e r p o l a t o r y  quadrature  form ula  em ploy ing  f u n c t i o n  v a lu es  and p o s s i b lyd e r i v a t i v e s  at  nodes in  the i n t e r v a l  []x ,x  ] , and va lues  o f  y a t  nonmeshn n+s Jp o in t s  are o b ta in e d  from the Hermite i n t e r p o l a t i o n  p o ly n om ia l  i n t e r p o l a t ­in g  the f i r s t  p i  - 1 d e r i v a t i v e s  o f  y a t  x n + i  f o r  i  = 0 , s .  Eachsuch  method can be w r i t t e n  in the formn + i = y + ‘ i . W ’ i  = 1, (5 )where 9 ( j , k )  € [ 0 , s ] ,  andyn+6 = i=0 j=0 t j ( e ) y( j )n+i* ( 6)I f  each  increm ent f u n c t i o n  has a degree  o f  p r e c i s i o n  a t  l e a s t  q -  1, then the method has o rd e r  r = m i n [ p ,q ] ,  where p = I! p #.43 .  ConvergenceThe f o l l o w i n g  theorem g iv e s  s u f f i c i e n t  c o n d i t i o n s  f o r  the con v erg en ce  o f  an s - b l o c k  method and in d i c a t e s  the o r d e r  o f  the accum ulated d i s c r e t i ­z a t io n  e r r o r .  Our methods s a t i s f y  th ese  c o n d i t i o n s  and hence are c o n v e r g e n t .  THEOREM. Let there  e x i s t  p o s i t i v e  c o n s ta n ts  L , M, and r such th a tg|cp ( x , £ , h )  -  cp ( x , r * ,h ) |  S L 2  |z -  z * | ,  1 = 1 ,  s ,  (7 )J=0 J Jf o r  ( x ^ j h )  and (x ,^z*,h) € R, and|di (x ,h )|  £ Mhr + 1 , i  = 1, . . . ,  s ,  (8 )f o r  ( x ,h )  € S. Then f o r  any h < m i n [ h Q ,2 /s ( s+ 1 ) l ] , the d i f f e r e n c ee q u a t io n s  (2 )  have a unique s o l u t i o n  { y  ) ,  d e f in e d  on I, , and th ere  i s  an nc o n s ta n t  N such th a t|yn - y ( x n ) l  * Nhr , n € i h .P r o o f .  Suppose a p a r t i a l  s o l u t i o n  s a t i s f y i n g  the d i f f e r e n c e  e q u a t io n s  (2 )  has been  found on the s e t  { o , l , . . . , n } ,  where n = i s  and i  <  m.E q u a tion s  (2 )  can be w r i t t e n  in  the form ^  = \| /(^ ) ,  whereiF or  a r b i t r a r y  ^  and the L i p s c h i t z  c o n d i t i o n  (7 )  im p l ie s  th a t5¡1 iW  - Ate*>ll i £J 1ihLH i  =where X = s ( s + l ) h L /2  <  1. Hence ^  i s  a c o n t r a c t i o n  mapping. S in ce  jr^i s  d e f in e d  f o r  a l l ^ y ,  i t  f o l l o w s  th a t  e q u a t io n s  (2 )  have a uniques o l u t i o n  . Thus the p a r t i a l  s o l u t i o n  can be c o n t in u e d  u n iq u e ly  throughout the s e t  I, .hA stan d ard  argument [ 5 ,  p . 12] shows th a t  the e r r o r s  e = |y - v ( x  )n 1 "'n J  v n 's a t i s f yei s + i  S ei s  +  i h L £ o e>ls+ i  +  Whr+1l  1, •••} s j 1 0 , . • » j m-1,These r e l a t i o n s  can be w r i t t e n  in the form  Ae„ £ b . f o r  Z = 0 . m-1~-~Z <-~>Z y ••• i  ywhere A and b . have e lem ents  ~  ~ Zaij  ^j[ j ^L, i, j 1, , S,...1 r+1b . -  ( l + ih L ) e i s  +  iMh , i  = 1,The in v e r s e  m atr ix  A 1 has the e lem entsc i j  "  ( ( l - s ( s + l ) h L / 2 ) 6 i j  +  i h L ) / ( l - s ( s + l ) h L / 2 ) ,  i ,  j = 1, ..., s.