Метод конечных разностей или метод сеток

AAAAeAIEA

Cia/eoaeueiaia /enei caaea/ oeceee e oaoieee i?eaiaeyo e
aeeooa?aioeeaeueiui o?aaiaieyi a /anoiuo i?icaiaeiuo (o?aaiaiey
iaoaiaoe/aneie oeceee). Onoaiiaeaoeany i?ioeannu ?acee/iie oece/aneie
i?e?iaeu iienuaathony o?aaiaieyie yeeeioe/aneiai oeia.

Oi/iua ?aoaiey e?aaauo caaea/ aeey yeeeioe/aneeo o?aaiaiee oaea?ony
iieo/eoue eeoue a /anoiuo neo/ayo. Iiyoiio yoe caaea/e ?aoatho a
iniiaiii i?eaeeae?iii. Iaeiei ec iaeaieaa oieaa?naeueiuo e yooaeoeaiuo
iaoiaeia, iieo/eaoeo a ianoiyuaa a?aiy oe?ieia ?ani?ino?aiaiea aeey
i?eaeeae?iiiai ?aoaiey o?aaiaiee iaoaiaoe/aneie oeceee, yaeyaony iaoiae
eiia/iuo ?aciinoae eee iaoiae naoie.

Nooue iaoiaea ninoieo a neaaeothuai. Iaeanoue iai?a?uaiiai eciaiaiey
a?aoiaioia, caiaiyaony aeene?aoiui iiiaeanoaii oi/ae (oceia), eioi?ia
iacuaaaony naoeie eee ?ao?oeie. Aianoi ooieoeee iai?a?uaiiai a?aoiaioa
?anniao?eaathony ooieoeee aeene?aoiiai a?aoiaioa, ii?aaeae?iiua a oceao
naoee e iacuaaaiua naoi/iuie ooieoeeyie. I?iecaiaeiua, aoiaeyuea a
aeeooa?aioeeaeueiia o?aaiaiea e a?aie/iua oneiaey, caiaiythony
?aciinoiuie i?iecaiaeiuie, i?e yoii e?aaaay caaea/a aeey
aeeooa?aioeeaeueiiai o?aaiaiey caiaiyaony nenoaiie eeiaeiuo eee
iaeeiaeiuo aeaaa?ae/aneeo o?aaiaiee (naoi/iuo eee ?aciinoiuo o?aaiaiee).
Oaeea nenoaiu /anoi iacuaatho ?aciinoiuie noaiaie. E yoe noaiu ?aoathony
ioiineoaeueii iaecaanoiie naoi/iie ooieoeee.

Aeaeaa iu aoaeai ?anniao?eaaoue i?eiaiaiea eoa?aoeeiiiiai iaoiaea
Caeaeaey aeey au/eneaiey iaecaanoiie naoi/iie ooieoeee a e?aaaie caaea/a
n iaiaeii?iaeiui aeaa?iiie/aneei o?aaiaieai.

IINOAIIAEA CAAeA*E

Ionoue o ian anoue aeaa?iiie/aneia o?aaiaiea :

U = f

Caaeaiiia ia iaeanoe G={ (x,y) : 00) caienuaaaony a
neaaeothuai aeaea :

i (k+1) M (k)

aijYj + aijYj = fi , i=1,2…M

j=1 j=i+1

(k)

aaea Yj – jay eiiiiiaioa eoa?aoeeiiiiai i?eaeeaeaiey iiia?a k. A
ea/anoaa ia/aeueiiai i?eaeeaeaiey auae?aaony i?iecaieueiue aaeoi?.

Ii?aaeaeaiea (k+1)-ie eoa?aoeee ia/eiaaony n i=1

(k+1) M (k)

a11Y1 = – a1jYj +f1

j=2

(k+1)

Oae eae a110 oi ionthaea iaeae?i Y1. E aeey i=2 iieo/ei :

(k+1) (k+1) M (k)

a22Y2 = – a21Y1 – a2jYj + f2

j=3

(k+1) (k+1) (k+1)
(k+1)

Ionoue oaea iaeaeaiu Y1 , Y2 … Yi-1 . Oiaaea Yi
iaoiaeeony ec o?aaiaiey :

(k+1) i-1 (k+1) M
(k)

aiiYi = – aijYj – aijYj + fi
(*)

j=1 j=i+1

Ec oi?ioeu (*) aeaeii , /oi aeai?eoi iaoiaea Caeaeaey /a?acau/aeii
i?ino. Iaeaeaiiia ii oi?ioea (*) cia/aiea Yi ?aciauaaony ia ianoa Yi.

