At th is Juno lu re i t i# worth g atb erlo g to ,^ t 3r t>io mohea## derived in th is clm pter vhioh arc lik e ly to be of in te r n a t, and of p ra e tie a i value in eolvirzg the three types o f problem mentioned in se c tio n 4*1#
The moat sl^piifL cant e x p lic it seheces are and (4 # 4 .l) Wiich
are aocurate to ord*gr h , and (4#3*2), (4«4#3) #md (4*4#8) which are accu rate to ord er bf # Although i t i s gem ivdly accepted th a t the most im o rie n t e x p lic it cehems i s the Lax V endroff method (4*4*3), i t seems
th a t formulae such as (4*3*2) and (4 .4 .8 ) might w ell be used a t e x tre m itie s o f regions #6ere ths cen tred form ula (4*4*3) i s n o t a p p lic a b le .
Im p lic it g chôme# of note are (4*3*3)# (4*3*8), scourate to o rd er h , and (4*3d1), (4*3*12), (4*4*5), (4*4*10), (4*4*7) accurate to o rd er h*« hixccpt fo r schemes (4*3*3) and (4*3.11) (Wendroff*s method) a l l arc fa c tc ria a b le schemes* The accurt.oy o f the fa c to ris e d e o lc ic s (4*3*8), (4*4*3), (4*4*7) i s o rd er h a t the i n i t i a l s te p , whs 4&s schemes (4*3*12), (4*4*10) arc in c o n siste rit w ith the d if f e r e n tia l system (4*1*1) over the
f i r s t step* Care must be taken
in
incorporateVie
conditions
o f s problem in such a way th a t the o v e ra ll accuracy o f fa c to ris e d schem ehas no aeeuraciy a t th e f i r s t step* This p o in t i s f u lly
explained w ith regard to souere (4*3*12) in chapter I I , and w ith regard to scheme (4*4*7) in clwqitcr III*
The o rig in a l a lte r n a tin g d ire c tio n êùhmms discussed in a general paper by DOUGlv^d and oami fs ] d id not lo se accuracy a t the interm ediate s te p , and so no boundary m odification was req u ired w ith Vie se seliemes.
'
ro'i,
Let ue now ooaeider wiiioh eoheaee ere most e il- e d f o r eeeh problem#
For the C&uohy problem, only e x p lic it aol^emee s :^ ap p licab le
einoe no boundary d a ta ie given# I t would a ^ ie a r th a t the Lax Vendroff method (4*4*3) la th e rw ct agitab le#
For th e in itia lH w m d a ry value problem the aclicmee most eui'^ed are theee baaed only cm forw ard d l;fe :c n c e e . In th is re sp e c t the
a l t e r n tin g d ire c tio n method (4.3*12) would appe »r to be the beat# For the mixed in i im l-boundar) value probl33, n e a rly al!t the soaemes are ^ p .llc a b le , w ith the ccheae (4 .4 .7 ) coTabinlng maxlmuc s t a b i l i t j t w ith second o rd er accuracy#
/ o ÿ .
^6 YarifcM. CoeX'iiclant»
I f the matriooft  and B depend on the space v& riahles x end y , we s^muld re tu rn to (4*1 *3} end use i t as a s ta r tin g p o in t fo r d eriv in g the approxim atia : d iffo ie n e e soheaes. I t i s sim p le r, however, to take th e co n stan t o o e ffie ic n t d iffe re n o e schemes obtained f r o s (4#2*l) and (4#2#2) and attem pt to modify them fo r the case o f v a ria b le c o e ffic ie n ts* In f a c t a l l the sc: er.ee accu rate to f i r s t o rd er m aintain th e ir accuracy#
Of the ache zes accurate to second o rd er, fa e to ris e d schemes (4#4#3) and (4#4#7) m aintain aeeur&sgr in the sane form, w heieas (4*3#2), the Lax V/endroff sciiene (4*4,3) # and Vendroff* s scheme (4,3*11) r e ta in th e ir accuracy only i f they a re w ritte n in a modified form* This c o n s is ts o f rep la cin g terms lik e >y AAxBAy and so on, o r by #va^%mting the sm trices a t some eV ier x ,y node# (T his m odification i s explained in cliapter I I fo r the c ig ^ t p o in t a lte rn a tin g d ire c tio n % Ihod) #
4*7 8W,jyAfaf
'I
We w ill now e m elder the e t e b i l l t y of the so! ewes mentioned in eeo tio n 4#5# I t ie eeauned th a t the seheaee have constant e o e z fio le n ts and th a t the boumtazy o o n d itio n s, i f any, are periodic# I f these
o onditions are s a tis f ie d then the F ourier tra n s fo re a tio of the spaoe v a ria b le s ean be made in th e u su al way and the r e s u lt
VbH « G (/?,y>e
obtained eheie G i s th e a m p lific a tio n m atrix of the p a r tic u la r scheme inder co n sidéra tie n and e re a rb itra ry re e l numbers, Again the co n d itio n fo r s t a b i l i t y , the ta x Richtmycr c o n d itio a r# q u ire s