(2#5#1) [ l - p ^ • !♦ pAf*e^pS?yei u * j •
On expansion in te rn s o f u and i t s d eriv a tiv e # a t the p o in t ( i , j,m) up to and including term s of 0 (h * ), the l e f t Hand s iie bLoo&aes
- !« • <4» ' ^
^ - f t ♦
^ < 4 } » - f " ^ '
- f
f t
- f -■ g » • * î^ « X “ f t l ÿ
- “ ' f “ K " f t ' - f < 4 5 * ' ’^ h ( f t ' J ^ whioh i s simply
[I ♦ ^pAfa ♦ jpB?y e ^ /B9yAf*3u* j
and th u s the aoeuracy o f (2 .5 .1 ) is deaonstrated in the case of the
m atrices A airi B depending on x*y and t . I t should be noted th a t the sftfttrices on the l e f t Wind side o f equation (2.5#1) a re evaluated a t t • m ♦ 1 whereas "hose on the r i ^ t hand side are evaluated a t t *
m.
Applying the above tec) m i que to each member of the fam ily o f formulae in se c tio n 2.3# i t can be v e rifie d th a t schemes o f t^pe ( i i i ) are s t i l l s a tis f ie d up to and including terms of order h* whereas those
53.
o f types (1) end ( i i ) nam have e tru n c a tio n e r r o r o f 0 (h * ). However Wendroff showed th a t i f the c o e ffic ie n t me tr ic e s A and B weie eve us ted
not e t ^ e p o in t ( i , j , s ) b u t a t the p o in t ( ie , je ,e ) (s # a ,a e 1) then h is scheme had second o rd er accuracy# Since the method
developed in ee etio n 2^2 d i to r s from Wendroffs fo r u le (even in the case of v a ria b le c o e ffic ie n ts ) by a term of 0(h*) i t follow s th a t the e ig h t p o in t enhenes o f "straig h t-th ro u g h * type w ill have th ird order tru n c a tio n e n o r, i f we evaluate the m a trices, not a t the p o in t ( i , J , s ) b u t a t the i>oint ( i e p , j e s ,s ) ( s ■ m,m e l ) # The signs in ( i t
1
&,s) are cliosen so t h o t j t h t o p o in t i s a t the ce n tre of thesquaiv o f nodes a t each tin s lev el# A sim ila r n o tif ic a tio n i s p o ssib le f o r the " la te r a l" schemes.
The s t a b ili ty o f a d iffe re n c e scheme w ith slowly' vta*ying c o e ffic ie n ts i s governed by the l^Oül am p lific a tio n matrix# din^ to LaX [2 0 ],
the d iffe re n c e scheme i s sta b le i f the norm o f the lo c a l am p lific a tio n m atrix does not exceed one# In the previous s e c tio n , i t was shown
th a t
. I provided
(2 .5 .2 ) AB . BA,
•ad ao (2 .5 .2 ) «ould appear to be the o o ad ltlsn fo r the e ig h t p o in t metiiods w ith v a ria b le c o e ffic ie n ts to be stable# This iw s tric tiv e co n d itio n was also req u ired by Lax and Wendroff in ord er th a t th e ir high accuracy e x p lic it scheme be s ta b le fo r the case o f varii&ble e o e ffio ie n te [22% I t i s hoped th a t even when (2#5#2) i s not
■ - '-i ' Sif>
s a tis f ie d , the asUuxis given abovs can b# used in the ease of v aria b le o oeffielsm ts# a % 5 ■ ■x’- _ r J l ■'--m t: ■■ C“- 1 ■ ■■ ’- • J j - ' J " : ' ; : %
63".
B a w i i ^ 4?: j:ia a .U i X w ?M tori»#d fon».
Conalder the ap p llcm tim o f th& fa c to rise d fa r» of the (2 ,2 ) member o f the fam ily o f aehemee to a num erieel p r ^ i e .• Thie form ula l e not v a lid a t nodes ehloh are influenced by val ics m the boundary planes x • 0 , y • 0 ( th a t i s a t the p o in ts O in the dW ram
F igure 2*6*1) •
t
0 o 0 o t 0 O O Q i 1>
0
X
7h# t o ta llt ,/ o f »qu#tion# obtained by uaing tio un», l i t form o ro r l o ln U on the g rid o f the qu.dM nt x ,y » 0 of the x ,y plane oan bo e r l t t a n in the fo ra
[ I » g(le ♦ • @(la * pA)H*jg"**
" [ I - A la - [1 • A la - pA)H*^ü* ♦ t( la ♦
(
2.
6.