Ioeaiei /enei a?eoiaoe/aneeo aeaenoaee, eioi?ia o?aaoaony aeey
?aaeecaoeee iaeiiai eoa?aoeeiiiiai oaaa. Anee ana aij ia ?aaiu ioeth, oi
au/eneaiey ii oi?ioea (*) o?aaotho M-1 iia?aoeee oiiiaeaiey e
iaeiiai aeaeaiey. Iiyoiio ?aaeecaoeey

iaeiiai oaaa inouanoaeyaony ca 2M – M a?eoiaoe/aneeo aeaenoaee.

Anee ioee/ii io ioey eeoue m yeaiaioia, a eiaiii yoa neooaoeey eiaao
ianoi aeey naoi/iuo yeeeioe/aneeo o?aaiaiee, oi ia ?aaeecaoeeth
eoa?aoeeiiiiai oaaa iio?aaoaony 2Mm-M aeaenoaee o.a. /enei aeaenoaee
i?iii?oeeiiaeueii /eneo iaecaanoiuo M.

Caieoai oaia?ue iaoiae Caeaeaey a iao?e/iie oi?ia. Aeey yoiai
i?aaenoaaei iao?eoeo A a aeaea noiiu aeeaaiiaeueiie, ieaeiae
o?aoaieueiie e aa?oiae o?aoaieueiie iao?eoe :

A = D + L + U

aaea

0 0 . . . 0
0 a12 a13 . . . a1M

a21 0
0 0 a23 . . . a2M

a31 a32 0
0 .

L = .
U= .

.
.

.
aM-1M

aM1 aM2 . . . aMM-1 0
0 0

E iao?eoea D – aeeaaiiaeueiay.

(k) (k) (k)

Iaicia/ei /a?ac Yk = ( Y1 ,Y2 … YM ) aaeoi? k-iai eoa?aoeeiiiiai
oaaa. Iieuecoynue yoeie iaicia/aieyie caieoai iaoiae Caeaeaey eia/a :

( D + L )Yk+1 + UYk = f , k=0,1…

I?eaaae?i yoo eoa?aoeeiiioth noaio e eaiiie/aneiio aeaeo aeaooneieiuo
noai :

( D + L )(Yk+1 – Yk) +AYk = f , k=0,1…

Iu ?anniio?aee oae iacuaaaiue oi/a/iue eee neaey?iue iaoiae Caeaeaey,
aiieiae/ii no?ieony aei/iue eee aaeoi?iue iaoiae Caeaeaey aeey neo/ay
eiaaea aii – anoue eaaae?aoiua iao?eoeu, aiiaua aiai?y, ?acee/iie
?acia?iinoe, a aij aeey ij – i?yiioaieueiua iao?eoeu. A yoii neo/aa Yi
e fi anoue aaeoi?u, ?acia?iinoue eioi?uo niioaaonoaoao ?acia?iinoe
iao?eoeu aii.

IINO?IAIEA ?ACIINOIUO NOAI

Ionoue Yi=Y(i) naoi/iay ooieoeey aeene?aoiiai a?aoiaioa i. Cia/aiey
naoi/iie ooieoeee Y(i) a naith i/a?aaeue ia?acotho aeene?aoiia
iiiaeanoai. Ia yoii iiiaeanoaa iiaeii ii?aaeaeyoue naoi/ioth ooieoeeth,
i?e?aaieaay eioi?oth e ioeth iieo/aai o?aaiaiea ioiineoaeueii naoi/iie
ooieoeee Y(i) – naoi/iia o?aaiaiea. Niaoeeaeueiui neo/aai naoi/iiai
o?aaiaiey yaeyaony ?aciinoiia o?aaiaiea.

Naoi/iia o?aaiaiea iieo/aaony i?e aii?ieneiaoeee ia naoea eioaa?aeueiuo
e aeeooa?aioeeaeueiuo o?aaiaiee.

Oae aeeooa?aioeeaeueiia o?aaiaiea ia?aiai ii?yaeea :

dU = f(x) , x > 0

iiaeii caiaieoue ?aciinoiui o?aaiaieai ia?aiai ii?yaeea :

Yi+1 – Yi = f(xi) , xi = ih, i=0,1…

eee Yi+1=Yi+hf(x), aaea h – oaa naoee v={xi=ih, i=0,1,2…}. Eneiiie
ooieoeeae yaeyaony naoi/iay ooieoeey Yi=Y(i).

