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_ `abcdefgbhdaijklmnop qrstkjq urj vljmnls w nxkjkomnls yzlmnroq |vwyq nq rumko lqqr ~nlmk nm ~rsn~lmk lo kjjrjjrok ~rnop no srsktks sloplpkq ln ku~nko~ nq qlss mk pnnop jsk ko mk kotnjro kom nq qksk~mk lo urjjr~mnro ~rk mlm qr s kjurj sljpkq~lsk qnslmnroq urj sropkj kjnrq m nq nq lo k~ksskom lo ktko ok~kqqlj pnksnok rktkj urj ksrjlmrj rjno q~nkomn~ ~rmnop nm l k rjmnsk mr ~roqnkj rmkj rmnroq

jlnmnrolss mk srsktks sloplpkq i i lo rjmjlo ltk kko mkrqm rslj lrop ~rmlmnrols q~nkomnqmq rktkj no jk~kom kljq kotnjro komq q~ lq lmsl lsk ~mltk v sq lo ltk k~r k rslj oljmn~slj lmsl lq kko mk jkukjjk kotnjro kom urj lo ~rmlmnrolsq~nkomnqmq lo kopnokkjq r k ru mk jklqroq urj mk q~~kqq ru mkqk kotnjro komq ljk mkqk qnsk lo ~sklo qoml nomkjl~mntk kk~mnro nm nknlmkukkl~ nomkpjlmnro ru tnqlsnlmnro lo prr r~ komlmnro lo rosnok ks o lnmnro l jn~ qmlolj snjlj ru o kjn~ls uo~mnrolsnm nq ltlnslsk no mkkotnjro komq mrpkmkj nm srmq ru nsmno uo~mnroq mlm rkjlmk k ~nkomsro ljjlq no ~rnsk ~rk smrp mk kjurj lo~k ujr srsktks ~rnsksloplpkq nq kmmkj lo q~nkomnqmq qns ukks rjk jr~mntk no urj noqmlo~k lmsl

k npsktks jrpjl nop sloplpk vmro lq k kjpk lq l tnlsk lsmkjolmntk mr lmsl ~mltk lo rmkj qn nslj kotnjro komq ko kmkok nmokjn~ls lo tnqlsnlmnro rskq vmro qljkq lo ru mk ltlomlpkq nm mk rmkj nomkjjkmk kotnjro komq komnrok lrtk jmkj vmro lq

lo lnmnrols ltlomlpk no mlm mk sloplpk nq tkj jn~ lo rkjus ltnopkko kqnpok ujr mk kpnoonop lq lo r k~mrjnkomk sloplpk kk urj rjk ~rskmk ~rtkjlpk ru mk ltlomlpkq ru vmro no q~nkomn~ ~rmnop

qnop vmro k ltk nsk komk mk ujl krj vy |vmro mko~nsyotnjrokom urj qrstnop vwyq qnop mk no nmk w nxkjko~k kmr |w ~jklmnop l qkm ru npsktks lqmjl~mnroq vy lssr q mk qkj mr rm q k~numk jr sk lo nsk kom okjn~ls qr smnroq ro l np sktks o rjkj mrrtk mk ur ~q ru mk qkj ll ujr mk kmlnsq ru mk nsk komlmnro mrljqmk jr sk q lo mk qrsmnro kmrq nm lq kko l prls urj rj kqnpo mlm~rk qrs k lq ~srqk mr qkr ~rk lq rqqnsk j mljpkm lnko~k ljkjkqklj~kjq r okk mr jrmrmk w qrstkjq lo lom mr sl nm mkw rj mk okjn~ls kmrq qk mr qrstk mk jkq smnop nq~jkmk jr sk

wk mr nmq kqnpo o kjn~ls ~rmlmnroq ~ljjnk rm no vmro nss qlss k ~ qsrkj mlo mk ql k ~rmlmnroq nsk komk no srsktks sloplpkqq~ lq i lo rjmjlo q nq~qqk no k p l sljpk ljm ru mk ~rkurj vwy qrstkjq klsq nm jrsk qk~n~lmnro lo tljnrq rrkknop mlqq rj mkqk ljmq ru mk ~rk k ~nko~ nq orm nrjmlom lo l npsktks sloplpkq~ lq vmro nq tkj kssqnmk rktkj mkjk nq l ljm ru mk ~rk kjkk ~nko~ nq nrjmlom mk ljm ru mk qrstkj kjk mk okjn~ls ~rmlmnroqljk ~ljjnk rm kmmkj kjurj lo~k ~lo k l~nktk rtnop mk mn k~jnmn~lsljm ru mk ~rk nomr kmkoqnro rskq urj vmro nsk komk no l ~rnsksloplpk k kmkoqnro rskq ~lo k knmkj pkokjls jrqk snk kjn~lsvmro |v rj ~qmr lk urj l ljmn~slj lsn~lmnro rj snjljuo~mnro o lnmnro ljlssks ~rmnop ~lo k qk mr qrstk l jr sk ulqmkj qjklnop mk rjsrl rtkj rjk mlo rok ~rmkj rm ljrl~kq ljkqk no vy

klss mk ujlkrj nmqksu qrs mlk ~ljk ru ljlssksnlmnro no q~ l lmlm mk qkj ~rk rkq orm okk mr ~lopk lm lss ko rtnop ujr l qkzkomnlsmr l ljlssks ~rmkj u ljlssksnlmnro nq kjurj k lolss kmlnsq ru rmmk jrsk lo mk kotnjro kom no n~ mk lsn~lmnro nss k jo ~lo kmlko nomr l~~rom no rjkj mr ~jklmk lo rmn ls ljlssks qr stkj urj mk jr sk u mk ljlssksnlmnro nq losk lmr lmn~lss mk ujl krj qrk knnsnm|lo kjlq kjurj lo~k qm k ql~jn~k k ksnktk rktkj mlm mk kokmq ru lmr lmn~ ljlssksnlmnro ljk rjk nrjmlom urj mk kkjn komlsjkqklj~kj ko~k k nsk kom ljlssksnlmnro no vy no q~ l l mlmmk qkj ~rk nq skum o~smmkjk

k kpno nm l nq~qqnro ro lm no ru qkj nomkjul~k l ujlkrjurj qrstnop vwyq nm w qrs ltk ko k jrtnk l uk klskqru vy no qk ko k nq~qq mk nskkomlmnro no rjk kmlns mr qkkr mk prlsq ljk l~nktk kurjk jkqkomnop qr k rjk notrstk ~lqk qmnkq nm vy nolss k rxkj ~ro~snop jk ljq lo qppkqmnroq urj mk ujmkjktksrkom ru vy

ha d

k lom mr kqnpo mk qkj nomkjul~k ru rj ujl krj urj qrstnop vwyq q~mlm nm nq rqqnsk mr rj nm w ro l np sktks nmrm ql~jn~nop knnsnm lm nq nm qrs k rqqnsk mr kq~jnk l jr sk no lq ~ kmlns lq nqok~kqqlj lo mk q~nkomnqm qrs ltk uss ~romjrs rtkj nq~jkmnlmnro lo qrsmnro lsprjnm q ko rjnop nm lmk lmn~q mk rkjlmrjq lo qrsqru lmk lmn~q ltk l rkjus nsmno qk lomn~q rj klsk lo kjkqqnro q~ lq

r = b − Ax~lo jkqsm no znmk nxkjkom ~ls~slmnroq k konop ro

lm mk qrsq jkjkqkom kjkurjk nm nq ulnj mr ql mlm mnq nq l npsktks

∆

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npjk irk urj qrstnop jrsk |

7

1 1

1

1

−4

npjk mko~ns urj mk kzlmnro no nookj orkq |

jkjkqkomlmnro k qkk mk ql k kltnrj urj mk qkj nomkjul~k ru vy nomkjul~k nss k l qkm ru ~slqqkq lo uo~mnroq k qkj qrs k lsk

mr mnsnk mk ujl krj ujr rm vmro jrpjl q lo nomkjl~mntk vmroqkqqnroq

k ~roqnkj mlm mk nrjmlom lq k~mq ru l mnsnm rj ujlkrj qnmlskurj npsktks rj nm w qrstkjq urj vwyq ~lo k qljnk lq urssr q

•k jrsk ~lo k qk~nk no lo nom nmntk lo k nsk l

•k qrsmnro kmr ~lo k nsk komk ~skljs lo klqns

•kqsmq lo jomnk qmlmnqmn~q qrs k klqns l~~kqqn sk urj rqmjr ~kqqnop lo lolsqnq

r ljkqq mkqk lqk~mq lo nq~qq rqqn sk qrsmnroq urj rj ujlkrj k~roqnkj mk urssr nop jrsk n~ rksq qmlmnrolj klm ~ro~mnro lq lonssqmjlmntk mkqm ~lqk

−∇2u = f, x ∈ Ω

∂u

∂n= 0, x ∈ ∂Ω

|

kjk Ω = [0, 1] × [0, 1] mk onm qzljk no w lo

∂Ωnmq rolj lo

fnq l pntko uo~mnro ntko mk okjn l slmmn~k pjn ru orkq

(xi, yj)urj

mk r lnoΩ

nq ~jklmk kjk i = 0, 1, . . . , nj = 0, 1, . . . , n

k ~kss skopmh = xi − xi−1 = yi − yi−1 = 1/n

nq lqqk mr k ~roqmlom qnop mk qmlolj ~komjk nxkjko~kq urj mk qk~rorjkj kjntlmntk no |

nksq

ui,j−1 + ui−1,j − 4ui,j + ui+1.j + ui,j+1 = −h2fi,j , i, j = 1, 2, . . . , n − 1|

urj mk nookj orkq no mk rlno kjk ui,j = u(xi, yj)lo

fi,j = f(xi, yj)

srop mk roljnkq mk kloo rolj ~ronmnro nss k qk mrksn nolmk tlskq no mk q~k l | ulssnop rmqnk mk r lno rj noqmlo~k lsrop mk

ynrolj l qk~rorjkj ljrn lmnro ru mk kloo ~ronmnro

nksq

2ui,n−1 + ui−1,n − 4ui,n + ui+1,n = −h2fi,n, i = 1, 2, . . . , n − 1.|7

r pntk lo njkqqnro ru lm l qrstkj nsk komk nm mk vy ujl krj srrq snk k qr mk ~rk urj qrstnop jrsk | ko f = sin(2πx)

no npjk

k kzlmnroq | lo |7 jkslmkq l rnom mr l uk ru nmq oknprjq m nq~rro mr olk mkqk jkslmnroq ~rmlmnrols rsk~skq rj qmko~nsq 7 qmko~ns ~lo k tnqlsnk lq no npjk lo ~lo k nomkjjkmk lq mk l~mnro ru