-1S in ce  A ‘  *  0 ,  i t  f o l l o w s  th a t  £  £ A_1b f o r   ^ = 0 ,  m-1. Def- LZz~ Z  ~ A  f ° r ^ “  0» •••> m -1, and s e t  6 q = 0 . The s ^ form  a non­d e c r e a s in g  sequence  andm er+1( i + l ) s  *  ( 1+ sh L * )e^ s +  M*h , A = 0 ,  . . . ,  m-1,6f o r  c e r t a i n  c o n s ta n ts  L* and M*. Another stan d ard  argument [ 5 ,  p . 12] showsei s  * (M*hr /sL*) (eLc^ ~ a  ^ - 1) , A  = o,  . . . ,  m.S in ce  e n £ f o r  n € 1^, the d e s i r e d  r e s u l t  f o l l o w s .4 .  S t a b i l i t yWe s h a l l  examine the s t a b i l i t y  o f  our methods by a p p ly in g  them to  the d i f f e r e n t i a l  e q u a t io n  y '  = X y , where X is  a com plex c o n s ta n t  w ith  Re(X) < 0 . The method (5 )  can be i n t e r p r e t e d  as an i m p l i c i t  o n e - s t e p  method w ith  s te p  s i z e  sh . S u b s t i t u t in g  y '  = Xy in e q u a t io n s  (5 )  and ( 6 ) ,  we o b t a inyn+s = R <shX> V  ( « )where R(|i) i s  a r a t i o n a l  a p p rox im a t ion  to  e^ . S in c e  R i s  a f u n c t i o n  o f  o n ly  the d eg rees  o f  p r e c i s i o n  o f  the increm ent f u n c t i o n s  and the o r d e r s  o f  c o l l o c a t i o n  p^ o f  the Hermite i n t e r p o l a t i o n  p o ly n o m ia l ,  the s t a b i l i t y  o f  e n t i r e  c l a s s e s  o f  methods can  be an a ly zed  s im u l ta n e o u s ly .  Let (P(j* • • • >PS) denote  the  c l a s s  o f  methods o f  the form (5 )  whose increm ent f u n c t i o n s  have d eg rees  o f  p r e c i s i o n  at  l e a s t  2p „  -  1, We s h a l l  c h a r a c t e r ­i z e  the s t a b i l i t y  o f  such c l a s s e s  in  terms o f  the f o l l o w i n g  c o n c e p t s .D e f i n i t i o n .  The c l a s s  ( P(), . . . , p  ) i s  A - s t a b l e  i f  |r (^)| <  1 f o r  Re (fi) <  0 .D e f i n i t i o n .  The c l a s s  (Pq , . . . , p ) i s  s t i f f l y  s t a b le  i f  J R (jjl ) J < 1 in the r e g io n s  { jjl : Re(p,) £ d} and {(i : D < R e(^ )  <  a ,  |lm(^i)| <  9 } .7D e f i n i t i o n .  The c l a s s  ( p _ , . . . , p  ) i s  s t r o n g l y  A- o r  s t r o n g l y  s t i f f l yu ss t a b le  i f  i t  i s  A- o r  s t i f f l y  s t a b le  and j R (jjl ) | “♦ 0 as Re(|i) -*We f i r s t  c o n s id e r  the c l a s s  ( p ,q )  o f  1 - b l o c k  methods em ploy ing  h ig h er  d e r i v a t i v e s .THEOREM. Let E ^ ( jj,)  denote  the ( i , j )  e n t r y  in the Pade ta b le  f o r  e ^ . Then R(|i )  = E^(|j,) f o r  the c l a s s  ( p , q ) .P r o o f .  