I?e ?aciinoiie aii?ieneiaoeee o?aaiaiey aoi?iai ii?yaea

d U = f(x)

iieo/ei ?aciinoiia o?aaiaiea aoi?iai ii?yaeea :

Yi+1 – 2Yi + Yi+1 = yi , aaea yi=h f i

fi = f(xi)

xi = ih

Aeey ?aciinoiie aii?ieneiaoeee i?iecaiaeiuo U’, U’’, U’’’ iiaeii
iieueciaaoueny oaaeiiaie n aieueoei /eneii oceia. Yoi i?eaiaeeo e
?aciinoiui o?aaiaieyi aieaa aunieiai ii?yaeea.

Aiieiae/ii ii?aaeaeyaony ?aciinoiia o?aaiaiea ioiineoaeueii naoi/iie
ooieoeee Uij = U(i,j) aeaoo aeene?aoiuo a?aoiaioia . Iai?eia?
iyoeoi/a/iay ?aciinoiay noaia “e?ano” aeey o?aaiaiey Ioanniia

Uxx + Uyy = f(x,y)

ia naoea W auaeyaeeo neaaeothuei ia?acii :

Ui-1j – 2Uij+Ui+1j + Uij-1 – 2Uij+Uij+1 = fij

2 2

hx hy

aaea hx – oaa naoee ii X

hy – oaa naoee ii Y

Naoi/iia o?aaiaiea iauaai aeaea iiaeii caienaoue oae:

CijUj = fi i=0,1…N

j=0

Iii niaea?aeeo ana cia/aiey U0, U1 … UN naoi/iie ooieoeee. Aai iiaeii
o?aeoiaaoue eae ?ciinoiia o?aaiaiea ii?yaeea N ?aaiiai /eneo oceia naoee
ieion aaeeieoea.

A iauai neo/aa iiae i – iiaeii iiieiaoue ia oieueei eiaeaen , ii e
ioeueoeeiaeaen o.a. aaeoi? i = (i1 … ip) n oeaei/eneaiiuie
eiiiiiaioaie e oiaaea :

NijUj =fi i I W

jIW

aaea noie?iaaiea i?ienoiaeeo ii anai oceai naoee W. Anee eiyooeoeeaiou
Nij ia caaenyo io i, oio?aaiaiea iacuaatho o?aaiaieai n iinoiyiiuie
eiyooeoeeaioaie.

Aii?ieneie?oai iaoo caaea/o o.a. caiaiei o?aaiaiea e e?aaaua oneiaey ia
niioaaonoaothuea ei naoi/iua o?aaiaiey.

U=U(x,y)

M b

M-1

Uij j

0 1 2 i
N-1 N=a x

Iino?iei ia iaeanoe G naoeo W . E caaeaaeei ia W naoi/ioth ooieoeeth
Uij=U(xi,yj) ,

aaea

xi=x0+ihx

yi=y0+jhy

hx = a/N ,

hy = b/M e o.e.

x0=y0

xi=ihx, yi=jhy, i=0…N

j=0…M

Iaeae?i ?aciinoiua i?iecaiaeiua aoiaeyuea a o?aaiaiea

DU = f

(o.a iino?iei ?aciinoiue aiaeia aeaa?iiie/aneiai o?aaiaiey).

Uxij = Ui+1j – Uij , Uxi-1j = Uij – Ui-1j

hx hx

Uxxij = Ui-1j – 2Uij + Ui+1j

?anniio?ei Uxxxxij eae ?aciinoue o?aoueeo i?iecaiaeiuo :

Uxxi-1j – Uxxij – Uxxij – Uxxi+1j

Uxxxxij = hx hx = Ui-2j –
4Ui-1j + 6Uij – 4Ui+1j + Ui+2j

hx
hx

Aiieiae/ii au/eneei i?iecaiaeioth ii y :

Uyyyyij = Uij-2 – 4Uij-1 + 6Uij – 4Uij+1 +Uij+2

Au/eneei niaoaiioth ?aciinoioth i?iecaiaeioth Uxxyy :

Uxxij-1 – Uxxij – Uxxij – Uxxij+1

(Uxx)yyij = hy hy =
Uxxij-1 – 2Uxxij +Uxxij+1 =

hy
hy

= Ui-1j-1 – 2Uij-1 + Ui+1j-1 – 2 Ui-1j – 2Uij + Ui+1j +
Ui-1j-1 – 2Uij+1 + Ui+1j+1

2 2 2
2 2 2

hxhy hxhy
hxhy

A neeo oiai /oi DU = f

eiaai:

Ui-2j – 4Ui-1j + 6Uij – 4Ui+1j +Ui+2j +

+ 2 Ui-1j-1 – 2Uij-1 + Ui+1j-1 – 4 Ui-1j – 2Uij +Ui+1j + 2 Ui-1j+1
-2Uij+1 + Ui+1j+1 +