npjk 7 jn urj jrsk | k pjkko ~nj~skq non~lmkq orkq no qmko~nsq

mk vwy lm l rnom qlss ros l uk oknprjq ljk no~sk no kl~ qmko~ns lo mk okj ru nxkjkom qmko~nsq okkk mr qk~nu mk jr sk urj mk rskrlno nq sn nmk kjkurjk l ~rotko nkom l mr q k~nu mk w urj mk vwyro l pntko pjn nq mr qk~nu mk nxkjkom qmko~nsq mr k qk mrpkmkj nm mkorkq no mk pjn kjk kl~ qmko~ns qrs k lsnk

o npjk 7 k qr mr ru mk rolj qmko~nsq urj mk kloo rolj~ronmnro lq kss lq mk qlk ru mk qmko~ns urj mk nookj orkq

iroqnkjnop klsk | lo mk w urj pntko no | 7 mk q k~n ~lmnroru mk jrsk notrstkq kq~jn nop mk slmmn~k pjn urj mk ~rmlmnrols r lnolo mk nq~jkmk urj ru mk vwy ro mlm pkr kmj q rmsnok lrtk rok lru kq~jnnop mk nq~jkmk jr sk nq no mkj q ru mk qmko~nsq m mkjk ljk rmkjlq rj no qmlo~k mkjk ljk qktkjls mrrsq mlm qk q k~nlsnksloplpkq rj l pjln~ls nomkjul~k mr qk~nu mk vwy rolj ~ronmnroq ~rmlmnrols r lno lo qr ro lqk ro qkj nom l nq~jkmk jrsknq pkokjlmk no mk mrrs lo qrstk nq rxkjq l tkj npsktks nomkjul~kurj jrsk qk~n~lmnro lo nkq mk kmlnsq ru nq~jkmnlmnro lo qrsmnrolsprjnm q k ltk ~rqko mr qk mk qmko~ns ljrl~

k qkm ru qmko~nsq urj l jrsk kokq l snoklj r kjlmrj urj q~lslj ksqrtkj mk slmmn~k pjn urj mk jrsk o mk ~lqk mlm lo ksn~nm q~k k nqqk lo lmk ru mk qrsmnro ~lo k ~rmk lsnop mk rkjlmrj mrmk qrsmnro ujr mk jktnrq nmkjlmnro u mk q~k k nq nsn~nm mk qmko~nsqurj lss kok l pkokjlmrj ru l snoklj qqmk rktkj nmkjlmntk kmrq mk qrsmnro ~lo k ljr n lmk qnop mk rkjlmrj nmrm urj nop mk snokljqqmk jsr kmrq 7 ljk klskq ru q~ kmrq

k qkm ru qmko~nsq urj l jrsk konop mk snoklj rkjlmrj qrs mkjkurjk k ko~lqslmk no l qnopsk r k~m yl~ qmko~ns qr s k lnjk nm mkqkm ru orkq no mk pjn urj n~ mk qmko~ns kokq mk l~mnro ru mk vwy nqr k~m ~lo k qk no rm ksn~nm lo nsn~nm qrstkjq lq k slnok lrtk

k ols qmk rqmjr~kqqnop nss qlss ros notrstk lolsqnq ru mk ooroq lumkj mk qrstkj nq onqk lq kss lq qmlmnqmn~q lrm kjjrj kqmn lmkq lomk okj ru nmkjlmnroq qmrjk jnop mk qrsmnro jr ~kqq q mk ooroq nss k ~rssk~mk no qr k lml qmj~mjk lo rmkj noq ru nourj lmnro l~rotko nkoms k qmrjk no qr k jkl lk r k~mq lq kss mnq slqm ljm llsqr k qkljlmk ujr mk jkqm ru mk mlqq mjrp l ~sklo nomkjul~k k ltkorm ln ~ lmmkomnro mr mk lmmkj ru rqmjr ~kqqnop no mk kljs qmlpkq ru

m nq ujl krj os qnsk srmmnop ~llnsnm nkq ljk nsk komk

jr mk klsk lrtk nm nq ktnkom mlm mk kqnpo ru l qkm ru lqmjl~mnroqnq ~j~nls urj l~nktnop mk qkj nomkjul~k k lom r k ru mk lqmjl~mnroqk ltk k~nk mr nsk kom l k ~rro mr rmkj ujlkrjq urj qrstnopvwyq 7 7 7 no qr k ~lqkq nm nxkjkom ol kq nsk qr klqmjl~mnroq ljk onzk mr mnq ujl krj k nss ur ~q ro mk urssr nop lqmjl~mnroq jn nks mko~ns lo mko~nskm kurjk k ~lo ls mk w mr l pntko jr sk k okk mr kok mk ~rmlmnrols r lno lq kss lq l slmmn~k pjn ~rtkjnop mk r lno nq nss k mk jrqk ru mk lqmjl~mnro ourj lmnro lrm pkrkmj ntnqnroljlkmkjq urj mk slmmn~k pjn lo qr ro nss k ~romlnok no mk pjn

j lqmjl~mnro nss ros losk slmmn~k pjnq n~ ~lo lslq k kq~jn k uo~mnrolss lqk ro mk pkr kmj lo mk ntnqnro ljl kmkjq ko~k ~rrjnolmkq urj lss orkq no mk pjn nss orm k qmrjk mq jk~nop mk krjjkznjkkom ru l pjn qnpon~loms k okk mr rj nm nq~jkmk ljr n lmnroq ru mk ooroq no mk vwy kqk ~lo k jkjkqkomk lq ksq rtkj mk pjn k ~jjkom nsk komlmnroru nq q~lslj n k nm qmrjkq rok tlsk urj kl~ ork no mk ~rjjkqronoppjn o mk umjk k l ~roqnkj tk~mrj ksq nm rjk mlo rok kpjkk ruujkkr no kl~ ork

k ljk qnop l qkljlmk lqmjl~mnro urj q~lslj ksq k~lqk k lom mr kl sk mr nok mk ks no nxkjkom lq lq kss lq kjurj ~kjmlno rkjlmnroqro ksq nq nss k nq~qqk no k~mnro !"#k $%&' nq mrpkmkj nm $%&'$ % mk rqm nrjmlom lqmjl~mnro no rjujl krj q kslnok lrtk mk qmko~ns kokq mk nq~jkmk ljr n lmnroru mk vwy lm rok rnom ru mk pjn q mk qmko~ns nss k mk k ~rrokomqk mr qk~nu l jrsk mk lqmjl~mnro qm ltk mk urssr nop jrkjmnkq

• qmko~ns qr s k klq mr ns q k~nu nop mk nontnls qmko~ns orkqlo mknj ~rk ~nkomq

•o lnmnro mr mk rx qkm orkq mk qmko~ns qr s lsqr ~romlno lo rmnrolsqrj~k mkj

•rm ~roqmlom lo tljnl sk rj uo~mnrols ~rk ~nkomq qrs k qrjmk o~mnrols ~rk ~nkomq qrs k ktlslmk pntnop mk ~rrjnolmkq ru lrnom lq ljp kom

•m qrs k rqqnsk mr l lo qmjl~m qmko~nsq lq kss lq smn s lqmko~ns nm l q~lslj no rjkj mr q~lsk mk qmko~ns

k jqm mjkk rnomq ljk nrjmlom mr pntk mk qkj q ~nkom rkj mr ns mkqmko~nsq urj mk vwy k slqm rnom kolskq mk qkj mr ~jklmk ~rsk qmko~nsq

(0,1) (1,1)

(0,0) (1,0)

npjk jn urj jr sk | k jk snokq lj mk ljmnmnro urj urjjr ~kqqrjq nsk mk lqk sk snokq lj kmkom ru prqm orkq k pjkko~nj~skq non~lmk qmko~nsq |mk sk ~nj~sk nq l prqm ork nsk mk okjq nojl~kmq non~lmk mk jr ~kqqrj w

jqm nsnop qnskj rokq lo slmkj rno nop mk mrpkmkj m lsqr pntkq mkknnsnm ru nsnop l snjlj ru ujkzkoms qk qmko~nsq n~ ~lo k lqmklo ~rnok mr urj mk ~rskmk qmko~ns urj l ljmn~ slj jrsk o mk vyujl krj k ltk l okj ru q~ jkkok qmko~nsq qkk qk~mnro !"# !nsk mk qmko~ns kq~jn kq mk nq~jkmk ljr n lmnro ru mk vwy urj rok ork mk r kjlmrj kok lss qmko~nsq urj mk jrsk kq~jn kq mk l~mnro ru mkvwy jrsk no~snop mk rolj ~ronmnroq urj lss orkq no mk ~rmlmnrols r lno k qk mk mkj $%&'$ % urj m nq lqmjl~mnro

k qmko~nsqkm nq l ~romlnokj no n~ smnsk qmko~nsq ~lo k qmrjk rpkmkj nm kl~ qmko~ns k qmrjk l kq~jnmnro ru l qkm ru orkq no mk pjn kjk mk qmko~ns qrs k lsnk k uklmjkq ru mk qmko~nsqkm ljk lq ur ssr q