A s im ple  c a l c u l a t i o n  shows th a tR(p.) = pp (M')/Qq (M<) swhere PpO-O and Qq (p*) are p o ly n o m ia ls  o f  o r d e r  p and q in  \ i . S in ce  R(|i) must be an ap p rox im a t ion  t o  e o f  o r d e r  p +  q , i t  f o l l o w s  th a t  R(|jl) must be the ( p ,q )  e n t r y  in the Pade t a b le  f o r  e ^ .COROLLARY. C lass  ( p ,p )  i s  A - s t a b le  and c l a s s e s  ( p ,p + l )  and (p ,p + 2 )  are s t r o n g l y  A - s t a b le  f o r  p ^ 1.P r o o f .  |e _ ( ijO|<  1 f o r  Re(p,) < 0 f  o r  i  = j  ^ 0 [ l ] , i  = j - U  0 ,and i  -  j - 2  ^ 0 [ 4 ] .  F i n a l l y , i t  i s  o b v io u s  th a t  ¡E . ,(ji)| -* 0 as ^JRe(|i) -» -03 f o r  i  < j .There are  i m p l i c i t  o n e - s t e p  methods p o s s e s s in g  s i m i la r  s t a b i l i t y  p r o p e r t i e s .  These in c lu d e  the method o f  Hermite [ 6 ]n+1 = y.P+ 2 i = l1 i  h Oi •y( i )p i ' nq— 2i = lh*£ q i  n+1 ( 10)where Oi . and |3 . are the i t h  c o e f f i c i e n t s  in  the numerator and denom inator p i  q io f  E , and the i m p l i c i t  R unge-Kutta p r o c e s s e s  d e v e lo p e d  by B utcher [ 2 ] ,  Ehle [ 4 ] ,  and Chipman [ 3 ] .  However, the methods in  the c l a s s  ( p ,q )  e n jo y8c e r t a i n  ad v an tag es .  They a t t a i n  the same o rd e r  o f  a c c u r a c y  as the scheme (10) w h i le  em ploying d e r i v a t i v e s  o f  low er o r d e r ,  and, u n l ik e  the Runge- Kutta  p r o c e s s e s ,  they  r e q u ir e  the s o l u t i o n  o f  o n ly  one n o n l in e a r  e q u a t io n  at each  time s t e p .We turn now to  2 -  and 3 - b l o c k  methods. C on s id er  the c l a s s  ( p , p , p ) .I t  can be shown th atr (m0  = ^ p ^ / ^ p ^ ’where P2 p (p,) i s  a p o ly n o m ia l  o f  o r d e r  2p in  (i. I t  f o l l o w s  th a t  | R(iil>) | = 1 f o r  u) € (-oo ,co) and | R (|i ) ( -  1 as ||jl| -» ® f o r  a l l  p ^ 1. Using the Routh- Hurwitz c o n d i t i o n s ,  we have v e r i f i e d  that  R(jj,) i s  r e g u la r  f o r  Re(M-) <  0 and hence th a t  ( p , p , p )  i s  A - s t a b le  f o r  p = 1, 2 ,  and 3 .  We c o n je c t u r e  th a t  t h i s  i s  true  f o r  a l l  p ^ 1.There are s t r o n g ly  A - s t a b le  and s t r o n g l y  s t i f f l y  s t a b le  2 -  and 3 - b l o c k  methods. For exam ple , the c l a s s  ( 1 , 1 , 2 )  is  s t r o n g l y  A - s t a b l e ,  and the c l a s s e s  ( 1 , 1 , 3 ) ,  ( 1 , 2 , 2 ) ,  ( 1 , 2 , 3 ) ,  ( 2 , 2 , 3 ) ,  ( 1 , 1 , 1 , 2 ) ,  and ( 1 , 1 , 2 , 2 )  are s t r o n g l y  s t i f f l y  s t a b l e .  F ig u re  1 shows the l o c i  in  the hX plane where j R | — 1 f o r  th ese  c l a s s e s  o f  methods. S in ce  the l o c i  are symmetric w ith  r e s p e c t  to  the r e a l  a x i s ,  o n ly  h a l f  o f  each  lo c u s  i s  p l o t t e d .  The r e g io n s  o f  s t a b i l i t y  l i e  to  the l e f t  o f  the l o c i .  