2 2 2 2
2 2

hxhy hxhy
hxhy

+ Uij-2 – 4Uij-1 + 6Uij – 4Uij+1 + Uij+2 = fij
(*)

Yoi o?aaiaiea eiaao ianoi aeey

i=1,2, … N-1

j=1,2, … M-1

?anniio?ei e?aaaua oneiaey caaea/e. I/aaeaeii neaaeothuaa :

x=0 ~ i = 0

x=a ~ xN=a

y=0 ~ Yo=0

y=b ~ YM=b

1) o=0 (eaaay a?aieoea iaeanoe G)

Caiaiei oneiaey

U = 0

x=o

Uxxx = 0

x=o

ia niioaaonoaothuea ei ?aciinoiua oneiaey

Uoj=0

U-1j=U2j – 3U1j (1`)

2) o=a (i?aaay a?aieoea iaeanoe G)

i=N

Ux = 0

x=a

Uxxx = 0

x=a ec oiai /oi Ui+1j –
Ui-1j = 0

2hx

UN+1j = UN-1j

UNj = 4 UN-1j – UN-2j (2`)

3) o=0 (ieaeiyy a?aieoea iaeanoe G)

j=0

Ui ,-1 = Ui1

Ui0 = 0 (3`)

yoi anoue ?aciinoiue aiaeia Uy = 0

y=o

U =0

y=o

4) o=b

i=M

U = 0

y=b o.a. UiM=0
(**)

?anieoai /a?ac ?aciinoiua i?iecaiaeiua Uxx + Uyy =0 e o/eouaay /oi j=M
e (**) iieo/ei

UiM-1 = UiM+1

Eoae e?aaaua oneiaey ia o=b eiatho aeae

UiM+1 = UiM-1

UiM = 0 (4`)

Eoiai iaoa caaea/a a ?aciinoiuo i?iecaiaeiuo ninoieo ec o?aaiaiey (*)
caaeaiiiai ia naoea W e e?aaauo oneiaee (1`)-(4`) caaeaiiuo ia a?aieoea
iaeanoe G (eee ia a?aieoea naoee W)

I?EIAIAIEA IAOIAeA CAEAeAEss

?anniio?ei i?eiaiaiea iaoiaea Caeaeaey aeey iaoiaeaeaiey i?eaeeaeaiiiai
?aoaiey iaoae ?aciinoiie caaea/e (*),(1`) – (4`).

A aeaiiii neo/aa iaecaanoiuie yaeythony

Uij = U(xi,yj)

aaea xi = ihx

yj = jhy

i?e /?i hx = a/N ,

hy = b/M

yoi anoue oaa naoee ii x e ii o niioaaonoaaiii , a N e I niioaaonoaaiii
eiee/anoai oi/ae ?acaeaiey io?aceia [0 , a] e [0 , b]

Iieuecoynue ?acoeueoaoaie i?aaeuaeouaai ?acaeaea caieoai o?aaiaiea

DU = f

eae ?aciinoiia o?aaiaiea. E oii?yaei/ei iaecaanoiua anoanoaaiiui ia?acii
ii no?ieai naoee W , ia/eiay n ieaeiae no?iee.

1 Ui-2j – 4 + 4 Ui-1j + 6 – 8 + 6 Uij – 4 + 4 Ui+1j +
1 Ui+2j + 2Ui-1j-1 –

4 4 2 2 4 2 2 4
4 2 2 4
2 2

hx hx hxhy hx hxhy hy hx hxhy
hx hxhy

– 4 + 4 Uij-1 + 2 Ui+1j-1 + 2 Ui-1j+1 – 4 + 4 Uij+1 + 2
Ui+1j+1 + 1 Uij-2 +

2 2 4 2 2 2 2
2 2 4 2 2
4

hxhy hy hxhy hxhy hxhy hy
hxhy hy

+ 1 Uij+2 = f ij aeey i=1 … N-1, j=1 … M-1

e U oaeiaeaoai?yao e?aaaui oneiaeyi (1`) – (4`), oae eae a eaaeaeii
o?aaiaiee naycaiu aianoa ia aieaa 13 iaecaanoiuo oi a iao?eoea A ioee/iu
io ioey ia aieaa 13-yeaiaioia a no?iea. A niioaaonoaee ni aoi?ui
?acaeaeii ia?aieoai o?aaiaiea:

(k+1) (k+1)
(k+1) (k+1)

6 – 8 + 6 Uij = – 1 Uij-2 – 2 Ui-1j-1
+ 4 + 4 Uij-1 –

4 2 2 4
4 2 2 2
2 4

hx hxhy hy hy hxhy
hxhy hy

(k+1) (k+1)
(k+1)
(k)