•nsm lnop qmko~nsq lo l qk~n~lmnroq ru qkm ru orkq mr mk qmko~nsqkm

•kjlmkq ro l ks lq l snoklj rkjlmrj

•r qmko~nsqkmq ~lo k ~rnok nomr rok no rjkj mr ns rok snokljr kjlmrj ujr smnsk qmko~nsqkmq

nq kqnpo lkq rj ujl krj tkj k n sk rj noqmlo~k nm nq klq mr n sk kom l qrstkj urj l jr sk nm nxkjkom ~ronmnroq ro nxkjkom ljmq ru mkrolj m nq lsqr rqqn sk mr nsk kom l qrstkj urj l jr sk kjk nxkjkomvwyq prtkjo nxkjkom ljmq ru mk ~rmlmnrols r lno

fbdbhg c hbhdak jr ~kqq ru ~jklmnop l ljlssks qrstkj ljmnmnronop mk jr sk nq kssoro no mk snmkjlmjk ~u ~ vljmnmnronop l jrsk ~lo k l ~lsskopnop mlq lo qkj nomkjl~mnro nq qlss jkznjk no mk jr~kqq j ljrl~ nq rktkj mr nskkom mk ujlkrj no q~ l l mlm l ljlssks

qrstkj ~lo k ~jklmk ujr l qkzkomnls qr stkj nmrm lo qkj nomkjl~mnro m l~lss ro mk kmr mlm nonmnlsnkq ljlssksnlmnro k~lqk ljmnmnronop nq roklmr lmn~lss k ltk mr koqjk mlm nm nq lq rmn ls lq rqqnsk lm nq kl~qpjn qrs ltk ljr n lmks mk ql k okj ru orkq lo mk ~ron~lmnro ~rqm qrs k non nk rjnop nm qmj~mjk slmmn~k pjnq nm nqjkslmntks klq mr us s mk jqm jkznjk kom

o npjk k qr l pjn ljmnmnrok urj urj jr ~kqqrjq nm qrk qmko~nsqurj l jrsk o rjkj mr ltrn ~ron~lmnro kl~ mn k l qmko~ns nq lsnklm lo nomkjols rolj nm nq ~rro mr ko sljpk kl~ qljm ru mk pjn nml slkj ru prqm orkq q~ mlm lss jkznjk lmkq ujr l oknprj ~lo k lk nm rok tk~mrj ~ron~lmnro k nm ru mk prqm ork slkj nss k kmkj nok mk qlk ru mk qmko~nsq noqk~mnop mk qmko~nsqkm k ~lol~znjk nourj lmnro lrm mk ~ron~lmnro ~rqm no kl~ njk~mnro qqkmlm lss qmko~nsq urj l jrsk ljk ~rnok nomr l qnopsk qmko~ns lo lppjkplmkqmko~ns rj kl~ ql~k n koqnro

i k kok mk ~ron~lmnro ~rqm tlskci

lqmk nm ru mk lppjkplmk qmko~ns no njk~mnro

i rj l pntko okj ru ql~k

n koqnroqk mk ~ron~lmnro ~rqm uo~mnrols urj rok jr ~kqq nss k

C(P ; k) =

k∑

i=0

ci ∗ ni

pi

|

kjk ninq mk okj ru orkq no njk~mnro

i lo

P = piki=1

nq mk okjru qljmq nomr n~ kl~ njk~mnro qrs k q snm q~ mlm

PT =∏

i pinq

mk mrmls okj ru jr~kqqkq ltlnslsk r rmlno mk rmn ls ljmnmnro k no n nk mk uo~mnrols | kurjk mk jkqkomk lsprjnm ~lo k qk mk qmko~nsqkm qm k oro ko~k mkjk nss k l qkzkomnls ljm ru mk ~rk mlm qm k kk~mk kurjk

mk jrsk ~lo k ljlssksnk k qkj qrs mj mr ltrn ~jklmnop ksqno mnq qkzkomnls ljm ru mk ~rk lq q~ ksq nss k psrls lo ko~k ljkznjk l sljpk lrom ru krj

ch abjrp klskq nss or qr r mk ujl krj ~lo k qk lo qm mk kjurjlo~k ru nsk komk qrstkjq k jkqkomk ~rk klskq nss no~skkmlnsq km mr k nq~qqk lq kmlnsq ru mk nsk komlmnro nss k ljkqqkno k~mnro iroqnkj mk klm r jrsk

ut(x, t) = uxx(x, t), x ∈ [0, 1], t ∈ R+,

|l∂u(x, t)

∂n= f(x), x = 0 ∧ x = 1, t ∈ R

+,|

u(x, 0) = g(x), x ∈ [0, 1].|~

k ls ~komjk nxkjko~kq urj mk ql~k kjntlmntk no |l lo l urjljnxkjko~k ljrn lmnro urj mn k n~ nksq mk q~k l

un+1i = un

i +∆t

∆x2(un

i−1 − 2uni + un

i+1)|

urj mk nookj orkq k qr mk vmro ~rk ru l qrstkj urj mnq jrsk nonpjk uk nrjm qmlmkkomq urj mk vy ujl krj lo ~rk urj mknonmnls ~ronmnro lo mk k loo rolj ~ronmnro uo~mnroq ljk r nmmk

# # # 3 #, ( #( , ( # # - , 3 2 4 , , $ # ## #

, 3 # # %

# # # $ %

npjk qnsk klsk k uo~mnroq lo ljk kok nq mkko mn k lo nq mk mn kqmk

nq ~rk lk qk ru mr qmko~nsq mlm ljk jkkok no mk ujl krj mk lo # qmko~nsq |qkk qk~mnro k ~ro~kmq ru lnopqmko~nsq lo smnsnop l qmko~ns nm l q~lslj ljk qk mr urj mk ~rskmkqmko~ns urj mk jrsk rmk mlm mk

∆xlkljnop no | nq no ~rjrjlmk no

mk # qmko~ns qr k okk ros smn s mk mn kqmk qnk∆t ko kns mk ~rskmk qmko~ns

k kjkqqnro , no mk ~rk nq mk ksn~nm mn k qmknop kjk mkqmko~nsqkm nq qk lq l snoklj r kjlmrj ro mk qrsmnro ujr mk jktnrq nmkjlmnro km k ~roqnkj mk ur ssr nop jr sk

ut = ∇ · (k(x)∇u) + f, x ∈ Ω, t ∈ R+,

|lu(x, t) = h(x, t), x ∈ ∂Ω, t ∈ R

+,|

u(x, 0) = g(x), x ∈ Ω.|~

qq k mlmΩ = [0, 1] × [0, 1]

mk onm qzljk no w nq mk ~rmlmnrolsrlno lo

x = (x1, x2)nq l rnom no w k kok mk ~rk ~nkom uo~mnro

k(x) mk qrj~k uo~mnro

f(x, t) mk w njn~skm rolj ~ronmnro

h(x, t)lo

mk nonmnls ~ronmnrog(x)

q~ mlm mk lolsmn~ls qrsmnro urj mk jrsk nq

u(x, t) = e−t sin(πx1) cos(πx2)|

u k qk l urjlj nxkjko~k urj mk mn k kjntlmntk lo ~komjk nxkjko~k urjmk kjntlmntk no ql~k no |l k pkm

ul+1i,j = H [ki,j−1/2u

li,j−1 + ki−1/2,ju

li−1,j

+ (1/H − ki,j−1/2 − ki−1/2,j − ki+1/2,j − ki,j+1/2)uli,j

+ ki+1/2,juli+1,j + ki,j+1/2u

li,j+1] + ∆tf l

i,j ,|

kjk H = ∆t/∆x2 yjkqqnroq q~ lq

ki−1/2,j

non~lmk mlm mk uo~mnrok(x)

qrs k ktlslmk lm mk nrnom kmkko xi−1,jlo

xi,j u

k(x)nq

pntko lq l q~lslj ks rtkj mk pjn lo ltkjlpk rukrtkj mk ql k qkp kom ~lo

" # # ( # ' 2 * #" ' 2 * #" ' 2 #" * ' 3 ' 2 #" ' 2 #" '

" % #4 "

, ( #( , ( # - , ( # " "

npjk irk urj nsnop mk qmko~nsq urj jrsk |l

~q 7

1000× 1000 qmk 7 77 7

qkk 7 7 1500× 1500

qmk 7 777 7 qkk 7 7 7

lsk iv mn k no qk~roq lo ~rjjkqronop qkk okjq urj mk klm~ro~mnro qrstkj

k qk noqmkl k ~rskmk ~rk urj mnq klsk nq no~sk no kon o npjk k qr mk ljm ru mk ~rk mlm nsq mk qmko~nsq o mnq ~lqk k ~loorm qk jkkok qmko~nsq ujr mk ujl krj m ltk mr ns mklolss k uo~mnro ' no mk ~rk nq

k(x) + ∆x/2 nsk ' nq

k(x) − ∆x/2

o lsk k snqm mk iv mn k urj mk qrstkj ko jo ro lo mlon lqk sno ~sqmkj urj nxkjkom okjq ru jr~kqqrjq lo mr nxkjkom pjnqnkq k lsqr snqm mk q kk okjq k ltk ros klqjk mk mn kru mk lno ~rmlmnrols srr rtkj lss mn k qmkq |mk mn ksrr k~snopqmljm lo no nmnlsnlmnro ~rqmq vljlssksnlmnro rjq znmk kss no mlm k ~loqkk ~srqk mr snoklj q kk jr jktnrq kkjnko~kq nm vmro k kk~mmk qrstkj mr jo znmk qsr s rj ~rljnqro k ltk mkjkurjk nsk komkl q~lslj qrstkj urj mk qlk jrsk no i nq qrstkj lq ls rqm or knnsnmlm lss ~lo ros qrstk mnq ljmn~ slj jrsk lo mkjkurjk nq qrqk mr pntk~srqk mr rmn ls k ~nko~ o lsk k qr mk iv mn k urj mk qrstkjnskkomk no i lq kss lq q kk okjq jkslmntk mr mk vmro qrstkj k iqrstkj joonop ro rok jr ~kqqrj kjurj q lrm mn kq ulqmkj mlomk vmro qrstkj joonop ro rok jr~kqqrj lo lsqr kjurj q lrm 7 mn kqulqmkj mlo mk vmro qrstkj joonop ro 7 jr ~kqqrjq

nsk mn kq qsrkj mlo i nq l qqmlomnls kjurj lo~k nm urj mk vmroqrstkj mk plno no k n nsnm nq qnpon~lom o lnmnro mk vmro kjurj lo~k l k qnml sk urj l jlopk ru lsn~lmnroq q~ lq kkjn komq nm qmko~nsq jrmrmnop ru qrstkjq lo q lsskj jrsk q rktkj mr qrstk sljpk q~lsk

! " #!$ % & !!#' ( %)* ( +#, - ./ ! ! ! ( $$ $ 0#! # (' !

