CHAPTER Secod Order Partia Differetia Equatio ad Fiite Differece Metods.. Itroductio A partia differetia equatio describes a reatio betwee a ukow fuctio ad its partia derivatives. Partia differetia equatios appear frequety i a areas of pysics ad egieerig. Moreover, i recet years we ave see a draatic icrease i te use of PDEs i areas suc as bioogy, ceistry, coputer scieces ad i ecooics. I fact, i eac area were tere is a iteractio betwee a uber of idepedet variabes, we attept to defie fuctios i tese variabes ad to ode a variety of processes by costructig equatios for tese fuctios. We te vaue of te ukow fuctio(s) at a certai poit depeds oy o wat appes i te viciity of tis poit, we sa, i geera, obtai a PDE. Te geera for of a PDE for a fuctio u(,y,z,,w) is F(,y,z,...,w,u, u,u y,u z,,u w,...,u y,...) = 0, (.) were,y,z,,w are te idepedet variabes, u is te ukow fuctio, ad u deotes te partia derivative u/. Te equatio is, i geera, suppeeted by additioa coditios suc as iitia coditios or boudary coditios. Te aaysis of PDEs as ay facets. Te cassica approac tat doiated te ieteet cetury was to deveop etods for fidig epicit soutios. Because of te iese iportace of PDEs i te differet braces of pysics, every ateatica deveopet tat eabed a soutio of a ew cass of PDEs was accopaied by sigificat progress i pysics. Tus, te etod of caracteristics iveted by Haito ed to aor advaces i optics ad i aaytica ecaics. Te Fourier etod eabed te soutio of eat trasfer ad wave propagatio, ad Gree s etod was istrueta i te deveopet of te teory of eectroagetis. Te ost draatic progress i PDEs as bee acieved i te ast 50 years wit te itroductio of uerica etods tat aow te use of coputers to sove PDEs of virtuay every kid, i geera geoetries ad uder arbitrary etera coditios. P a g e
Te fudaeta teoretica questio is weter te probe cosistig of te equatio ad its associated side coditios is we posed. Te Frec ateaticia Jacques Hadaard (865 96) coied te otio of we-posedess. Accordig to is defiitio, a probe is caed we-posed if it satisfies a of te foowig criteria. Eistece: Te probe as a soutio.. Uiqueess: Tere is o ore ta oe soutio.. Stabiity: A sa cage i te equatio or i te side coditios gives rise to a sa cage i te soutio. Equatios wit variabe coefficiets, equatios i copicated doais, ad oiear equatios caot, i geera, be soved aayticay. We sa terefore ave a etirey differet approac to sovig PDEs. Te etod is based o repacig te cotiuous variabes by discrete variabes. Tus te cotiuu probe represeted by te PDE is trasfored ito a discrete probe i fiitey ay variabes. Naturay we pay a price for tis sipificatio: we ca oy obtai a approiatio to te eact aswer, ad eve tis approiatio is oy obtaied at te discrete vaues take by te variabes. Te discipie of uerica soutio of PDEs is rater very recet. Te first aaysis (ad, i fact, aso te first foruatio) of a discrete approac to a PDE was preseted i 99 by te Gera-Aerica ateaticias Ricard Courat (888 97), Kurt Otto Friedrics (90 98), ad Has Lewy (905 988) for te specia case of te wave equatio. Icidetay, tey were ot iterested i te uerica soutio of te PDE (teir work preceded te era of eectroic coputers by aost two decades), but rater tey foruated te discrete probe as a eas for a teoretica aaysis of te wave equatio. Te Secod Word War witessed te itroductio of te first coputers tat were buit to sove probes i cotiuu ecaics. Foowig te war ad te rapid progress i te coputatioa power of coputers, it was argued by ay scietists tat soo peope woud be abe to sove uericay ay PDE. Tus, vo Neua evisioed te abiity to obtai og-ter weater predictio by odeig te ydrodyaica beaviour of te atospere. Tese epectatios tured out to be too optiistic for severa reasos: () May oiear PDEs suffer fro ieret istabiities; a sa error i estiatig te equatio s coefficiets, te iitia coditios, or te boudary coditios ay ead to a arge deviatio of te soutio. Suc difficuties are currety ivestigated uder te tite caos teory. () Discretizig a PDE turs out to be a otrivia task. It was discovered tat equatios of differet types soud be aded uericay differety. Tis probe ed to te creatio a ew brac i ateatics: uerica aaysis. P a g e
() Eac ew geeratio of coputers brigs a icrease i coputatioa power ad as bee accopaied by a icreased dead for accuracy. At te sae tie scietists deveop ore ad ore sopisticated pysica odes. Tese factors resut i a ever edig race for iproved uerica etods. A uerica soutio provides oy a approiatio to te eact soutio. Te ai idea of a uerica etod is to repace te PDE, foruated for oe ukow rea vaued fuctio, by a discrete equatio i fiitey ay ukows. Te discrete probe is caed a uerica scee. Tus a PDE is repaced by a agebraic equatio. We te origia PDE is iear, we obtai, i geera, a syste of iear agebraic equatios. We sa deostrate beow tat te accuracy of te soutio depeds o te uber of discrete variabes, or, aterativey, o te uber of agebraic equatios. Terefore, seekig a accurate approiatio requires us to sove arge agebraic systes. Tere are severa teciques for covertig a PDE ito a discrete probe. Te ost popuar uerica etods are te fiite differece etod (FDM) ad te fiite eeets etod (FEM). Bot etods ca be used for ost probes, icudig equatios wit costat or o-costat coefficiets, equatios i geera doais, ad eve oiear equatios. Tere is a o-goig debate o weter oe of te etods is superior to te oter. Our view is tat te FDM is siper to describe ad to progra (at east for sipe equatios). Te FEM, o te oter ad, is soeow deeper fro te ateatica poit of view, ad is ore feibe we sovig equatios i cope geoetries... Cassificatio of secod order partia differetia equatios Te study of partia differetia equatios is quite versatie. Terefore it akes sese to first caracterize te accordig to certai properties tat wi provide us wit guideies for ivestigatig te furter. We cosider te geera secod order partia differetia equatio of te for u u u u u A B C F, y, u,, 0. (..) = y y y Equatio (..) is said to be (a) Quasi-iear if te coefficiets A, B ad C are fuctios of, y, u, u ad u y. (b) Midy Quasi-iear if te coefficiets A, B ad C are fuctios of, y ad u. A iear partia differetia equatio ay be writte as Au Bu y Cu yy Du Eu y Fu G = 0 (..) P a g e