4 + 4 Ui-1j + 4 + 4 Ui+1j –

2 2 4 4 2 2
4 2 2

hxhy hx hx hxhy
hx hxhy

(k) (k)
(k) (k)
(k)

– 1 Ui+2j – 2 Ui-1j+1 + 4 + 4 Uij+1 – 2
Ui+1j+1 – 1 Uij+2 + fij

4 2 2
2 2 4 2 2
4

hx hxhy hxhy hy
hxhy hy

(k)

I?e /ai U oaeiaeaoai?yao e?aaaui oneiaeyi (1`) – (4`). Au/eneaiey
ia/eiathony n i=1, j=1 e i?iaeieaeathony eeai ii no?ieai eeai ii
noieaoeai naoee W. *enei iaecaanoiuo a caaea/a n = (N-1)(M-1).

Eae aeaeii ec auoaeceiaeaiiuo ?annoaeaeaiee oaaeii a yoie caaea/a
o?eiaaeoeaoeoi/a/iue o.a. ia eaaeaeii oaaa a ?aciinoiii o?aaiaiee
o/anoaotho 13 oi/ae (oceia naoee) ?anniio?ei aeae iao?eoeu A – aeey
aeaiiie caaea/e.

j+2

j+1

j-1

Iao?eoea iaoiaea iieo/aaony neaaeothuei ia?acii : ana oceu naoee
ia?aioia?iauaathony e ?aciauathony a iao?eoea Oae /oi ana oceu
iiiaaeatho ia iaeio no?ieo e iiyoiio iao?eoea iaoiaea aeey iaoae
caaea/e aoaeao o?eiaaeoeaoeaeeaaiiaeueiie .

j-2

i-1

i+1

i+2

i-2

Oaaeii caaea/e

IIENAIEA I?IA?AIIU.

Eiinoaiou eniieuecoaiua a i?ia?aiia :

aq = 1 – i?aaay a?aieoea iaeanoe G

b = 1 – eaaay a?aieoea iaeanoe G

N = 8 – eieee/anoai oi/ae ?acaeaiey io?acea [0,a]

M = 8 – eieee/anoai oi/ae ?acaeaiey io?acea [0,b]

h1 = aq/N – oaa naoee ii X

h2 = b/M – oaa naoee ii Y

Ia?aiaiiua :

u0 – cia/aiey naoi/iie ooieoeee U ia k-ii oaaa

u1 – cia/aiey naoi/iie ooieoeee U ia (k+1)-ii oaaa

a – iannea eiyooeoeeaioia oaaeiia

Iienaiea i?ioeaaeo? :

procedure Prt(u:masa) – ia/aoue ?acoeueoaoa

function ff(x1,x2: real):real – aica?auaao cia/aiea ooieoeee f a ocea
(x1,x2)

procedure Koef – caaea?o cia/aiey eiyooeoeeaioia

Aeaenoaea :

Aa??ony ia/aeueia i?eaeeaeaiea u0 e n o/?oii e?aaauo oneiaee aaae?ony
au/eneaiea n i=2 … N , j=2 … M. Ia eaaeaeii eoa?aoeeiiiii oaaa
iieo/aai u1 ii u0. Ii aeinoeaeaiee caaeaiiie oi/iinoe eps>0 au/eneaiey
i?ae?auathony. E ana yeaiaiou iao?eoeu A, eioi?ua eaaeao ieaea aeaaiie
aeeaaiiaee iieo/atho eoa?aoeeiiiue oaa (k+1) , a oa yeaiaiou eioi?ua
eaaeao auoa aeaaiie aeeaaiiaee (eneeth/ay aeaaioth aeeaaiiaeue)
iieo/atho eoa?aoeeiiiue oaa k.

I?eia/aiea : i?ia?aiia ?aaeeciaaia ia ycuea Borland Pascal 7.0

Ieienoa?noai iauaai e i?ioanneiiaeueiiai ia?aciaaiey ?O

Ai?iiaaeneee ainoaea?noaaiiue oieaa?neoao

oaeoeueoao III

eaoaae?a Aeeooa?aioeeaeueiuo o?aaiaiee

Eo?niaie i?iaeo

“?aoaiea aeaa?iiie/aneiai o?aaiaiey iaoiaeii Caeaeaey”

Eniieieoaeue : nooaeaio 4 eo?na 5 a?oiiu

Ieeoeei E.A.

?oeiaiaeeoaeue : noa?oee i?aiiaeaaaoaeue

?uaeeia A.A.

Ai?iiaae 1997a.