1)
♦ A la ♦ pB)k5**. 1 ( le * jB )(Ia ♦ pA)k5**. K la - pA)kf
"-m
56.
where the a e tr ix epereto: e Kx, ere the q x q b i .ek m trie e e
Hx
r
—
I«qit
—la q Za q . m • " la q • . U being the q x q b lo ^ t a e tr ix “ le •In In . •In •end I the u n it m atrix o f o rd er nq*. The q u a n titie s k f , k£ , k , are v ecto rs involving bound ary v alu es piok&d up by 9 , , 9y, reppm ctively mt s ta tio n m in tin e# U* re p re se n ts th - to ta lit y o f v e c to r fu n ctio n s
each of ord er n e t the q* p o in ts on the plane t « nk soennad rcwwiso# f
The t o t a l i t y o f eq u atio n s obtained by using the fo o t jrise d fo re o f (2 ,2 ) and soannln^ p o in ts by rows oan be w ritte n in the foru
[I - i ( U ♦
. [I - «(!« - p,‘.)iu,u" ♦ c
(
2.
6.
2)
[ I - i( ln ♦ pA)tttjU*‘** « [ I - A la -
where C and D depend on the boundaxy valu es of u on the planes X # 0 , y m 0# r^ liain atio n o f gives
(2 .6 ,3 ) ( I • i( la ♦ î» )lh r]ri - H ie ♦ pA)Hx
• [I • j(lm - pB)%r][I- i
(la- pA)Hjc,U* e [I - ^(in • pB)lÿ]C
♦ [ I • *(la ♦ j^)lfy]D Comparison o f (2#4#1) and (2#6#3) shows th a t i f
j-y .
(2.6.4)
C • •,ldî***eeàS"* .4#liS D - b,k#**.W dl*' *b,kg***b»k% bm ^ *b.b# ■ If »'4 @f# ■a
then b$ m •0(T# 4 pB) be » •Q(lm ♦ pA) , “f."b i « Q (ill - pB) b# # Q(Xfi • pA)
b> « vC(lii ♦ pB)(Xa ♦ pA) b# • k ( l # • pB)(l# • pA) whexw
•ad
Q . (4pB)“ * (l ♦ j* ) . ( i • 1 ,2 ,...6 )
\ • - ( I - P * )'* ( l ♦ pB)b^ (1 . 1 ,2 ,...6 )
S im ilarly i f we ch#w## to ua# the D,R* type fbwt r ia a tio n then
(
2,
2)
beeonea( I - K in ♦ pB)Hy V***«[I - s(I*-»A)H*:tJ*Aa|!B * * (la ♦ p8)K [I* ♦ ^ ) [ I - A in ♦ 2i#V **M la - pB)[I - « (!• -
Wiere K rep reaen ta the l a s t six term# on the r ig h t h:st: sH e o f (2#6#l)« The eeheme (2 ,2 ) was oliomen to de^onatrate th is mo ifie a tio m
beoauee in a follow ing aeo tio n i t i s applied to a ainpl^ p roblem and oofitpmred w ith the Lex wendroff method*
:6s.
2 .7 An AAtgraaUv# ; fOtoxliftUPD.I n S M tion (2 * l) the P.! « and t#2« type f a e to ris a tla n a vere
Intraiuoed# Both ere desi^pned to sim plify the so lu tio n o f #n im p lic it sohmem o f the typo
(2 .7 .1 ) HU"** ■ 9 f
where H end 0 may eeeh be w ritie n as tli# product H . II#He, G . GfGa
o f two sim pler eprre to rs* The F#R. method fo r th* s d u ti c n o f (2*7*1) then tak es the form
II» v"** ■ 0*b" %u"** B c»v"**
whereas the D*l# method takes the form H»V"** m 0»U"
#%U"** B#V"** * yC#U" Where
fi ♦ yiîf • eHb*
Both o f these methods re q u ire m o dification a t bouad/ijry - lan es (see e e e tio n 2*6) * In many eases th is o d ifio a tio n may bo v»ry complex o r may n o t even e x is t (a s in the case fo r the " la te r a l" imd "diagonal" schemes developed in se c tio n 2*3)• However a th ird t pe o f
f a c to ris a tio n e x is ts Wiloli re q u ire s straig h tfo rw ard a o iii ic a tio n and involves very l i t t l e e x tra work#
For (2*7*1) th is 1?*JA/'D!>0V [4] typa ftio to ria a tio n (D-type) becomes
H,VWO$ m CrOW
.''•■■■' J 7.
l a the n o ta tio n o f eootion 2#7, the D-type e p l l t i l n j of the (2 ,2 ) member of the fam ily o f aciioaea tak es the form
[I -