were A, B, C, D, E, F ad G are fuctios of ad y ad do ot vais siutaeousy. We assue tat te fuctio u ad te coefficiets are twice cotiuousy differetiabe i soe doai R. Te cassificatio of secod order partia differetia equatios is suggested by te cassificatio of quadratic equatio of coic sectios i te aaytic geoetry. Te equatio A By Cy D Ey F = 0 (..) is yperboic, paraboic, or eiptic accordig as B 4AC is positive, zero, or egative. Te cassificatio of secod order equatios is based upo te possibiity of reducig te equatio (..) by coordiate trasforatio to caoica or stadard for at a poit. I te case of two idepedet variabes, a trasforatio ca aways be foud to reduce te give equatio to te caoica for i te give doai. However, i te case of severa idepedet variabes, it is ot, i geera, possibe to fid suc a trasforatio. To trasfor te equatio (..) ito a caoica for, we itroduce a cage of idepedet variabes. Let te ew variabes be ξ = ξ(, y), η = η(, y) (..4) Assuig tat ξ ad η are twice cotiuousy differetiabe ad tat te Jacobia J ξ = η ξ η y y is ozero i te regio uder cosideratio, te ad y ca be uiquey deteried fro te syste (..4). Usig te cai rue, we ave u = u ξ u η ξ η u = u ξ u η y ξ y η y u = u ξ u ξ η u η u ξ u η ξξ ξη ηη ξ η u = u ξ ξ u ( ξ η ξ η ) u η η u ξ u η y ξξ y ξη y y ηη y ξ y η y u = u ξ u ξ η u η u ξ u η yy ξξ y ξη y y ηη y ξ y η yy Substitutig tese vaues ito te equatio (..) yieds 4 P a g e
A u B u C u D u E u F u = G * * * * * * * ξξ ξη ηη ξ η (..5) were A = Aξ Bξ ξ Cξ * y y * B = Aξ η B( ξ η y ξ yη ) Cξ yη y C = Aη Bη η Cη * y y * D = Aξ Bξ y Cξ yy Dξ Eξ y * E = Aη Bη y Cη yy Dη Eη y (..6a) (..6b) (..6c) (..6d) (..6e) * F * G = F (..6f) = G (..6g) Oe ca obtai te foowig reatiosip betwee A, B, C ad * * * A, B, C : * * * B 4 A C = J ( B 4 AC). It is apparet tat, uder tis cage of variabes, te sig of wit respect to B * 4A * C *. B 4AC reais ivariat Te cassificatio of equatio (..) depeds o te coefficiets A, B ad C. We sa, terefore rewrite te equatio (..) as Au Bu Cu = H (, y, u, u, u ) y yy y (..7) ad te equatio (..5) as * * * A u B u C u = H * ( ξ, η, u, u, u ) ξξ ξη ηη ξ η (..8) To reduce equatio (..7) to caoica for first assue tat oe of A, B ad C, is * * zero. Let ξ ad η be ew variabes suc tat te variabes A ad C i te equatio (..5) vais. Fro equatios (..6a), (..6c), we ave A = Aξ Bξ ξ Cξ = 0 (..9a) * y y C = Aη Bη η Cη = 0 (..9b) * y y Tese equatios are of sae type, terefore we ca write te i te for 5 P a g e
Aζ Bζ ζ Cζ = 0 y y or, ζ ζ A B C ζ = 0 y ζ y (..0) were ζ stads for eiter te fuctio of ξ or η. Aog te curve ζ = 0= costat, we ave dζ = ζ d ζ dy = 0 dy ζ Tus, =. d ζ Terefore, te equatio (..0) ca be writte as y y dy dy A B C = 0. d d (..) Te roots of tis equatio are dy B B 4AC = (..a) d A ad dy B B 4AC =. (..b) d A Tese equatios are te ordiary differetia equatios for te faiy of curves i te ypae aog wic ξ =costat ad η = costat ad are kow as te caracteristic equatios. Te itegras of tese equatios are caed caracteristic curves. Sice te equatios are first order ordiary differetia equatios, te soutios ay be writte as Hece, te trasforatio φ (, y) = c, φ (, y) = c, ξ = φ (, y), η = φ (, y) c = costat, c = costat. wi trasfor te equatio (..7) to a caoica for. 6 P a g e
Now, we cassify te equatio (..) as a) Hyperboic: If B 4AC > 0, te equatio (..) is caed a yperboic type partia differetia equatio. I suc case, te equatios (..a) ad (..b) yied two rea ad distict faiies of caracteristics. So, te equatio (..8) reduces to u = ξη H (..) * H were H =. Tis for is caed te first caoica for of te yperboic * B equatio. If te ew idepedet variabes α = ξ η, β = ξ η are itroduced, te equatio (..8) is trasfored ito u u = H (,, u, u, u ) α β. αα ββ α β Tis for is caed te secod caoica for of te yperboic equatio. Hyperboic partia differetia equatios usuay arise we waves or vibratios occur i a pysica syste, ad ateatica odeig of suc a syste ivariaby ivoves te soutio of a yperboic equatio. Soe ost coo eapes of secod order yperboic equatios wic are used for uerica iustratio i te tesis are:. Wave Equatio i poar co-ordiates.. Teegrap Equatio.. Va der Po Equatio. 4. Dissipative Equatio etc. b) Paraboic: If B 4AC = 0, te equatio (..) is caed a paraboic type partia differetia equatio. I tis case equatios (..a) ad (..b) coicide. Tus tere eists oe rea faiy of caracteristics, ad we obtai oy a sige itegra ξ = costat (or η = costat). Te, equatio (..8) reduces to uξξ = H( ξ, η, u, uξ, uη ) or uηη = H4( ξ, η, u, uξ, uη ) (..4) Tis is caed te caoica for of te paraboic equatio. Paraboic equatios describe eat fow ad diffusio processes. 7 P a g e
c) Eiptic: If B 4AC < 0, te equatio (..) is caed a eiptic type partia differetia equatio. Cosequety, te quadratic equatio (..) as o rea soutios, but it as two cope cougate soutios. Sice ξ ad η are cope, we itroduce ew variabes α = ( ξ η ), β = ( ξ η ), i so tat ξ = α iβ, η = α iβ. Usig trasforatio, equatio (..8) becoes u u = H (,, u, u, u ) 5 α β. αα ββ α β Tis is caed te caoica for of te eiptic equatio. Eiptic partia differetia equatios are usuay associated wit steady- state or equiibriu probes i wic tere is ore ta oe idepedet variabe. Eigevaue Based Cassificatio of Partia Differetia Equatios If tere are idepedet variabes,,...,, a geera iear partia differetia equatio of secod order as te for u u Lu = a b cu = 0 i i i, = i i= i (..5) Te coefficiet atri of igest order derivatives is A = [ a ]. Te caracteristic equatio of A wi ave roots ad eigevaues. Caracterizatio of te iear partia differetia equatio (..5) is based o foowig: If ay eigevaue is zero, te partia differetia equatio (..5) is paraboic. If a eigevaues are o-zero ad oe eigevaue as opposite sig, te partia differetia equatio (..5) is yperboic. If a eigevaues are ozero avig sae sig, te partia differetia equatio (..5) is eiptic. i 8 P a g e
Te sipest eapes of tree types are: (i) Wave Equatio u u = t (..6) B 4AC = 4 > 0. Hece, te equatio (..6) is yperboic. (ii) Heat Fow Equatio u u = t (..7) Here, B 4AC = 0. Hece, te equatio (..7) is paraboic. (iii) Lapace equatio u u y u = = 0 (..8) Here, B 4AC = -4 < 0. Hece, te equatio (..8) is eiptic. (iv) u u = 0 y (..9) Here, B 4AC = 4. Hece, te equatio (..9) is eiptic for > 0, paraboic for = 0, yperboic for < 0 (v) u uyy = uz (..0) Here, 0 0 A = 0 0. 0 0 0 Te caracteristic roots of A are,, 0. Hece te equatio (..0) is paraboic. (vi) u u u = 0 (..) yy zz Here 0 0 A = 0 0 0 0 Te caracteristic roots of A are,,. Hece te equatio (..) is eiptic. 9 P a g e