vjrsk qnk jomn k ~ vmro/i 7~ vmro

/i

1000× 1000 qmkq 7 7

1500× 1500 qmkq 7 7

lsk iv mn k no qk~roq urj mk qrstkj nsk komk no i lq kss lqqkk jkslmntk mr mk vmro qrstkj joonop ro rok lo 7 jr ~kqqrjq

vjr sk qnk i vmro1000× 1000

mn k qmkq 7 7

lsk 7 n nop ru mk ~rmlmnrols srr urj lo kkjn kom nm ro s ql~kkkokom qrj~k lo wnjn~skm rolj ~ronmnro uo~mnroq k okjq ljkiv mn k no qk~roq

jrsk q mk qkj okkq kmmkj kjurj lo~k umkj nq~qqnop vy no pjklmkjkmlns k nss jkmjo mr mnq kkjn kom lo qr r kmmkj kjurjlo~k ~lok l~nktk

o mk klsk nq~qqk kjk rm mk qrj~k uo~mnro lo mk wnjn~skmrolj ~ronmnro ljk mn kkkokom kqk uo~mnroq ljk nsk komk lqqmlolj vmro uo~mnroq lo qm mr k kk~mk urj lss jksktlom orkq lmkl~ mn kqmk nq nq l lrj kjurj lo~k rmmskok~ ru mk qrstkj iroqnkj no qmkl mk jrsk kjk mk wnjn~skm lo qrj~k uo~mnroq ljk ql~kkkokom ros u mk iv mn k ru mk mn ksrr nq klqjk k pkm mkokjq pntko no lsk 7 o mnq ~lqk mk vmro qnslmrj nq l~mlss ulqmkjmlo mk lmrjq lonsk komk i qnslmrj m qrs k ormk mlm mknonmnlsnlmnro lqk n~ nq r nmmk no mnq mn nop nq qnpo n~loms sropkj urjmk vmro qnslmrj mlo mk i qnslmrj m mnq nq ru skqq nrjmlo~k komk qnslmrj joq urj l srop mn k k ~ro~sk mlm mk okk mr ~lss mk qrj~klo wnjn~skm uo~mnroq noqnk mk mn k srr urj lss jksktlom orkq ~roqmnmmkqmk lno rmmskok~ ru mk ~jjkom nsk komlmnro nq nqqk nss k ljkqqkno k~mnro

dc ho mk urssr nop qk~mnroq k jrtnk lo rtkjtnk ru mk ~slqqkq lo uo~mnroqnskkomk no vy k ~roqmjlno rjqksu mr nq~qqnop mk lqn~ sn~ qkjnomkjul~k ru mk ujl krj o k~mnroq lo 7 k nq~qq qr k ru mkrjk ltlo~k ljmq ru vy nq~qqnro ru mrrsq mlm ljk lnos urj nomkjolsqk ljk kro mk q~r k ru mnq mkm

rtk k nq~qqk mk k lqmjl~mnroq no vy kqk lqmjl~mnroq ljkjklsnk lq ~slqqkq no vmro iroqkzkoms mkjk ljk ~slqqkq ol k $ # ( # lo ( #( o lnmnro vy ~roqnqmq ru l qkm ru mnsnm~slqqkq lo uo~mnroq rj ljmq ru mnq nq~qqnro l pkokjls okjqmlonop ru q k~nls kmrq lo rkjlmrj rtkjsrlnop no vmro nss k kok~nls rjlo rtkjtnk ru mnq mrn~ sklqk jkukj mr urj noqmlo~k 7 qk~ 7 7 rj

k ~slqq ~romlnoq l kq~jnmnro ru mk ~rmlmnrols r lno lq kss lqr mnq qrs k ~rtkjk nm l slmmn~k pjn ijjkoms rqlk r lnoq

- ?

6∆x

∆y

[xmin, ymin]

(0, 0) (n, 0)

(n, m)(0, m)

[xmax, ymax]

npjk w slmmn~k pjn

no lo okj ru ql~k n koqnroq ljk qrjmk o klsk no w nq pntkono npjk r kq~jn k mk pjn uss

[xmin, ymin]

[xmax, ymax] lo knmkj

(∆x∆y)

rj(n, m)

qm k oro k ~roqmj~mrj '' '' ru mk ~slqq mlkq mk pkr kmj nourj lmnro lo mk okj ru ntnqnroq lq ljpkomq mrpkmkj nm mk okj ru ql~k n koqnroq 6 k pkr kmj nourj lmnro nq pntko lq

([xmin, xmax], [ymin, ymax]) k ntnqnro

ljlkmkj ~lo lsqr k pntko lq l qnopsk nomkpkj o mlm ~lqk kl~ nkoqnro nss k ntnk mk qlk okj k pkokjlsnlmnro mr

nn koqnroq nq qmjlnpmurjlj

Grid

Methods:

nsd: number of space dimensions

division: list, length=nsd

geometry: list, length=nsd

fields: list of field−references

corners(): returns iterator

boundary(): returns iterator

innerPoints(): returns iterator

range(): returns index−range

partition(stencillist): parallelize grid

and attached fields.

Members:

_ _init_ _(div,nsd,geo) : constructor

npjk msnok ru mk ~slqq

k rqm nrjmlom kkjq lo kmrq ru ljk snqmk no npjk rmk mlmk qmrjk mk ∆x, ∆y, . . .

tlskq lq l snqm no mk kkj o lnmnro mk ntnqnroljlkmkjq ljk njk~ms ltlnslsk lq lo / urj pjnq no rok mr lo mjkk ql~kn koqnroq lq mnq nq rumko okkk no mk o kjn~ls lsprjnm q

q nq~qqk no k~mnro kl~ qmko~ns no l qmko~nsqkm okkq l kq~jnmnro ru lqkm ru orkq no mk pjn kjk mk qmko~nsqrs k lsnk k jrqk ru mk kmrq " lo - no nq mr jrtnk q~ kq~jnmnroq rjnoqmlo~k - nss pntk lo %% urj mk non~kq ru lss nookj orkq no mk pjn n nsljs" lo pntk mk non~kq urjrolj lo ~rjokj orkq jkq k~mntks mqrs k ormk mlm ros orkq mlm ljk ro

mk qn~ls rolj ljk ~roqnkjk rolj orkq lo mlm nomkjnrj orkq lq jkmjok - ljk mk ~rsk kom mr mk rolj rjm urjnpkjrjkj kmrq nq qr km nop mlm nss k ~roqnkjk urj ujmkj ktksr kom k kmrq - lo " l~~km lo ljpkom mlm ~lo k q k~nk mr qksk~m l qqkm ru mk r lno k ljp kom qrs k pntko no mk ql k urj lq mk pkr kmj lo nu nm nq orm jkqkom mk rskpjn nq lqq k k " kmr ~lo no lnmnro ~jklmk lo nmkjlmrj urjorkq no qnk rj rmqnk l ~nj~sk nm l pntko ~komjk lo jlnq rjk klskqru mk qk ru mkqk kmrq nss k pntko no mk ~lqk qmnkq no k~mnro mmr non~lmk mk qoml k jrtnk rok klsk kjk l ~nj~sk ro mk ul~k rul 7w rlno " * * #

6 k kmr pntkq mk nok jlopk urj orkq no mk pjn rj mk pjn

no npjk urj noqmlo~k jkmjoq mk snqm k slqm kmr no npjk nq k snok # no l qrstkj ljmnmnroq mk pjn urj ljlssks ~rmlmnroq lqk ro mk no qmlo~kru # pntko lq ljp kom o lnmnro lss ksq kok rtkj mk pjn n~ ljk jkukjko~k no mk kkj # nss k ljlssksnk mk kmr nolss mk qmko~nsqkm pntko lq ljpkom nss k rnk l~~rjnop mr mk ljlssksnlmnro ru mk pjn o kxk~m mnq qnopsk kmr notr ~lmnro nq mk ros qkjnomkjul~k mr ljlssks ~rmnop no vy

k kkjq lo kmrq no mk ~slqq ( # ljk rmsnok no npjk klqn~ tkjqnro ru mk ~roqmj~mrj urj l qmko~ns r k~m l~~kmq mjkk ljp komq ' ( # $# k ljp komq pntko kjk ljk lsqr mk kulsm ljpkomq urj mk ~roqmj~mrj lo ~lo mkjkurjk k r nmmk

_ _init_ _(nsd,varcoeff,source, nodes)

Members:

Methods:

Stencil

nsd: number of space dimensions

varcoeff: True/False − var. coefficients

source: Source value/function

stencil_coeff: dictionary

addNode(index,value): add node

addSource(value): add source

_ _add_ _(stencil): add operator

_ _sub_ _(stencil): subtract operator

_ _iadd_ _(stencil): inplace−add

_ _rmul_ _(val): multiply with scalar

convert_to_varcoeff()

npjk msnok ru mk ( #~slqq

kjk l mrn koqnrols qmko~ns nmk ~rk ~nkom $# lo lqrj~k mkj ru nq ~jklmk u tljnl sk~rk ~nkomq ljk jkzkqmk mk qrj~k mkj~lo k pntko lq l uo~mnro lq kss

' ( # rkq ~lo k lk mr l qmko~ns no

mr lq njk~ms ro no qmlomnlmnro qnopmk krj ljp kom rj nm mk2 kmr k ork nq q k~nk lqlo rx qkm m sk lo mk ~rk ~nkom nq pntkolq knmkj l q~lslj tlsk rj l uo~mnro ' ( # )

7

* *! * +

' 2

"# ' 2 ' k ljp kom qrs k qk~nkqnop mk qoml ru vmro n~mnroljnkq

q qro no npjk ( # qrjmq lnmnro qmjl~mnro lo nosl~k lnmnro rkjlmrjq nm lormkj qmko~ns n sk komk nm mk qk~nls kmrq '''' ''"'' lo'''' sqr l qmko~ns ~lo k q~lsk smnsn~lmnro nm l q~lslj nsk komk nm ''#'' nolss lo noqmlo~k ru ( # nm k ~rk ~nkomq l k ~rotkjmk mr ltljnlsk ~rk ~nkomq qmko~ns lm lo mn k qnop mk kmr '' m mk ~rotkjqk nq orm rqqnsk

smrp ( #( jrls nq mk rqm nrjmlom lqmjl~mnro no rj ujl krj mk nsk komlmnro ru mk ~slqq nq jlmkj qrjm lo qmjlnpmurjlj jrmk rnom ru tnk ru nsk komlmnro nm nq lqn~lss l sl~krskj urj qmko~nsqkzn k nm l sn nmk qkm ru r kjlmrjq o rmsnok ru mk ~slqq nq qro nonpjk

k ~roqmj~mrj ru ( #( mlkq rok ljp kom l pjn , ( #( k kmr ( # nq qk mr l qmko~nsq lo mk l~~rlonop nmkjlmrjqmr lo noqmlo~k ru ( #( nmlsk nmkjlmrjq ~lo k rmlnok ujr mk kmrq ro mk pjn no qmlo~k , ( # ' - " , ( # "' "

k tlsk jkmjok ( # nq lo nomkpkj nok tlsk n~ ~lo kqk slmkj ro mr jkmjnktk l jkukjko~k mr mk lk qmko~ns ujr mk qmko~nsqkm rjm nq jrqk ( #( nskkomkq mk qk~nls kmrq '' '' lo'' '' n~ lssr q qzljk jl~km r kjlmrjq mr k qk nmnoqmlo~kq ru ( #( #' # , , ' #

StencilListMembers:

Methods:

collection: list of stencils

where: list of iterators

_ _init_ _(grid) : constructor

addStencil(st,it): add stencil and

iterator

_ _call_ _(field): apply to field

_ _iadd_ _(stencillist): inplace−add

_ _mul_ _(field): alias for call

_ _getitem_ _(index): get stencil for

index

_ _setitem_ _(index,st): replace stencil

for index.

grid: reference to a grid

k rqm nrjmlom r kjlmrj qrjmk ( #( nq mk ~lss rkjlmrj k~lss rkjlmrj nsk komk nm mk q k~nls'' ##'' kmr nq l uklmjk ru r k~m rjnkomlmnro no vmro n~ lkq lo no qmlo~k rumk ~slqq kltk snk l uo~mnro nq nq qkmr nsk kom mk snoklj r kjlmrj kltnrjru mk qmko~nsqkm km , k lo noqmlo~k ru( #( lo k lo no qmlo~k ru $ # k qkj ~lo qk mk qoml , mr ~lss

'' ##'' nm lq ljp kom nq rkjlmrj lsq mjrp mk snqm ru qmko~nsq lo urjkl~ ru mk lsnkq mk qmko~ns lm kl~ orkpntko mk ~rjjkqronop nmkjlmrj komkjk ljk tljnlsk ~rk ~nkomq lo ko mk~rk ~nkom urj l qmko~ns ork ~romlnoq rjkmlo rok uo~mnro lss uo~mnroq ljk ktlslmk nm mk ~rrjnolmkq urj mk ~jjkompjn ork lq ljp kom k jkqsmq ljk mkolk mrpkmkj lo smnsnk mk tlskru mk ks no mk pntko ork k nsk komk ~lss rkjlmrj qrjmq ros $ #lq rkjlo

r k rjk kl~m mk qkm ru qmko~nsq lo nmkjlmrjq no mk qmko~nsqkm nq ro s mjltkjqk ro mk jqm notr ~lmnro rj qqkzkom ~lssq mk nourj lmnro nq lqqkskno l qljqk lmjn qmj~mjk n~ ~lo k lsnk njk~ms mr ksq mq nksnop ulqm rkjlmnroq u lo ~rk ~nkom rj qrj~k uo~mnro okkq mr k lmk urjnoqmlo~k k mr mn k kkoko~ mk lmjn qmj~mjk okkq mr k jklqqksk k kmrq %( lo ( %( qkml lp no mk qmko~nsqkm noqmlo~k mlm mjnppkjq jklqqksnop ru knmkj lss lml rjlml urj qrj~k uo~mnroq ros ro mk okm notr ~lmnro ru mk ~lss rkjlmrj

k ljnmkmn~ smn sn~lmnro rkjlmrj n sk komk nm mk q k~nls kmr ''#'' nq lo lsnlq urj mk ~lss rkjlmrj k jklqro urj m nq nq mlm nomk ~romkm ru k p nmkjlmntk kmrq q~ lq jsr qrstkjq nm nq ~rro mrjnmk , mq kk nop mk lmjntk~mrj lolsrp no mk qoml

k slqm kmr komnrok no npjk nq '''' nq nsk komqmk nosl~k ljnm kmn~ lnmnro 3 r kjlmrj n~ nq qk mr ~rnok mrnoqmlo~kq ru ( #( nomr rok r k~m , ( #( # ,4 ( #( # 4 , 4 , , 3 4 ~rro qlpk ru mnq r kjlmrj nss k qkko no k~mnro

q kq~jnk lrtk $ # nq qk mr rs q~lslj ksq rtkj pjnq r rs mkokjq $ # qkq l rokn koqnrols v ljjl qmrjk no mk kkj k noqmlo~k ru mlm mk ks nq kok urj nq qmrjk no o mk~lqk mlm mk n koqnro ru mk pjn nq sljpkj mlo rok mk jkqlk uo~mnrolsnm nov nq qk mr ~jklmk lo lnsnlj lml qmj~mjk mlm jkukjko~kqmk lml ljjl m ~lo k nokk nm smnn koqnrols non~kq l~~rjnopmr mk pjn jnm kmn~ r kjlmnroq ~lo k ~ljjnk rm rjk k ~nkoms ro mkrokn koqnrols lml nsk l smnn koqnrols qmj~mjk nq ~rotkonkom korjnop nm mk qmko~nsq k kkjq lo kmrq ru $ # ljk rmsnok nonpjk

k ~roqmj~mrj no $ # mlkq lo no qmlo~k ru lq ljp kom

$ #

Members:

Methods:

_ _setitem_ _(index,val): set value

_ _getitem_ _(index): get value

_ _add_ _(field): add operator

_ _sub_ _(field): subtract operator

fill(func): fill with values from func.

_ _init_ _(grid) : constructor

plot(): create 2D/3D plot

inner(field): innerproduct

_ _rmul_ _(val): multiply w. scalar

Field

shapedata: reference to data

data: numeric array, 1D

grid: reference to grid

_ _mul_ _(val): multiply operator

as left operand.

npjk msnok ru mk $ #~slqq

$ # qrjmq l ~rssk~mnro ru r kjlmrjq~rros qk urj tk~mrjq k~mrj rkjlmnroq ro ksq ljk ~ljjnk rm njk~ms romk okjsnop v ljjl qnop k ~nkomr kjlmrjq nsk komk no v kqkr kjlmrjq no~sk lnmnro qmjl~mnro lornom nqk smnsn~lmnro nm lormkj $ # lo smnsn~lmnro nm l q~lslj ss mkqkr kjlmnroq ljk nsk komk nm mk ~rjjkqronop q k~nls kmrq lq qkko no npjk lo ko~k ~lssk qnop mk ~rroljnm kmn~ rkjlmrjq $ # $ # 3 * 6 zljk jl~km ormlmnro ~lo k qk mr l~znjk lo qkm tlskq urj nontnls orkq

kjk ljk mjkk rjk kmrq no npjk njqm mkjk nq ## n~ ssq mkks nm tlskq lqk ro l uo~mnro

## k ljpkom mr mk qsnk uo~mnro nss k ~rrjnolmkq urj kl~ ork no mkpjn pntko lq l snqm rmk mlm mk ~rrjnolmkq nss lsqr k pntko lq l snqm urj rokn koqnrols pjnq jmkj noqmlo~kq ru $ # ltk mk kmr n~~rmkq mk nookj jr~m ru mr ksq lo # n~ ~jklmkq l pjln~lssrm urj rok rj mrn koqnrols ksq

ljlssks noujlqmj~mjk nq ok~kqqlj no rjkj mr mnsnk l ljlssks ~rmkj kjk ljk qktkjls nxkjkom ljrl~kq mr nskkomnop ljlssksnq qkk k p 77 o vy k ltk qk mk k ul~mr qmlolj urj ljlssks ~rmnop v 7 n~ nq nks ltlnslsk lo qk kjk ljk qktkjlsvmro rskq mlm jrtnkq jlkjq mr v qr k ru mk ljk v n ~nkomn ~ v lo vlj lqk ro nmq qnsn~nm lonmq l nsnm mr ~ron~lmk o kjn~ls ljjlq no l tkj k ~nkom lookj k ltk~rqko mr qk vlj vlj ro s nsk komq l sn nmk m nrjmlom qqkm ruv rj noqmlo~k orosr ~nop qko lo jk~kntk nq orm nsk komk sqr nm nqorm rqqnsk mr qk v ~ron~lmrjq nm vlj rktkj urj rj jr k~m nm nq kssqnmk q mk l~mls nsk komlmnro ru mk lmr lmn~ ljlssksnlmnro nq znmk notrstk lo lsqr oktkj krqk mr mk qkj njk~ms k r orm kjk no~sk lnq~qqnro ru mnq mrn~

k nss or nq~qq qr k ru mk rmkj mrrsq mlm ljk nsk komk no mk ujl krj nq no~skq qr k nsk komk qmko~nsq qrk nmkjlmntk qrstkjq lo l

npjk qmko~ns rnk urj l k loo rolj ~ronmnro

mrrs urj nsk komnop k loo rolj ~ronmnroq kqk mrrsq ljk rm rumko qk lo ~lo qkjtk lq klskq ru r vy ~lo k qk nolss k lsqrnq~qq mk nmkjlmrjq mr k qk no qmko~nsqkmq !"#rj qmko~nsq ljk nsk komk lo nkomnm qmko~ns l qmko~ns urj mk lsl~k rkjlmrj l rnk lsl~k qmko~ns lo l qmko~ns urj w njn~skm rolj ~ronmnroq ss qmko~nsq ljk nsk komk lq q~slqqkq ru mk ( # ~slqq

k ~slqq nskkomq l qmko~ns nm mk ~rk ~nkom no mk ~komjklq mk ros ork k ~roqmj~mrj mlkq l qnopsk ljl kmkj mk okj ru ql~kn koqnroq

k nq~jkmk lsl~k rkjlmrj n~ jkqsmq ujr qnop ~komjk nxkjko~kqro mk lsl~k rkjlmrj

∇2 nq rumko ko~romkjk no lsn~lmnroq k nq~jkmk

urj kkoq ro mk ntnqnro ljlkmkj ru mk pjn lq kss lq mk okj ruql~k n koqnroq k ltk nsk komk l # ~slqq mlm nsq mk qmko~nsurj mk lsl~k r kjlmrj no lo okj ru ql~k n koqnroq kjk mk ntnqnroljlkmkj no mk pjn l k nxkjkom urj kl~ n koqnro o noqmlo~k ru nq mk jkznjk ljp kom urj mk ~roqmj~mrj

k ~slqq # n sk komq l tkjqnro ru mk nq~jkmk lsl~k rkjlmrjkjk mk ~rk ~nkom urj mk ~komjk ork nq kjr lo mk rmkj ~rk ~nkomq ljksmnsnk nm mk notkjqk ru mk ~rk ~nkom urj mk ~komjk ork no mk jkp sljnq~jkmk lsl~k r kjlmrj nq nq urj noqmlo~k mk qmko~ns mlm nss lklj kol l~rn nmkjlmnro nq lsnk mr mk q~k k urj mk lsl~k kzlmnro