The r e g io n s  o f  i n s t a b i l ­i t y  in the l e f t  h a l f - p l a n e  are remarkably sm a l l .  The c o r r e s p o n d in g  r e g io n s  f o r  Gear s £53 s t i f f l y  s t a b le  l in e a r  m u l t is t e p  methods are la r g e r  by s e v e r a l  o r d e r s  o f  m agnitude.95. C on clud ing  RemarksWe have p r e s e n te d  a new c l a s s  o f  b l o c k  i m p l i c i t  o n e - s t e p  methods p o s s e s s in g  d e s i r a b l e  s t a b i l i t y  p r o p e r t i e s .  S in ce  they  are i m p l i c i t ,  t h e i r  im plem entation  i s  n o n t r i v i a l  and r e q u ir e s  the developm ent o f  e f f e c t i v e  and co n v e rg e n t  i t e r a t i v e  p ro ce d u re s  and p r a c t i c a l  e r r o r  e s t im a t io n  schemes. These problem s are b e in g  s t u d ie d  and w i l l  be d i s c u s s e d  e ls e w h e r e .10R e fe re n ce s1. G. B i r k h o f f  and R. S. V arga, " D i s c r e t i z a t i o n  e r r o r s  f o r  w e l l - s e t  Cauchy p rob lem s,  I , "  J .  Math, and P h y s . , v .  4 4 ,  1965, p p . 1 -2 3 .2 . J .  C. B u tch er ,  " I m p l i c i t  Runge-Kutta p r o c e s s e s , "  Math♦ Comp. , v .  18, 1964, pp . 5 0 -6 4 .3. F. H. Chipman, N um erica l S o lu t i o n  o f  I n i t i a l  Value Problems Using A -S ta b le  Runge-Kutta P r o c e s s e s , R esearch  R eport  CSRR 2042, D ept, o f  A. A. C. S . ,  U n iv e r s i t y  o f  W a te r lo o ,  W a te r lo o ,  O n ta r io ,  1971.4 .  B. L. E h le ,  On Pade A pp rox im ations  t o  the E x p o n e n t ia l  F u n ct ion  and A -S ta b le  Methods f o r  the N um erical S o l u t i o n  o f  I n i t i a l  Value P rob lem s, R esearch  R eport  CSRR 2010, D ept, o f  A. A. C. S . ,  U n iv e r s i t y  o f  W a te r lo o ,  W a te r lo o ,  O n ta r io ,  1969.5 . C. W. Gear, N um erica l I n i t i a l  Value Problem s in  O rdinary  D i f f e r e n t i a l  E q u a t io n s , P r e n t i c e - H a l l ,  Englewood C l i f f s ,  New J e r s e y ,  1971.n6. C. H erm ite , "Sur la  form ule  d ' i n t e r p o l a t i o n  de L a g r a n g e ,"  J . fu r  Reine u. Angew. M a th . , v .  84, 1878, p p . 7 0 -7 9 .7. J .  B. R o s s e r ,  "A Runge-Kutta f o r  a l l  s e a s o n s , "  SIAM R ev . , v .  9,1967, pp . 4 1 7 -4 5 2 .8. L . F. Shampine and H. A. W atts ,  " B lo c k  i m p l i c i t  o n e -s t e p  m eth od s ,"  Math. Comp., v .  23 , 1969, pp. 731 -740 .11F ig u re  1. S t a b i l i t y  r e g io n s  f o r  2 -  and 3 - b lo c k  methods.^ecui^^Cla^s^icatiot^DOCUMENT CONTROL DATA - R & D( S e c u r i t y  d a s  s i  l i e  a t  i o n  o f  t i t l e ,  b o d y  o7 a b s t r a c t  a n d  i n d e x i n g a n n o t a t i o n  m u s t  be  e n t e r e d  w h e n  th e  o v e r a l l  r e p o r t  i s  c l a s s i f i e d )ZB. R E P O R T  S E C U R I T Y  C L A S S I F I C A T I O NUNCLASSIFIED°doo râ ïn a  tAe (Î1 'S Tc i4 cn5eraf ’áffSFát oryU n iv e r s i ty  o f  I l l i n o i s  Urbana, I l l i n o i s  618013.  