(vii) u uyy uzz = utt (..) 0 0 0 0 0 0 Here, A =. 0 0 0 0 0 0 Te caracteristic roots of A are,,, -. Hece, te equatio (..) is yperboic. Type of Coditios Iitia Vaue Probe: A iitia vaue probe is oe i wic te depedet variabe ad possiby its derivatives are specified iitiay (i.e. at tie t = 0) or at te sae vaue of idepedet variabe i te equatio. Iitia vaue probes are geeray tiedepedet probes. Iitia Vaue Probes ca furter be cassified as Pure Iitia Vaue Probes or Iitia Boudary Vaue Probes. Pure Iitia Vaue Probe (Caucy Probe): i) u u = < <, t 0 t aog wit te iitia coditio u(,0) = f ( ), < <. ii) u u = < <, t 0 t aog wit te iitia coditios u(,0) = f ( ) u (,0) = g( ), t < < a) Iitia Boudary Vaue Probe : i) u u = t 0, t 0 0 P a g e
aog wit te iitia coditio u(,0) = f ( ) 0 < ad te boudary coditio u(0, t) = α ( t), u (0, t ) = α ( t ), t 0. ii) u u = t 0, t 0 aog wit te iitia coditio u(,0) = f ( ) 0 ad te boudary coditios u(0, t) = α ( t), u(, t) = α ( t), t 0. Boudary Vaue Probes: A boudary vaue probe is oe i wic te depedet variabe ad possiby its derivatives are specified at te etree of te idepedet variabe. For steady state equiibriu probes, te auiiary coditios cosist of boudary coditios o te etire boudary of te cosed soutio doai. Tere are tree types of boudary coditio: i. Diricet boudary coditio: Diricet boudary coditio is a type of boudary coditio, aed after Joa Peter Gustav Leeue Diricet (805 859). We iposed o a ordiary or a partia differetia equatio, it specifies te vaues a soutio eeds to take o te boudary of te doai. For eape if a iro rod ad oe ed ed at absoute zero te te vaue of te probe woud be kow at tat poit i space. Te questio of fidig soutios to suc equatios is kow as te Diricet probe. It is aso caed first type boudary coditio. ii. Neua boudary coditio: Te Neua boudary coditio is a type of boudary coditio, aed after Car Neua. We iposed o a ordiary or a partia differetia equatio, it specifies te vaues tat te derivative of a soutio is to take o te boudary of te doai. For eape if oe iro rod ad eater at oe ed te eergy woud be added at a costat rate P a g e
but te actua teperature woud ot be kow. It is aso caed secod type boudary coditio. iii. Mied boudary coditio: Te iear cobiatio of Diricet ad Neua boudary coditios specified o te boudary is kow as ied boudary coditio. Mied boudary coditios are aso kow as Caucy boudary coditios. A Caucy boudary coditio iposed o a ordiary or a partia differetia equatio specifies bot te vaues a soutio of a differetia equatio is to take o te boudary of te doai ad te ora derivative at te boudary. It is aso caed Robi s boudary coditio... Matri agebra We preset ere soe of te basic resuts of Matri teory, wic ep i te study of te uerica soutio of partia differetia equatios by fiite differece etod. Te fiite differece etod yieds a syste of iear ad/or o-iear siutaeous agebraic equatios tat ca be deoted by a atri syste. Te iterative etods for sovig suc syste of equatios deped o certai properties of atrices. Let A = [ ] be a atri of order ( ) te A is said to be: a i Rea Matri if te eeets a of A are a rea ubers. i Cou atri if A as oy oe cou i.e. ( ) i size. We te uquaified ter Vector is used, it eas te cou atri. Row atri if A as oy oe row i.e. ( ) i size. Nu atri if every eeet of A is zero. Siguar if det( A ) = 0 ad No-Siguar if det( A) 0 i.e. if tere eists a uique atri B suc tat AB = I = BA, were B = A is te iverse of A ad I is te Idetity Matri. Diagoa if its oy o-zero eeets ie o te diagoa. Sparse if ay of its eeets are zero. Syetric if A T = A, i.e. if ai = a for a i ad i. P a g e
Skew-Syetric if A = A T i.e. if ai = a for a i ad i. Tri-diagoa if a i = 0 for i >. Upper triaguar if a i = 0 for i >. Lower triaguar if a i = 0 for i <. Diagoay doiat if = i a a for a i. ii Strog diagoay doiat if i > = i a a for a i. ii Weak diagoay doiat if = i i a a i =,,..., ii ad for at east oe i i a ii > = i a i Bock diagoa if A = D 0 0 D D P a g e
were eac D ( =,,, ) is a square atri of order k wit k k k =. Two atrices A ad B are caed coutative if AB = BA. Tey te possess te sae set of eigevectors (discussed ater i tis capter). Two square atrices A ad B of te sae order are said to be siiar atrices if a o-siguar atri S ca be deteried suc tat B = S AS. Siiar atrices ave sae rak ad sae eige vaues. Ortogoa if A = A T i.e. A A = I = AA Nora if AA H H Uitary if AA = A A = I. H T T. H = A A, were A H deotes te cougate traspose of A. Heritia if A H = A H, ad Skew-Heritia if A = A. No-egative atri (Positive) if A O( > O ), were O is te u atri i.e. if a 0( > 0) for a i ad. i H Positive Defiite, if > 0 H v Av 0 for ay vector v 0. H v Av, were v = ( v) T, ad Positive Sei-defiite if If A is o-siguar ad positive defiite, te B = A H A is Heritia ad positive defiite. If A is syetric ad positive defiite, te its eige vaues are a positive. Perutatio Matri if A cosists of eacty oe o-zero eeet, aey, uity, i eac row ad eac cou. Eape: 0 0 0 0 0 0 0 0 0 0 0 0 is a perutatio atri. Reducibe if tere eists a perutatio atri P suc tat T F G PAP = O H 4 P a g e
were O is te u atri ad F, G ad H are square sub-atrices (Youg [86] ad Varga [8]). A atri is said to be irreducibe atri if it is ot reducibe. Irreduciby diagoay doiat atri if A is irreducibe atri ad weaky diagoay doiat. Teore: If A is irreduciby diagoay doiat atri, te det( A) 0 ad oe of te diagoa eeets of A vaises. Teore: If A is a Heritia, stricty diagoay doiat atri wit positive rea diagoa etries, te A is Positive defiite. Rak of a atri: A atri A is said to be of rak r if it as at east oe o-zero ior of order r, ad every ior of order iger ta r is zero. Trace of a atri: Te Trace of a square atri is te su of te eeets o its ai diagoa. Bad widt of a atri: For a atri A = [ a i ], if a i = 0 for i > w, te w is caed te bad widt of te atri A. Suc a atri is caed a Baded Matri of bad widt w. Tridiagoa atrices are baded atrices of bad widt. Eige vaues ad Eige vectors Defiitio: A eige vaue (or Caracteristic root) of a atri A is a rea or cope uber λ wic for soe ozero vector v satisfies te equatio ( A λ I) v = 0. (..) Ay ozero vector v satisfyig te above equatio (..) is caed a eige vector of te atri A correspodig to eige vaue λ. I order for (..) to ave a otrivia soutio for vector v te det( A λi ) ust be zero. Hece, ay eige vaue λ ust satisfy det( A λ I) = 0. (..) Equatio (..) wic is caed te caracteristic equatio of A, is a poyoia of degree i λ wic ca be writte as a0 aλ a λ ( ) λ = 0. Sice te coefficiet of λ is ot zero, te above equatio as aways roots (cope or rea) wic are te eige vaues of te atri A, aey λ, λ,, λ (ot ecessariy a distict), eac of te possessig a uique correspodig eige vector. 5 P a g e