∇2u = f

lq pntko no | k slqm qmko~ns k ltk nskkomk nq l w njn~skm rolj ~ronmnro

no mk ~slqq % #4 nq nq l qmko~ns mlm rkq orm ltk orkq m rkq ltk l orokjr qrj~k mkj mlm jkjkqkomq mk kqqkomnls rolj~ronmnro ko~k mk ljp komq mr mk ~roqmj~mrj ljk mk okj ru ql~kn koqnroq lo mk rolj ~ronmnro lq l uo~mnro "" " "!"nsk no jno~n sk klq mk nsk komlmnro ru l kloo rolj ~ronmnro~lo k ~lsskopnop lm sklqm urj smnsk nkoqnroq k k loo ~ronmnro nqnomkjjkmk lq lo lnsnlj ~ronmnro qk mr ksn nolmk orkq mlm ulss rmqnk mk~rmlmnrols rlno ujr qmko~nsq lm mk rolj lq nq~qqk no k~mnro n~ orkq lo ktko mk okj ru orkq mr k ksn nolmk tljnkq urj mk

5 " , " # *6 # $# " * , 3 #

# " " # #

" * , 3

npjk 7 irk urj mk l~r n nmkjlmnro

nxkjkom ljmq ru mk rolj lo qr nm nq l ~rsk mlq mr nsk kom mkqmko~nsq urj mk k loo ~ronmnro lo mk nmkjlmrjq mlm ljk okkk urj kl~ rumk npjk qr q mjkk nxkjkom klsk rolj qmko~nsq urj mk kjrolj mk jnpm rolj lo mk srkj skum ~rjokj q k qkk rok ork nqjkrtk ujr mk kj rolj qmko~ns lormkj rok ujr mk jnpm roljqmko~ns nsk mr orkq ljk jkrtk ujr mk ~rjokj qmko~ns

r qnsnu m nq mlq l uo~mnro nm mk ol k 2 4 nqnskkomk nq uo~mnro ~jklmkq l qkm ru qmko~nsq mrpkmkj nm mk l~~r lonop nmkjlmrjq n~ nskkomq mk k loo ~ronmnro ro l ljm rj mkrsk rolj ru l pjn k qmko~ns mlm qr s k rnk lm mk roljqm k pntko lq lo ljpkom mr mk ~jklmk uo~mnro jmkj nm okkq l pjn mk k loo ~ronmnro lq l ~roqmlom rj l uo~mnro lo l qk~n~lmnro ru mkrolj k rolj qrs k q k~nk lq urj mk " kmr no k ~jklmk uo~mnro jkmjoq l ok qmko~nsqkm nm lss mk rnk qmko~nsqlo mknj nmkjlmrjq k rolj qk~n~lmnro ~lo k qk mr qksk~m l ljm rumk rolj ru mk pjn kjk mk ~ronmnro qrs k qk o mnq l nxkjkom rolj ~ronmnroq ~lo k nskkomk ro nxkjkom ljmq ru mk roljylskq nss k jrtnk no mk ~lqk qmnkq no k~mnro

k qmko~nsqkm jr~k 2 4 ~lo knmkj k lkmr lormkj qmko~nsqkm n~ ~romlnoq qmko~nsq urj mk jkqm ru mk r lno rjqmko~nsq lo nmkjlmrjq urj rmkj ljmq ru mk r lno ~lo k lk mr mk jr~kqmko~nsqkm k nosl~k lnmnro rkjlmrj ro qmko~nsqkm lq ~jklmk urj mkjrqk ru lnop mnq ~jklmk qmko~nsqkm mr lormkj k nqmnop qmko~nsqkm no mkqkj~rk "! kpljnop njk~m rj ksn~nm qrstkjq mkjk nq jklss ormnop mr nsk kom k~lqkl qmk rj lo nmkjlmnro no lo ksn~nm kmr nss qns ls mk qmko~nsqkm mrqr k ks ru ooroq o klsk ru l njk~m qrstkj lq pntko no k~mnro

kjk ljk mr nmkjlmntk qrstkjq nsk komk no mk ujl krj mk l~rn nmkjlmnro lo mk iroplmk jlnkom nmkjlmnro ltlnslsk lq mk uo~mnroq " 5 " lo 5 5 rmk mlm mkqk qrstkjq ljk jrtnklq klskq ru r snoklj qrstkjq ~lo k nsk komk no vmro no pkokjls lovy no ljmn~slj rqm jklsrjs lsn~lmnro nss jkz njk jk~ronmnronop rumk snoklj qqmk n~ nq orm nsk komk no mkqk klskq

o npjk 7 k qr mk ~rk urj mk l~rn nmkjlmnro ntko l qmko~nsqkm

5 , " # *6 # $# " * , #

# " " # #

, 3 * 3

npjk irk urj mk iroplmk jlnkom nmkjlmnro

, l ks urj mk nonmnls pkqq ru mk qrsmnro lo l ks urj mk jnpmloqnk no mk jrsk " mk l~rn nmkjlmnro nq kjurj k no rjkj mr o loljr n lmk qrsmnro urj ~rotkjpko~k ~jnmkjnro ~lo k q k~nk lq l urjmljpkom rj nm mk krj # kjk nq lsqr l ~rn~k ru qnop ljkslmntk ~rotkjpko~k ~jnmkjnro qkmmnop mk lp # mr omlm ~lqk mk orj ru mk jkqnls nq ntnk mk orj ru mk jnpmlo qnk kurjk knop ~k~k lplnoqm mk mrskjlo~k k kulsm tlskq urj mk mrskjlo~klo mk jkslmntk ~rotkjpko~k lp ljk

1.0e−5lo $# jkqk~mntksklj mlm ktko mrp k ljk rjnop nm ~slqqkq nsk komk no vmro

rqm ru mk ljnm kmn~ rkjlmnroq ljk ~ljjnk rm njk~ms ro mk okjsnoplml ljjlq kjk mkqk rkjlmnroq ljk nsk komk no l srsktks ~rnsksloplpk qlss i o mnq l k~kom okjn~ls k ~nko~ ~lo k l~ nktk sqr ormk r o~smmkjk mk nsk komlmnro nq nq nq l~nktk mnsnnopmk rkjlmrj rtkjsrlnop ~ro~km mlm nq ~rro mr r k~mrjnkomk sloplpkq mrkok mk l~mnro ru ~rro ljnm kmn~ r kjlmrjq urj lo r k~m k ol n~mnop no vmro nq lsqr lo nrjmlom ul~mrj

o npjk k qr mk ~rk urj mk iroplmk pjlnkom qrstkj lq nsk komk no 5 5 k qnpolmjk ru mk uo~mnro nq kl~ms mkql k lq urj mk l~rn qrstkj q~ mlm mk qrstkjq ~lo k nomkj~lopk

kjk ljk mr qmjn nop ul~mq mlm qrs k komnrok jkpljnop mk ~rkq nonpjk 7 lo njqm nm nq rjmnsk mr ormk mk jkqkslo~k ru mk ~rk mrqkr ~rk smrp lm k qr kjk nq jkls rjnop ~rk lq nsk komkno mk ujl krj nm srrq tkj ~ snk qkr ~rk urj mk ql k lsprjnm qno lo mkmrr k nrjmlo~k ru mnq nq mlm mk ~rk nq klq mr okjqmlo lo mlm nm nq znmk klq mr nsk kom rmkj qrstkjq nu r ltk qkr ~rk urjmk lsprjnm ltlnslsk lq lo ls rqm snok snok mjloqslmnro mr vmro ~rknq rqqnsk k qk~ro rnom nq mlm mkjk nq or jkukjko~k mr ljlssksnq nomk ~rk q mk ~rrokomq qk ljk nokjkoms ljlssks mkjk nq or okk mr~roqnkj ljlssksnq ksn~nm s no mk qrstkjq

nolss k rs snk mr jk lj mlm lsmrp mk r k~m , lq qrqk mr k l qmko~nsqkm lo mk r k~mq lo " ksq mkjk nq ormnop no mk nsk komkqrstkjq mlm jkznjk mnq o r k~mq mlm qrjm mk ljnm kmn~ r kjlmnroqqk lq kss lq mk nookj jr~m lo mk ~r rkjlmnroq ~lo k qk nm mkqrstkjq ko~k nu , nq l lmjn lo lo " ljk tk~mrjq mk ql k qrstkjq ~lo k

npjk ijklmk lo qk l #

qk njk~ms no l q~lslj ~romkm o l ljlssks ~romkm ros mk ljlssks tkjqnroqru , lo mk nookj jr~m qm k nsk komk kurjk mk qrstkjq ~lo kqk nm lmjn lo tk~mrjq no mnq ~lqk nq nq l uklmjk lk rqqnsk qnop l ol n~lssm k sloplpk q~ lq vmro "!!" !!o mk # rsk ru vy k ltk nsk komk l uk rjk qkus mrrsq k rqm nrjmlom ljk mk nmkjlmrjq k qk no qmko~nsqkmq mr kq~jn k qkmq runon~kq no pjnq kjk l qmko~ns qr s k lsnk

sk nnsnm jkpljnop r qmko~nsqkmq ~lo k ~jklmk nq rok ru mk lno qmjkopmqru rj ujl krj k~lqk nm rxkjq l rkjus mrrs urj q k~nu nop lo qrstnop lsljpk ~slqq ru jrsk q m ~lo k l n ~sm mlq mr qksk~m l qqkm ru non~kqujr mk slmmn~k pjn urj l qmko~ns lm sklqm no mk smnn koqnrols ~lqk kol oromjnt nls qqkm nq ~lssk urj ko~k k ltk nsk komk l qkm ru nmkjlmrj~slqqkq mlm qrs pntk mk jkz njk k nnsnm nsk lnomlnonop l jkslmntksqnsk qkj nomkjul~k o lnmnro mk nmkjlmrjq k ltk nsk komk loskmk jkznjk komq ru ljlssksnlmnro k nsk komk nmkjlmrjq ljk mk lqnq urjmk kmrq no mlm jkmjo qkmq ru non~kq |qkk k~mnro lo urj mk 2 4 uo~mnro rj l sljpk jlopk ru ~lqkq mkqk kmrqqrs k q ~nkom m urj ktko rjk k nnsnm mk ~slqqkq jkqkomk kjk ~lok qk njk~ms rj mk klskq jkqkomk no mnq lkj k nss lkjk mrmk nomkjul~k jrtnk lo mk ~jklmk uo~mnro urj k loo ~ronmnroq