R E P O R T  T I T L EBLOCK IMPLICIT ONE-STEP METHODS4.  D E S C R I P T I V E  N O T E S  ( T y p e  o í  r e p o r t  a n d  i n c l u s i v e  d a t e s )5- A U T H O R ( S )  ( F i r s t  n a m e ,  m i d d l e  i n i t i a l ,  l a s t  n a m e )Ü. S. Watanabe6-  R E P O R T  D A T EJune, 19727 a .  T O T A L  N O .  O F  P A G E S117 b .  N O .  O F  R E F S  88 a .  C O N T R A C T  O R  G R A N T  N O .  DAAB“ 0 7“ 6 7~C“ 0 19 9 JONR N 00014 -67 -A -0305-00199 a .  O R I G I N A T O R ' S  R E P O R T  N U M B E R ( S )b .  P R O J E C  T  N O . R-5739 b .  O T H E R  R E P O R T  N O ( S )  ( A n y  o t h e r  n u m b e r s  t h a t  m a y  b e  a s  s i  g n e dthis repc^Lu-ENQ 72-22341 0 .  D I S T R I B U T I O N  S T A T E M E N TThis document has been approved f o r  p u b l i c  r e l e a s e  and s a l e ;  i t s  d i s t r i b u t i o n  u n l im ite d .i s1 1 .  S U P P L E M E N T A R Y  N O T E S1 3 .  A B S T R A C T1 2 .  S P O N S O R I N G  M I L I T A R Y  A C T I V I T YJ o in t  S e r v i c e s  E l e c t r o n i c s  Program througl U. S. Army E l e c t r o n i c s  Command and O f f i c e  o f  N aval ResearchA new c l a s s  o f  b l o c k  i m p l i c i t  o n e -s t e p  methods f o r  o r d in a r y  d i f f e r e n t i a l  e q u a t io n s  i s  p r e s e n te d .  The methods are based  on quad ra tu re  and g en era te  fu n c t io n  v a lu es  a t  nonmes quadrature  p o in t s  through Iiermite i n t e r p o l a t i o n .  A g e n e r a l  con v erg en ce  theorem  f o r  b l o c k  i m p l i c i t  methods i s  p r o v e d ,  and the s t a b i l i t y  o f  the new c l a s s  o f  methods i s  a n a ly zed . The c l a s s  i s  shown t o  c o n ta in  A - s t a b l e ,  s t i f f l y  s t a b l e ,  s t r o n g l y  A - s t a b l e ,  and s t r o n g l y  s t i f f l y  s t a b l e  methods.liDD 473Security ClassificationSecurity Classification1 4K E Y  W O R D SL I N K  A L I N K  B L I N K  CR O L E . W T R O L E W T R O L E W TN um erical Methods f o r  O rdinary D i f f e r e n t i a l  E quationsMethods Based on QuadratureHermite I n t e r p o l a t i o nA -S ta b le  MethodsS t i f f l y  S ta b le  MethodsS tr o n g ly  A -S ta b le  MethodsS tr o n g ly  S t i f f l y  S ta b le  MethodsS t i f f  O rdinary D i f f e r e n t i a l  E quations1Security Classification

Block Implicit One-Step Methods

IDEALS @ Illinois

https://www.ideals.illinois.edu/bitstream/handle/2142/74774/B9-573.pdf?sequence=2&isAllowed=y

Block Implicit One-Step Methods

Abstract

Similar works

Full text

Available Versions

Illinois Digital Environment for Access to Learning and Scholarship Repository

IDEALS @ Illinois