Defiitio: If A is a atri te te spectra radius of A is defied as ρ ( A ) = a λ, were S A is te set of a eige vaues of A (i.e. te spectru of A). We ote te foowig facts about eige vaues ad eige vectors: Eige vectors correspodig to distict eige vaues of A are ieary idepedet. Eige vaues of a triaguar atri A are te diagoa etries. If λ 0 is a eige vaue of a osiguar atri A ad v is te correspodig eige vector i.e. Av = λv te A v = v, wic eas λ is a eige vaue of A. A ad A λ ave sae eige vectors. T Sice det( A λi) = det( A λi), A ad A T ave sae caracteristic poyoia ad ece sae eige vaues. A syetric atri of order as rea eige vaues ad utuay ortogoa eige vectors. Eige vaues of syetric ad Heritia atrices are rea. Eige vaues of skewsyetric ad skew-heritia atrices are purey iagiary. For every square atri A, te su of its eige vaues equas its trace. For every square atri A, te product of its eige vaues equas its deteriat i.e. det( A) = N λi i = were λ, λ,..., λ N are eige vaues of A. Every square atri satisfies its caracteristic equatio. Tis is referred to as Cayey Haito Teore. λ S A Teore: Te eige vaues of a tridiagoa atri a b c a b O c a b A = O c a b c a NXN 6 P a g e
sπ are λs = a bc cos, s = () N ; were a, b ad c ay be rea or cope. Tis cass of atrices arises cooy i te study of stabiity of te fiite differece processes, ad a kowedge of its eige vaues eads iediatey ito usefu stabiity coditios. Teore: (Youg [86]) A atri A is irreducibe if ad oy if = or give ay two distict itegers i ad wit i,, te a 0. Usefu teores o bouds for eige vaues: Gercgori s First Teore: Te eige vaue of te square atri A wic as argest oduus, caot eceed te argest su of te odui of te eeets aog ay row or ay cou. Teore (Gercgori s Circe Teore or Brauer s Teore): Let Ps be te su of te oduii of te eeets aog s t row ecudig te diagoa eeet a ss. Te eac eige vaue of A ies iside or o te boudary of ateast oe of te circes λ a = P. Teore: If p of te circes of Gercgori s circe teore for a coected doai tat is isoated fro te oter circes, te tere are precisey p eige vaues of te atri A witi tis coected doai. I particuar, a isoated Gercgori s circe cotais eacty oe eige vaue. Teore: (Perro ad Frobeius Teore) (Varga[8]) Let A O atri, te i) A as a positive rea eige vaue equa to its spectra radius. ii) To ρ ( A), tere correspods a eige vector v > 0. iii) ρ ( A) icreases we ay etry of A icreases. iv) ρ ( A) is a sipe eige vaue of A. i ss be a irreducibe Perro ad Frobeius Teore ca aso be eteded for reducibe atrices as foows: Teore: Let A O be a atri, te i) A as a o-egative rea eige vaue equa to its spectra radius. Moreover, tis eige vaue is positive uess A is reducibe ad te ora for of A is stricty upper triaguar. s 7 P a g e
ii) ρ ( A) does ot decrease we ay etry of A is icreased. Teore: If A = [ a ] is a Heritia stricty diagoay doiat or irreducibe i diagoay doiat atri wit positive rea diagoa etries, te A is positive defiite. Vector ad Matri ors A or serves as a easure of te size or agitude of a give atri. Vector or: Te o-egative quatity v is a easure of size or egt of a vector v satisfyig te foowig aios: a) v > 0 for v 0 ad v = 0 v = 0. b) cv = c v for ay arbitrary cope uber c. c) v w v w for vectors v ad w (triage iequaity). Soe iportat vector ors are: i. Absoute Nor ( or): v =... = v v v v ii. Eucidea Nor ( or): i= i v ( ) / =... = v v v v i i= / iii. Maiu Nor ( or): 8 P a g e v = Ma v i were v, v,..., v are copoets of vector v. i Matri or: Te or of a atri A is a rea o-egative uber wic satisfy te foowig properties: a) A = 0 if ad oy if A = O. b) ca = c A for ay cope uber c. c) A B A B for ay atrices A ad B.
d) AB A B for ay atrices A ad B. Soe iportat atri ors are defied as foows: i. Te Eucidea or (Frobeius or): Eucidea Nor of a atri A = [ a ] is give by i A / = ai. i, ii. Te Spectra or (Hibert or or L or): H A = [ ρ ( A A)] / were H ρ( A A) a = λi i= H. te eige vaues of A A is te spectra radius of A H A, ad λ, λ,..., λ are If te atri A is Heritia, te H ρ( A A) = ρ( A ) = ρ( A) so tat A = ρ( A). iii. Te L or: A = a ai i (aiu absoute cou su). i A = a a (aiu absoute row su). i iv. Maiu Nor ( L or): A = a a. i, i Sice atrices ad vectors appear togeter ofte, a reatio betwee teir ors is required to be itroduced. A atri or A is said to be copatibe wit a vector or v, if Av A v for a v 0. Te atri or copatibe wit te vector or ca be costructed as 9 P a g e
A = sup A v 0 v wic is equivaet to A = sup v = Av. Teore: (Evas [47]) If A is a atri of order, te A ρ( A). () Defiitio: A sequece of atrices A, ( ) i A A = 0. () () A, A,... is said to coverge to a atri A if Defiitio: Let A be a rea or cope atri. Te A is coverget (to zero) if te sequece of atrices A, A, A,... coverges to te u atri O ad is diverget oterwise. Teore: If A is a atri, te A is coverget if ad oy if ρ ( A) <, were ρ ( A) is te spectra radius of A..4. Iterative etods for iear ad o-iear systes i. Liear Syste Te appicatio of fiite differece etod o differetia equatio to obtai te uerica soutio yieds a syste of siutaeous equatios wic ca be represeted i atri otatio as A = b (.4.) were A is a ( ) o-siguar coefficiet atri, (i case of fiite differece etod, is te uber of iterior es poits), b is a cou vector correspodig to boudary coditios, is a soutio vector. We te iear syste is sa eoug to be ecooicay stored i te eory of a coputer, it is geeray ost efficiet to use a direct etod to obtai te soutio. Large sparse systes, o te oter ad, wic are frequety associated wit te uerica soutio of partia differetia equatios, ca geeray be soved ore efficiety use a iterative etod, provided tat fu advatage is take of te sparseess of te coefficiet atri. Wie Direct etods produce te eact soutio after a fiite uber of steps, iterative etods yied te eact soutio as a iit of sequece of approiate soutios. Startig wit a iitia approiatio to te actua soutio ad te obtaiig better ad better 0 P a g e