kjk ljk urj nxkjkom nmkjlmrjq nsk komk no mk # rsk njqm mkjk nq l pkokjls ~slqq ol k # n~ nq qk mk rmkj rjkqk~nlsnk nmkjlmrjq m ~lo lsqr k qk njk~ms k ol k jkk~mq mk ul~mmlm mk ksk komq jr~k ljk non~kq pntko lq mskq npjk qr q r l# nq ~jklmk lq kss lq qlsk qlpk jp komq ljk mk okjru ql~k n koqnroq lo mk jlopk ru mk mskq pkokjlmk qk~nk mk no n lo ln ~rjokjq qnop # mk qkj ~lo pkokjlmkmskq urj lo rqlk jkpnro no l pjn no rjkj mr lqqnpo l ljmn~ sljqmko~ns mr mnq jkpnro k n koqnro ru mk r rkq orm okk mr k mk ql klq mlm ru mk pjn l w jkpnro no l 7w pjn ~lo k qksk~mk skmmnop lookj ru tlskq no mk no n lo ln ~rjokjq qk~nk k mk ql k o klsk nq qro no npjk kjk l jkpnro no mk slok | nqqksk~mk no mkj q ru 7w non~kq n~ ~rs k qk mr qksk~m l jkpnro ujr l7w pjn

o lnmnro mr # mk urssr nop nmkjlmrj ~slqqk ljk nsk

komk # urj nookj orkq "# urjrolj orkq lo # urj ~rjokj orkq k qnpolmjk urjmk ~roqmj~mrj nq mk ql k urj lss mkqk nmkjlmrjq mk okj ru ql~k n koqnroq mk no n msk lo mk ln msk

nolss mkjk nq l q k~nls nmkjlmrj nskkomk urj ~nj~ slj jkpnroq nqnmkjlmrj nq qk~nls no mlm nm jlq lo ru mk rmkj nmkjlmrjq lo jkqmjn~mq mknon~kq mr mrqk knmkj rmqnk rj noqnk l ~nj~sk rj qkjk nm pntko jlnq lo~komjk nsk lss mk rmkj nmkjlmrjq ro s jr~k qkmq ru non~kq mk ~nj~sknmkjlmrj okkq mr jkslmk mr qr k pjn no rjkj mr ~rmk mk nqmlo~k kmkkol pntko rnom lo mk ~komjk ru mk ~nj~sk k ~slqq nq ol k # lo mk ~roqmj~mrj jkznjkq urj ljpkomq lo nmkjlmrj jrtnnop mk non~kq l pjn mk ~komkj lo mk jlnq o lnmnro mk njk~mnro ~lo k qk~nk lqknmkj rj nq mk kulsm

bfehk or jkqkom mjkk ~lqkq mlm jrtnk l kmmkj okjqmlonop ru r vy~lo k qk o mk jqm ~lqk k notkqmnplmk mk ~rotkjpko~k rjkj ru nxkjkomonmk nxkjko~k q~kkq urj l ltk kzlmnro no mr ql~k nkoqnroq kqk~ro ~lqk ~ro~kjoq lormkj lsn~lmnro ru mk ltk kzlmnro no mk qnslmnroru smjlqron~ l~rqmn~ls ltkq ~rros oro lq smjlqro o mnq ~lqk mkltk kzlmnro nq qrstk no mjkk ql~k n koqnroq o mk slqm ~lqk k jktnqnm mkklm ~ro~mnro jr sk no kzlmnroq |l~ nm rjk ur~q ro kjurj lo~k

iroqnkj mk ltk kzlmnro

utt = ∇2u, (x, t) ∈ Ω × R+,

u(x, 0) = f(x), x ∈ Ω,

ut(x, 0) = 0, x ∈ Ω.

|

rolj ~ronmnroq no rj kkjn komq nss k knmkj mk rrpkokrq wnjn~skm ~ronmnro

u(x, t) = 0, (x, t) ∈ ∂Ω × R+,

|rj mk rrpkokrq k loo ~ronmnro

∂u(x, t)

∂n= 0, (x, t) ∈ ∂Ω × R

+.|

jmkj k lqq k mlmΩ = [−1, 1] × [−1, 1]

qnop l qk~rorjkj nxkjko~kno mn k k rmlno mk qk nnq~jkmk urj

un+1 − 2un + un−1

∆t2= ∇2un,

|7 n~ pntkq mk ksn~nm q~kk

un+1 = 2un − un−1 + ∆t2L∆un.|

kjk L∆u

jkjkqkomq mk nq~jkmnlmnro ru mk lsl~k r kjlmrj u k ls mk ~komjk nxkjko~k urj mk lsl~k rkjlmrj lq k n urj mk

klm ~ro~mnro nq~qqk no k~mnro |qkk k p kzlmnro | mk kjjrj qrsmkrjkmn~lss k ru qk~ro rjkj no mn k lo ql~k

e = O(∆t2, h2).kjk lo

no mk jkqm ru mnq qk~mnro k lqq kh = ∆x = ∆y.

noqmkl qnop l rnom

−103

1/6 1/6

1/61/6

2/3 2/3

2/3

2/3

npjk rnom qmko~ns ru mk ~rl~m urjmrjkj q~k k urj mk lsl~kr kjlmrj no w

q~k k urj mk lsl~k rkjlmrj |qkk ~ 7 k rmlno mkrjkmn~lss lq~k k nm kjjrj

e = O(∆t2, h4) k ~rk ~nkomq no mk rnom qmko~ns nss klq no npjk

irotkjpko~k urj mk jrsk ~lo k njrtk ktko ujmkj qnop l npkjrjkj kmr urj mk mn k kjntlmntk k nss qk l urjmrjkj qmjkmr ~ urj mk mn k kjntlmntk no rjkj mr pkm lo kjjrj kqmn lmke = O(∆t4, h4)

rj mk jrsk nm l w njn~skm rolj ~ronmnro k kk~m mr qkk ~srqk mr

rmn ls ~rotkjpko~k rjkj no mk kkjn komq rj mk jr sk nm l k loorolj ~ronmnro k nss qk mk 2 4 uo~mnro no vy n~ nsk komq l qk~rorjkj ljrn lmnro ru mk rolj ~ronmnro ko~k mkrjkmn~lss k rmlno l qk~rorjkj q~kk urj mk rsk r lno ktko nu l urjmrjkj rnom q~k k nq qk urj mk nookj orkq no mk r lno

rj rm mk wnjn~skm lo k loo rolj ~ronmnro ~lqkq k ~lo~rrqk mk lolsmn~ls qrsmnro ru | lo kjntk mk nonmnls uo~mnro

f(x)lo

mk rolj ~ronmnro k kjjrj ~lo mko k ~rmk lq mk q rtkj lssmn k qmkq ru mk nq~jkmk

L2

orj ru mk nxkjko~k kmkko mk lolsmn~ls loljr n lmntk qrsmnroq smnsnk nm ∆t

e = (∆t∑

n

h2∑

i,j

(uni,j − an

i,j)2)1/2.

kjk ui,j

nq mk ljrn lmnro ru mk qrsmnro loai,j

nq mk lolsmn~ls qr smnrono mk orkq ru mk pjn k kjjrj kqmn lmk urj rj jr sk ~lo k jnmmko nomk urj

e = A∆tα + Bhβ.u k lqq kα = 2

loβ = 4

lo ~rrqk∆t ≤ h2

k pkm

e = A∆t2 + Bh4 = A(h2)2 + Bh4 = Ch4.

ko~k nu∆t

nq ~rqko q~ mlm∆t ≤ h2

k kk~m mr rqkjtk urjmrjkj ~rotkjpko~k ko l urjmrjkj ljr n lmnro nq qk no ql~k ktko nu l qk~rorjkj q~kk nq qk urj mk mn k kjntlmntk rj qk~rorjkj ljr n lmnrono rm ql~k lo mn k k pkm

e = Ch2

~rrqnop∆t ≤ h

lo ko~k k kk~m mr rqkjtk qk~rorjkj ~rotkjpko~kno mnq ~lqk o mk ql k l urjmrjkj ~rotkjpko~k nq k k~mk urj mk

hm m m

k k 7k 7k k 7 k k k k 7k k k k k k 7 7k 7

klqm qzljk m i k k kklqm qzljk m

α 7 7

lsk y jjrj lo ~rotkjpko~k jlmk urj mk ltk kzlmnro nm wnjn~skm rolj ~ronmnro nm ∆t = 2.5e− 05

q~ mlm∆t ≤ h2

urj mk q lsskqmh y npm

mn k qmkq kjurj k

hm m m

k k7 7k k k 7 7k 7k k k 7k k k k 7k k k

klqm qzljk m i 7k 7k7 k klqm qzljk m

α 7

lsk y jjrj lo ~rotkjpko~k jlmk urj mk ltk kzlmnro nm wnjn~skm rolj ~ronmnro nm ∆t = 2.5e − 03

q~ mlm∆t ≤ h

urj mk q lsskqmh y npm

mn k qmkq kjurj k

O(∆t4, h4)q~k k n~ nq rmlnok ~rnonop mk rnom q~kk no ql~k

nm l urjmrjkj qmj kmr no mn k lqk ro mk nq~qqnro lrtk k k lqq k lo kjjrj rks ru mk urj

e = Chα

urj rj kkjn komq lo qk l snoklj sklqm qzljk m mr ~rmkC

loα

ujrmk kkjn komls lml k qk mk urssr nop slksq urj mk kkjn komq % urjmk q~k k lqk ro mk ~komkjk nxkjko~k no mn k lo mk rnom qmko~ns urjlsl~k % urj mk q~k k lqk ro mk rnom qmko~ns urj lsl~k lo mkql k nxkjko~k no mn k lo % kormkq mlm mk urjmrjkj qmjq~k k no mn k lo mk rnom qmko~ns urj lsl~k nq qk k jkqsmq ru rjkkjn komq ljk pntko no lskq q kk~mk urj mk wnjn~skm rolj~ronmnro jrsk k pkm ~srqk mr qk~rorjkj ~rotkjpko~k urj mk % q~k kko ∆t ≤ h

lo urjmrjkj ~rotkjpko~k urj % lo % ko ∆t ≤ h2

rj % k pkm ~srqk mr urjmrjkj ~rotkjpko~k lsqr ko ∆t ≤ hurj mk

w njn~ skm jrsk nsk mk ~rotkjpko~k jlmk ru % nq ~srqk mr no mnq ~lqk k ~rotkjpko~k jlmk urj mk kloo rolj ~ronmnro jrsk k

pjlkq urj mk q~k kq qnop urjmrjkj ljr n lmnro no ql~k k~lqk mkk loo ~ronmnro nq nsk komk nm ros l qk~rorjkj ljr n lmnro rktkj nm nq nomkjkqmnop mr rqkjtk mlm k pkm mnjrjkj ~rotkjpko~k urjrm % lo % no mnq ~lqk koktkj ∆t ≤ h

nq non~lmkq mlmmk qk~rorjkj ljr n lmnro ru mk kloo rolj ~ronmnro rkq ormkpjlk mk ~rotkjpko~k rjkj lq ~ lq mk mkrj qppkqmq