approiatios fro a coputatioa cyce repeated tie ad agai, for acievig a desired accuracy. Tus, i iterative etod, te aout of coputatio depeds o te degree of accuracy required. For arge syste of equatios, iterative etods produce soutios faster ta te direct etods as te prograig ad data adig is uc siper for iterative etods ta for te direct etods. Iterative etods ca be eteded to sove te syste of o-iear equatios. However, te oy disadvatage of Iterative etods is te probe of seectig a good iitia vector to start te iterative process, wereas o iitia vector is required wit Direct etods. Iterative etods ca furter be cassified ito two casses: Siutaeous cass ad Successive cass. Siutaeous cass: I te siutaeous iterative etods, a te eeets of te approiate soutio are odified at te sae tie i.e. (k)t iterate is a fuctio of kt ad earier iterates oy. Hece, te order i wic te soutios are cacuated is ot sigificat. For eape, Jacobi Iteratio Metod is a siutaeous etod. Successive cass: I te successive iteratio etods, te approiate soutio vaues are odified usig te atest avaiabe vaues of te iterates oe after te oter i.e. as soo as we get a ew vaue, we discard te od vaue ad use te ew vaue so tat soe of te eeets of te (k)t iterate are fuctios of (k)t ad earier iterates. I tis case, te orderig of te es poits is terefore, very iportat. For eape Gauss Sieda Iterative Metod is a successive etod. A iterative tecique to sove a iear syste of te for (.4.) starts wit a iitia ( 0) ( 0) () () approiatio to te soutio ad geerates a sequece of vectors,,, wic coverge to. Suc iterative etods covert te give syste A = b ito a equivaet syste of te for = Bc were B is a square atri of order ad c is a vector wit copoets. Te coice of ( 0) is argey arbitrary ad te sequece of iterates is geerated fro (k) (k) = B c. (.4.) A sufficiet coditio for te covergece of (.4.) is B <. (.4.) Te iteratio procedure is cotiued uti soe criterio for stoppig is satisfied; two suc criteria are ad ( ) k k < ε for soe k (.4.4) P a g e
( k ) ( k ) < ε ( k ) (.4.5) were ε > 0 is soe toerace. Te usua or used is te L or for vectors = a ; r =,,...,. r r Te absoute error criterio (.4.4) is suitabe for use we te variabes ( i i =,..., ) are a of te sae order of agitude, wie te reative error criterio (.4.5) is suitabe for use we te i ave widey differig orders of agitude. Let te atri A be o-siguar wit te decopositio A = D L U were D is te ai diagoa eeets of A, L ad U are stricty ower ad upper ( ) triaguar atrices respectivey as sow beow: a 0 0 0 0 0 a 0 0 0 D =, 0 0 0 0 0 a 0 0 0 0 0 a 0 0 0 0 L ad = a 0 0 0 a a a 0 U 0 a a a 0 0 a a = 0 0. 0 a 0 0 0 0 Te, (.5.) ay be writte as (D L U) = b. We ow defie te foowig basic iear statioary iterative etods: a) Te Jacobi (J) Iterative Metod Te Jacobi Iteratio etod is derived as (k ) (k) D = L U b or (k ) (k) = D L U D b P a g e
or (k ) (k) = H c were H D ( L U ) k = 0,,,... =, caed te Jacobi Iteratio Matri, ad c = D b. Tis is aso caed te etod of siutaeous dispaceet. b) Te Gauss-Seide (GS) Iterative Metod Gauss-Sieda Iteratio etod is derived as: (k ) (k) D L = U b (k ) (k) or = ( ) ( ) D L U D L b or (k ) (k) = H c were H ( D L) = U is caed te Gauss-Sieda Iteratio atri, ad c = D L b. Sice, aii 0, det( L D) = aa... a 0 L D is o-siguar. Te Gauss-Sieda etod as coputatioa advatage over te Jacobi etod tat it does ot require te siutaeous storage of vaues k ad k i te course of te coputatio. Tus, Gauss-Sieda etod is aso caed etod of successive dispaceets. It is obvious tat te Gauss-Sieda etod coverges faster ta te Jacobi-Iteratio etod. c) Te Jacobi Over-Reaatio (JOR) Iterative Metod It is a geeraisatio of te Jacobi etod, give by i.e. ( ) ( ω ) D = ω L U b D (k ) (k) (k) (k ) D (k) is weigted su of ( L U ) b ad (k) D. Tis yieds (k ) (k) = H c were ω ( ω) H = D L U I, c ω = D b. Te atri H is caed JOR-Iteratio Matri. Te quatity paraeter. Tis etod is cosistet provided ω 0. ω is caed te reaatio We ω =, te JOR etod reduced to Jacobi-Iteratio etod. If ω >, it is caed over-reaatio etod. If ω <, it is caed uder-reaatio etod. P a g e
d) Te Successive Over-Reaatio(SOR) Iterative Metod Te successive over-reaatio iterative etod is te geeraizatio of te Gauss-Sieda etod, give by Tis yieds ( ω ) (k ) (k ) (k) (k) D = ω L U b D. ( ) ω ω ω ω D L (k ) = U D (k) b or (k ) (k) = H c were = ( ω ) ω ( ω) H D L U D ad c = ω D ω L b. Te quatity ω is caed te reaatio factor. If ω =, te SOR etod reduces to te Gauss-Sieda Iteratio etod. If If < ω < 0 < ω <, we ave over-reaatio etod., we ave uder-reaatio etod. Tis odificatio of te Gauss-Sieda iteratio etod wi aow us to coose ω, so tat te soutio coverges faster, ad a optiu covergece rate ca be acieved, if te optia ω is cose. Covergece aaysis of iterative etods Teore: Te iteratio etod of te for (k ) (k) = H c k = 0,,,... coverges to te eact soutio for ay iitia vector if H <. Teore: A ecessary ad sufficiet coditio for a iterative etod (k ) (k) = H c k = 0,,,... to coverge for a arbitrary iitia approiatio te spectra radius of H. (0) is tat ( H ) ρ <, were ( H ) ρ is Defiitio: Te rate of covergece of a iterative etod (k ) (k) H c =, k = 0,,,... 4 P a g e
is give by ν og ρ ( H ) = 0. Defiitio: For ay coverget iterative etod (k ) (k) = H c k = 0,,,... te average rate of covergece R H after iteratios is defied by og R H H = were H is te spectra or of H. If R ( H ) R ( H ) =, te H is reativey faster for iteratios ta H. Teore: Let A be a syetric atri wit positive diagoa eeets. Te te SOR etod coverges, if ad oy if, A is positive defiite ad 0 < ω <. Note: i. Te optia ω, deoted as ω, for aiu rate of covergece is give by ω opt =. µ opt ii. Rate of covergece of te Gauss-Sieda Iteratio etod is twice te rate of covergece of Jacobi Iteratio etod. Teore: If A is a stricty diagoay doiat atri, te bot te Jacobi Iteratio scee ad Gauss-Sieda Iteratio scee coverge for ay startig vector. Teore: If A is syetric ad positive defiite, te its Gauss-Sieda iteratio atri as spectra radius ess ta uity i.e. i tis case te Gauss-Sieda iterative etod aways coverges. Teore: If te Jacobi etod coverges, te te Jacobi Reaatio etod coverges for 0 < ω. Teore: Let A be a irreducibe atri wit weak diagoa doiace, te a) Te Jacobi etod coverges ad te Jacobi reaatio etod coverges for 0 < ω. b) Te Gauss-Sieda etod coverges ad te SOR etod coverges for 0 < ω. 5 P a g e
ii. No-Liear Syste Te deteriatio of soutio of o-iear equatio is of iterest ot oy to ateaticias but aso to scietists ad egieers. No-iear equatios wic caot be soved agebricay are soved usig uerica teciques. Te ost coo procedure for sovig o-iear equatios is to foruate te as a iterative sequece of iear equatios to wic we ca sti appy soutio etods for iear probes. a) Newto-Rapso Metod Cosider te o-iear equatio f ( ) = 0. Let k be a approiatio to te root of tis equatio. Let be a icreet i suc tat is a eact root. Te, k f ( ) = 0 k (.4.6) Epadig i Tayor series about te poit k, we get f ( k ) f ( k ) ( ) f ( ξ ) = 0, k < ξ < (.4.7)! Negectig te secod powers of, we obtai f ( ) f ( ) = 0, f ( k ) wic gives =. f ( ) Hece, we obtai te iteratio etod k k k f ( k ) = k k = k, k = f ( ) 0,,,... k (.4.8) Defiitio: A sequece of iterates { } is said to coverge to te root ξ if i ξ = 0 or i k = ξ. k k Defiitio: A iterative etod is said to be of order p or as te rate of covergece p, if p is te argest positive rea uber for wic tere eists a fiite costat C 0 suc tat ε C ε p k were ε k = k ξ is te error i te k t iterate. Te costat C is caed te asyptotic error costat ad usuay depeds o te derivatives of f ( ) at = ξ. 6 P a g e k k k
To estabis te covergece of te Newto-Rapso etod, fro (.4.7) we ca write ad usig (.4.8), we get Puttig ε = So, we ave k k f ( ) were C =. f ( ), we get f ( ξ) k = ( k ). f ( ) f ( ξ) = ε k. f ( ) as Tis sows tat eac error is rougy proportioa to te sequece of te previous error. Tis eas tat te uber of correct decia paces rougy doubes wit eac approiatio ad ece Newto-Rapso etod as a quadratic rate of covergece. Teore: Assue tat f ( ) 0 ad f ( ) does ot cage sig i te iterva [ a, b] f ( a) suc tat f ( a). f ( b ) < 0. Te if ( b a) ad, te te Newtof ( a ) < f ( b) ( b a) f ( b ) < Rapso etod coverges for a arbitrary iitia poit i te iterva [ a, b]. b) Geeraised Newto-Rapso etod Let = α be a root of f ( ) = 0, wit utipicity, te it is aso a root of f ( ) = 0 wit utipicity, of f ( ) = 0 wit utipicity ad so o. Hece, if te iitia approiatio is sufficiety cose to te root, te epressios 0 wi ave te sae vaues. 0 f ( k ) f ( ξ ) ( k ) = ( k ) f ( ) f ( ) k ε ε k Cε k k k For = α to be a root of f ( ) = 0 wit utipicity, te iteratio foruae is give by k f ( 0) f ( 0) f ( 0) 0, 0 ( ), 0 ( ),... f ( ) f ( ) f ( ) 0 k 0 k 0 7 P a g e
wic is caed te geeraised Newto-Rapso foruae. It reduces to te Newto- Rapso foruae for =. c) Te No-Liear Over Reaatio Metod (NLOR) Let us cosider a syste of N o-iear equatios wit cotiuous first derivative ters ad itroduce a reaatio factor ω. Te NLOR etod is give by f p were f pq =. u Tis etod is a successive iteratio etod. Wit ω =, it is caed te geeraized Newto-Rapso etod. I additio, if f is iear te tis etod reduces to te SOR etod. Te covergece criteria for te NLOR etod is te sae as tat for te SOR etod wit te coefficiet atri A repaced by te Jacobia atri of te equatio f ( u, u,..., u ) = 0, give by J ( f ( f p u k ) k pq ) = = D k Lk U k, ) q were, D k : is te diagoa atri of Jacobia atri, p, q = ( N L : is te ower tridiagoa atri of Jacobia atri, k q u u u f ( k ) = k k f ( ) f ( u, u,..., u ) = 0, p = () N p ( k ) ( k ) ( k ) ( k ) ( k ) N = u ( k ) ( k ) ( k ) f u u un N ω f ( u, u,..., u ) (,,..., ) ω f ( u, u,..., u ) (,,..., ) ( k ) ( k ) ( k ) ( k ) ( k ) N = u ( k ) ( k ) ( k ) f u u un ω f ( u, u,..., u ) (,,..., ) ( k ) ( k ) ( k ) ( k ) ( k ) N N N = un ( k ) ( k ) ( k ) f NN u u un k p N p 8 P a g e
U : is te upper triaguar atri of Jacobia atri. k For covergece, te Jacobia atri at eac stage of te iteratio ust ave te sae properties as i te SOR etod..5. Fiite differece etod Te fiite differece teciques are based upo te approiatios tat perit repacig differetia equatios by fiite differece equatios. Tese fiite differece approiatios are agebraic i for, ad te soutios are reated to grid poits. Tus a fiite differece etod basicay ivoves tree steps:. Dividig te soutio ito grids of odes.. Approiatig te give differetia equatio by fiite differece equivaece tat reates te soutios to grid poits.. Sovig te differece equatios subect to te prescribed boudary coditios ad/or iitia coditios. f() P B A 0-0 0 Figure.5. Estiates for te derivative of f() at P by usig forward, backward ad cetra differeces. Give a fuctio f() sow i figure.5., we ca approiate its derivative, sope or taget at a poit P by te sope of te arcs PB, PA, or AB, for obtaiig te forwarddifferece, backward-differece, ad cetra- differece foruas respectivey. 9 P a g e
Were forward-differece forua is give by f '( ) 0 f ( ) f ( ) 0 0, backward-differece forua is give by f '( ) 0 f ( ) f ( ) 0 0, ad cetra-differece forua is give by f f ( ) f ( ) 0 0 '( 0 ). Te approac used for obtaiig above fiite differece equatios is Tayor series 4 f ( 0 ) = f ( 0 ) f '( 0 ) f ''( 0 ) f '''( 0 ) O( ) (.5.) 6 =, (.5.) 6 4 ad f ( 0 ) f ( 0 ) f '( 0 ) f ''( 0 ) f '''( 0 ) O( ) 4 were O ( ) is te error obtaied by trucatig te series. Fro (.5.) ad (.5.), we obtai 0 0 0 f ( ) f ( ) = f '( ) O f ( 0 ) f ( 0 ) f '( 0 ) = O( ) (.5.) wic is te cetra-differece forua. Note tat O( ) is te oca trucatio error. Te forward-differece ad backward-differece foruas coud be obtaied by rearragig (.5.) ad (.5.) respectivey, ad we ave ad f ( ) f ( ) = for forward differece (.5.4) 0 0 f '( 0 ) O f ( ) f ( ) = for backward differece. (.5.5) 0 0 f '( 0) O 0 P a g e
We ca fid tat te trucatio errors of tese two foruas are of order O ( ). Upo addig (.5.) ad (.5.), we get f ( 0 ) f ( 0 ) = f ( 0 ) f ''( 0 ) O 4 f ( 0 ) f ( 0 ) f ( 0 ) f ''( 0 ) = O ( ). (.5.6) Higer order fiite differece approiatios ca be obtaied by takig ore ters i Tayor series epasio. Net, cosider a fuctio U(,t) defied i te regio {(, t ) : 0, t 0} Ω < < >. t-directio (,t ) k k.. 0 4 Figure.4. N (N) -directio To appy te differece etod to fid te soutio of te fuctio U(, t), we divide te soutio regio i -t pae ito equa rectages or eses of sides ad k. So tat =, =0()N ad t = k, 0 < < J, N ad J beig positive itegers ad (N)=. We repace te regio Ω by a set of grid poits (, t ) as sow i te figure.5.. Let te vaues of U (, t ) at te grid poit (, t ), be deoted by U. Te Tayor series epasios of fuctio U (, t ) about te poit (, t ) are give by P a g e