7

hm m m

k k k k k 7 k 7 7k 7 7k k k k k k 7k 7k 7k

klqm qzljk m i 7 k k kklqm qzljk m

α

lsk y jjrj lo ~rotkjpko~k jlmk urj mk ltk kzlmnro nm k loorolj ~ronmnro nm ∆t = 2.5e − 05

q~ mlm∆t ≤ h2

urj mk q lsskqm

h y npm mn k qmkq kjurjk

hm m m

k 77k7 k k k 7 k 7 7k 7 k k 7k 7 k 7 k k k k 7 77k

klqm qzljk m i k 7k kklqm qzljk m

α 7

lsk y jjrj lo ~rotkjpko~k jlmk urj mk ltk kzlmnro nm k loorolj ~ronmnro nm ∆t = 2.5e − 03

q~ mlm∆t ≤ h

urj mk q lsskqmh

y npm mn k qmkq kjurj k

" "!!"iroqnkj mk nq~jkmk urj pntko no | nq q~k k ~lo k jkjnmmko lq

un+1 = (2 + ∆t2L∆)un − un−1|

o mk klskq k ltk qkko qr ulj mk qmko~nsqkm lq lslq rkjlmk ro lqnopsk ks ru ooro tlskq

un no rjkj mr ~rmk lo lmk qnop knmkjksn~nm kmrq rj lo nmkjlmntk qrstkj jr | nm nq ~sklj mlm k okk mrrj nm mr nxkjkom tkjqnroq ru

u mk ~jjkom lo mk jktnrq mn k qmk

mr ~rmk tlskq urj mk okm mn k qmk q l qmko~nsqkm nq ros lsk mr r kjlmkro l qnopsk ks k okk mr qk mr qmko~nsqkmq rok urj mk mkj

2+∆t2L∆

lorok urj

−1 k urssr nop k~kjm ujr mk ~rk qr q mk lqnq ru mk ltkqrstkj

* * , ( #( 4 ( #(

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k ~rskmk ~rk urj mk qnslmrj nq rjk notrstk k~lqk nm ~romlnoqmk rmnro ru qksk~mnop knmkj w njn~ skm rj kloo rolj ~ronmnroq no mkqnslmnro lq kss lq ~rmlmnro ru mk kjjrj orj o lnmnro k nq~rtkjkjnop mk jr ~kqq mlm mk rnom q~k k urj lsl~k rkq orm lmr lmn~lssnks urjmrjkj ~rotkjpko~k o rjkj mr l~nktk urjmrjkj ~rotkjpko~k mk q~k k qm k lsnk mr mk rnk kzlmnro

Mutt = ∇2u|

kjk Mnq kok mk qmko~ns qro no npjk ~~rjnop mr mk

nxkjko~k kmkko mk qrsmnro ru | lo | nq ru rjkjO(h4)

qr mk qrsmnroru | qrs qmnss p ntk lo kjjrj ru urjm rjkj ko ~rljk mr lolsmn~lsqrsmnroq ru | jnmnop | lq

utt = M−1∇2u,|

k ~lo jr~kk lq kurjk mr rmlno mk nq~jkmk q~k kun+1 = 2un − un−1 + ∆t2M−1L∆un.

| q

M−1nq orm oro lo k pkokjlss ltrn notkjmnop l lmjn njk~ms k

jnmk mk jrsk lqMun+1 = (2Mun + ∆t2L∆)un −Mun−1.

| wk mr mk

Mmkj ro mk skumlo qnk mnq nq or sropkj lo ksn~nm q~k k

rjmolmks mk ~ronmnro okj ruM

nq q lss qr mk jrsk ~lo k qrstkkxk~mntks nm rm mk l~rn lo iroplmk pjlnkom qrstkjq nq~qqk nok~mnro 7

112

112

112

112

23

npjk k ltkjlpnop lqq qmko~nsM

mlm qm k qk mrpkmkj nm mk rnom q~k k urj lsl~k

o 7 onmk kskkom qnslmnro ru smjlqro nq nq~qqk rj smjlqroqnslmnro no mk lo r mk onmk ksk kom kmr ~lo k kok~nls kmr ~rsk pkrkmj ru mk r rktkj urj rqm ru mk klskq jkqkomkno 7 qk ru mk onmk ksk kom kmr nq orm l jkznjkkom kjkurjk klom mr nsk kom l onmk nxkjko~k qnslmrj urj smjlqro no snoklj knl qnop vy

k snoklj rks urj smjlqro nq qn nslj mr mk ltk kzlmnro nq~qqk nomk jktnrq qk~mnro

utt(x, t) = c2∇2u(x, t), (x, t) ∈ Ω × R+,

u(x, 0) = 0, x ∈ Ω,

ut(x, 0) = 0, x ∈ Ω,

u(x, t) = T (x, t), (x, t) ∈ ∂ΩT × R+,

∂u

∂n= −

1

c

∂u

∂t, (x, t) ∈ ∂ΩT c × R

+.

|

kjk cnq mk q kk ru qro

∂ΩTnq mk jkpnro ro mk rolj urj mk mjloq

~kj mlm pkokjlmkq mk smjlqro ltkq lo T (x)nq mk mjloq~kj uo~mnro

o mk jkqm ru mk rolj∂ΩT c ,

lo lqrjnop rolj ~ronmnro nq qk no mk urj ru l k loo ~ronmnro

k qrstkj urj mk smjlqro jrsk nq nsk komk lsrop mk qlk snokqlq mk ltk kzlmnro nq~qqk lrtk qnop mk ~komkjk nxkjko~kq urj mklsl~k rkjlmrj jkqsmnop no l rnom qmko~ns no mjkkn koqnrols ql~k rjmk mn k kjntlmntk mk qlk ~komjk nxkjko~k lq no | nq qk k no m nqjrsk nq mk n ru w njn~skm lo k loo ~ronmnroq ro nxkjkom ljmq rumk rolj o lnmnro mk lqrjnop rolj ~ronmnro no~skq mk ksu n~ kloq mlm k okk qr k k~lonq mr no~sk noqmlo~kq ru $ # no

mk uo~mnro qsnk mr 2 4 k mjloq~kj no rj qnslmnro nq q k~nk lq l ~nj~sk ro mk ul~k rumk r lno nm ~komjk no rjnpno lo l pntko jlnq k qk l kqqks uo~mnrolq mk pkokjlmnop uo~mnro urj mk jkqqjk nqmjnmnro ro mk mjloq~kj lopkokjlmk l pntko okj ru sqkq ro mk mjloq~kj

Ω

∂p

∂n= 0

∂p

∂n=

−1

c

∂p

∂t

∂ΩT

Ω

T c

npjk wr lno lo rolj ~ronmnroq urj mk smjlqro kkjn kom k qr mk slok no mnq pjk

!

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k lqrj nop rolj ~ronmnro mlm k nss qk lq mk k loo ~ronmnro urj mk jkqm ru mk rolj no~skq mk mn k kjntlmntk ru mk ooroks

∂u

∂n= −

1

c

∂u

∂t, x ∈ ∂ΩT c .

| qnop l jqmrjkj nxkjko~k urj mk mn k kjntlmntk k rmlno

∂u

∂n= −

1

c∆t(un − un−1).

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fnq or jkrtk ujr mk qmko~ns k

lom mr ~jklmk mk qmko~nsqkm ( lo 4 q~ mlm ( ~lo k lsnk mr mk qrj~kks $ lo 4 mr mk rolj ks "$ lo mk jkqsmq lk mrpkmkj nm , mr jrtnk lo lmk | nm mk rjnpnols ~lss qoml , 3 ( $ 3 4 "$ k ~lo l~ nktk mnq no mk urssr nop l ( nq mk nkomnm urj lss nookj orkq lo kjr urj rolj orkq 4 nq mk nkomnm urj rolj orkq lo kjrurj nookj orkq lo , qkq mk jrsk qmko~ns urj nookj orkq lo kjr urj mkrolj , ( #( , " # ", ( # -

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7

~q 7

1000× 1000 mn k qmk 7 7 7

q kk 7 1500× 1500

mn k qmk 7 77 7q kk 7 7 7

lsk iv mn k no qk~roq lo ~rjjkqronop q kk okjqvjrsk qnk jomn k jks ~ vmro jks 7~ vmro

1000× 1000 mn k qmkq 7 7

1500× 1500 mn k qmkq 7 7 7

lsk iv mn k no qk~roq urj mk mn k srr no mk qrstkj nskkomk noi lq kss lq qkk jkslmntk mr mk vmro qrstkj joonop ro rok lo 7jr ~kqqrjq

o lsk k snqm mk iv mn k urj mk lno ~rmlmnrols srr ru mkqrstkj k~snop nonmnlsnlmnro irljk mr mk okjq urj mk ql k jrsk no k~mnro mkjk nq l qnpo n~lom kjurj lo~k plno kpljnop ljlssksk ~nko~ k lplno rqkjtk tkj prr qkk ko ~rljk lplnoqm mk ijrpjl mlm qrstkq mk ql k jrsk mk kjurj lo~k plno ujr mk jkjnmknq ktko rjk tnqn sk r mk vmro qrstkj joonop ro rok iv nq qm 7 mn kq qsrkj mlo mk i qrstkj nsk mk kjurj lo~k ru mk rjnpnols qrstkj lq mn kq qsrkj mlo i ko rjk jr ~kqqrjq ljk qk mk vmro qrstkj ~loqrstk mk jrsk qnpon~loms ulqmkj mlo mk q~lslj i qrstkj r rmlno mk kjurjlo~k no~jklqk k l mr nsk kom ~rk mlm nq rjk n ~sm mr jkllo qk qr k skqq nomnmntk ~roqmj~mq o lnmnro k l mr ss mk qrj~klo rolj ksq no l tk~mrjnk lookj n~ nq skqq k nsk mlo lqqnponoptlskq rnom rnom ktkjmkskqq mk ljrl~ qk nq pkokjls lo ~lo klsnk mr lo qrstkjq jrmrm k nm vy mr l~nktk kmmkj kjurj lo~k

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kjk ljk qktkjls nqqkq mlm qrs k ljkqqk urj mk ktksr kom ruumjk tkjqnroq ru vy k rqm nrjmlom ru mkqk ljk mk urssr nop qrjm urj npkjrjkj kmrq no mk kmrq mlm nks rolj lo nomkjnrj

7

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cag k vw klpk mm u qrj~kurjpk okm jlqkm $ '&%& %% % nqro

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