4 U± = U( ±, t ) = U ± U U ± U O ( ), 6 (.5.7) k k 4 U = U(, t ± k) = U ± kut U tt ± U ttt O ( k ). 6 (.5.8) Usig te Tayor series epasios (.5.7) ad (.5.8), we write approiate epressios U U U U U U =, (.5.9) U U U U = O( k ). (.5.0) k k O t Siiary, fro (.5.7) ad (.5.8) te approiatios for U U O ( ) U ad U tt are U U U U U U =, (.5.) U U U U U U =. (.5.) k k O tt ( k ) A tese differece foruas are etreey usefu i fidig uerica soutios of te first or secod order partia differetia equatios. Suppose U represets te eact soutio of a partia differetia equatio L(U) = 0 wit te idepedet variabes ad t, ad u be te eact soutio of te correspodig fiite differece equatio F ( u ) = 0 at te grid poit (, t ). Defiitio: Te differece U u discretizatio) error. is caed te cuuative trucatio (or Tis error ca geeray be iiized by reducig te es sizes ad k. However, tis error depeds ot oy o ad k, but aso o te uber of ters i te trucated series wic is used to approiate eac partia derivative. Aoter kid of error is itroduced we a partia differetia equatio is approiated by a fiite differece equatio. Defiitio: If te eact fiite differece soutio u is repaced by eact soutio U of partia differetia equatio at te grid poit (, t ), te te vaue F ( U ) is caed te oca trucatio error (LTE) at grid poit (, t ). P a g e
Let us ow iustrate tis cocept by cosiderig a sipe eape of a yperboic partia differetia equatio u tt = u (.5.) Te eact soutio U of partia differetia equatio (.5.) satisfies U U U U U U O k = k O( ) or U U U U U U = LTE k Negectig te trucatio ters, we ay rewrite above equatio as ( ) ( λ ) U = λ U U U U (.5.4) k were λ = is te es ratio paraeter. It is obvious tat te differece scee (.5.4) represets te differetia equatio (.5.) correct to order k ad i t ad variabes, respectivey. We say tat te k. Tus differece scee as order of accuracy or te trucatio error of order as 0, k 0 te oca trucatio error teds to 0 ad te differece ad differetia equatios becoe equivaet. Defiitio: Te fiite differece scee ad te partia differetia equatio are said to be cosistet if oca trucatio error teds to zero as ad k ted to zero. I geera fiite differece equatios caot be soved eacty because te uerica coputatio is carried out oy up to fiite uber of decia paces. Cosequety aoter kid of error is itroduced i te fiite differece soutio durig te actua process of coputatio. Tis kid of error is caed roud-off error, ad aso depeds upo te type of coputer used. I practice te actua coputatioa soutio is u, but * ot u, so tat te differece u u is te roud-off error at te grid poit (, t ). Te roud-off error depeds aiy o te actua coputatioa process ad te fiite differece itsef. I cotrast to discretizatio, te error roud-off caot be ade sa by aowig ad k to ted to zero. Tus te tota error ivoved i te fiite differece aaysis at grid poit (, t ) is ( ) * * U u = U u u u = discritizatio error roud-off error * P a g e
Decreasig te trucatio error by usig a fier es ay resut i icreasig te roudoff error due to te icreased uber of aritetic operatios. A poit is reaced were iiu tota error occurs for ay particuar agorit usig ay give word egt. Tis is iustrated i figure.5.. Figure.4. Error as a fuctio of te es size. Defiitio: Te fiite differece scee is said to be stabe if te roud off errors are sufficiety sa for a as, tat is te growt of roud-off error is cotroed. I oter words, if te errors decay ad evetuay dap out, te uerica scee is said to be stabe. If, o te cotrary, te errors grow wit tie te uerica soutio diverges fro te true, correct aswer ad tus te uerica scee is said to be ustabe. A eutray stabe scee is oe i wic errors reai costat as te coputatios are carried forward. By covergece, we ea tat te eact soutio of partia differece equatio approiates te eact soutio of partia differetia equatio as ad k bot approac to zero. I oter words, as te step-size srik, te fiite differece scee soutio ust iprove, utiatey covergig to te correspodig soutio of te origia partia differece equatio at every poit of te doai. 4 P a g e
La (954) proved a rearkabe teore wic estabises te reatiosip betwee cosistecy, stabiity ad covergece for te fiite differece agorit. Teore: (La s Equivaece Teore) Give a we posed iear iitia vaue probe ad a fiite differece approiatio to it tat satisfies te cosistecy criterio, stabiity is te ecessary ad sufficiet coditio for covergece. Aaytic treatet of stabiity Tere are two stadard ways of ivestigatig te boudedess of te soutio of te fiite differece equatios. I oe, we use a fiite Fourier series ad i oter, we epress te equatio i atri for ad eaie te eige vaues of a associated atri. i) Vo Neua stabiity Metod Stabiity, i geera, ca be difficut to ivestigate, especiay we equatio uder cosideratio is o-iear. Te stabiity of uerica scees ca be ivestigated by perforig vo Neua stabiity aaysis. For tie-depedet probes, stabiity guaratees tat te uerica etod produces a bouded soutio weever te soutio of te eact differetia equatio is bouded. Vo Neua stabiity etod is based upo a fiite Fourier series. It epresses te iitia errors o te ie t = 0 i ters of a fiite Fourier series ad te eaies te propagatio of errors as t. Cosider a costat coefficiet partia differetia equatio i oe space diesio ad a cosistet differece approiatio to it. For differece equatios wit costat coefficiets, te error ay be epaded i fiite epasio i fiite Fourier series. Te error ca be writte as N 0 i = A e β = 0 ε, = 0() N (.5.5) were β = π /( b a), were b a is te egt of te iterva o wic te fuctio is defied. A are te Fourier coefficiets wic are deteried fro te N equatios (.5.5). We ow seek a soutio of te fiite differece equatios for error i te for iβ e we t = 0. We assue tat = were ξ is a arbitrary rea or cope uber. N iβ ε Aξ e (.5.6) = 0 Sice, for iear oogeeous equatio, te su of idepedet soutios is aso a soutio, it is eoug if we cosider a sige ter 5 P a g e
iβ ε = ξ (.5.7) A e were β is a rea uber, ad A is a arbitrary costat. Te error is bouded as icreases ( t ), provided tat ξ. (.5.8) Tis coditio is foud to be ecessary ad sufficiet for te stabiity of te fiite differece agorit. Te uber ξ is caed te apificatio factor of te scee. ii) Te atri etod Te atri etod of aaysis is appicabe to iitia boudary vaue probes, but it ca oy be used for yperboic equatios we te differece scee as bee epressed i two-tie eve for. Suppose tat te differece scee is i te geera for DU = BU CU b (.5.9) were b is a vector wic depeds o te boudary coditios ad D, B, C are square atrices of order N, were N is te uber of es poits at eac tie eve. I te case of a differetia equatio wit costat coefficiets, te atrices D, B, C are costat; i te case of a variabe coefficiet probe, te atrices D, B, C are evauated at ties (),, (-) respectivey. I costat coefficiet probes te atrices D ad C are usuay equa ad o-siguar. Writig (.5.9) i te for 6 P a g e = U D BU D CU D b it foows tat a perturbatio Z 0 of te iitia coditios wi satisfy Tis ay be writte wic is of te for =. (.5.0) Z D BZ D CZ (.5.) Z D B D C Z = (.5.) Z I O Z E = WE were E =[ Z, Z ] T ; W is caed te apificatio atri. It foows tat E W. E (.5.) (.5.4) were. deotes a suitabe or. Equatio (.5.) is a two-tie eve equatio ad te Vo Neua ecessary coditio for te stabiity of a differece scee suc as (.5.9), usig a costat tie step ad ettig, is
W. (.5.5) Te success of te atri etod for te secod order yperboic equatios tus depeds o beig abe to obtai a suitabe estiate for W. We ave see above tat i discussig te stabiity of a differece scee, we wis to kow weter te roots of te poyoia equatio of degree viz. p ( ξ ) = v ξ v ξ... v ξ v = 0 (.5.6) 0 (were v i ay be costats or fuctios of soe paraeters) satisfy te coditio ξ. It is difficut to fid a te roots of (.5.6) ad te test te coditio ξ. I suc cases, te iterva of stabiity or te vaues of te paraeters for wic stabiity ca be acieved, ca be deteried by usig te trasforatio z ξ = z wic aps te iterior of te uit circe ξ = oto te eft af of te pae z 0, te uit circe ξ = oto te iagiary ais ad te poit ξ = oto z = 0 as sow i figure.5.4. Te trasfored caracteristic equatio is k k a0z a z... a k = 0. (.5.7) We ca ow appy te Rout-Hurwitz Criterio wic gives te ecessary ad sufficiet coditios for te roots of te caracteristic equatio (.5.7) to ave egative rea parts, ad ece ξ <. 7 P a g e Figure.5.4
Teore (Rout Hurwitz Criterio): Let k k p z = a0z az... a k ad D a a a a a a a a 0 a a a 5 k 0 4 k k = 0 a0 a ak4 0 0 0 0 were a 0 for a. Te, te rea parts of te roots of p( z ) = 0 are egative if ad oy if te eadig pricipa iors of D are positive. We ave for k =, a0 > 0, a > 0 for k a0 a a =, > 0, > 0, > 0 for k a0 a a a aa aa 0 =, > 0, > 0, > 0, > 0, > 0 as te ecessary ad sufficiet coditios for rea parts of te roots of (.5.7) to be egative. Tese coditios give te stabiity iterva for te differece etod. Defiitio: A epicit fiite differece scee is ay scee tat ca be writte i te for U ' = a fiite su of ' U wit '. A fiite differece scee wic is ot epicit is kow as ipicit. Ipicit etods require a etra coputatio, ad tey ca be uc arder to ipeet. Te ai advatage of ipicit fiite differece etods is tat tere are o restrictios o te tie step. Sovig Tridiagoa Systes To use ay ipicit scees, we ust kow ow to sove tridiagoa systes of iear equatios. Tridiagoa systes are particuary easy (ad fast) to sove usig Gaussia eiiatio. It is coveiet to sove suc systes usig te foowig otatio: 8 P a g e
d u = b d u = b d u = b 4 d = b were di, u i ad i are, respectivey, te diagoa, upper, ad ower atri etries i row i. A coefficiets ca ow be stored i tree oe-diesioa arrays, D, U, ad L, istead of a fu two-diesioa array A. Te soutio agorit (reductio to upper triaguar for by Gaussia eiiatio foowed by back-sovig) ca ow be suarizedas foows: (i) For k =,,, (k = pivot row) k a) = ( = utipier eeded to aiiate ter beow), d b) dk dk uk k = (ew diagoa etry i et row), c) bk bk bk = (ew rs i et row). (ii) b d = (start of back-sove). (iii) For k =,,, (back-sove oop) k b = u d k k k. k Tridiagoa systes arise i a variety of appicatios of fiite differece etod for sovig partia differetia equatios..6. Suary of te tsesis: Motivatio: Hig-order accurate fiite differece scees are iportat i scietific coputatio because tey offer a eas to obtai accurate soutios wit ess work ta ay be required for etods of ower accuracy. A copact differece scee is oe tat utiizes te patc of ces iediatey surroudig te ode at wic a uerica approiatio is beig ade ad does ot eted furter. Hyperboic probes, coo i ecaics ad egieerig appicatios ca be soved to secod order usig sipe (or cetra) fiite differece scees tat are osty copact. Fourt order fiite differece scees ave becoe quite popuar as agaist te oter ower order accurate scees 9 P a g e
wic require ig es refieet ad ece are coputatioay iefficiet. O te oter ad, te iger order accuracy of te fourt order copact etods cobied wit te copactess of te differece steci yieds igy accurate uerica soutios o reativey coarser grids wit greater coputatioa efficiecy. Te ig order copact etod wic we cosider ere is differet i tat te goverig differetia equatio is used to approiate te ower order derivative ters wit te ibeddig tecique. Te scee is difficut to deveop due to te eed of etesive agebraic aipuatio, especiay for o-iear probes. However, oce ig order etod deveoped, it ca be icorporated easiy i appicatio. Rea word uerica data is usuay difficut to aayse. Ay fuctio wic woud effectivey correate te data woud be difficut to obtai ad igy uwiedy. To tis ed, te idea of te cubic spie was deveoped. Usig tis process, a series of uique cubic poyoias are fitted betwee eac of te data poits, wit te stipuatio tat te curve obtaied be cotiuous ad appear soot. A cubic spie approiatio is preseted wic is suited for ay fuid-ecaics probes. Tis procedure provides a ig degree of accuracy ad eads to a accurate treatet to te iitia ad te boudary coditios. Te preset study is otivated by te above aaysis ad is appied o o iear yperboic probes. A uber of approaces to derive suitabe digita coputer agorits ave bee deveoped. Fiite Differece Metods are oe eas of obtaiig approiate soutios to ordiary or partia differetia equatios. Oter etods icude fiite eeets, fiite voues, spectra etods, various spie approiatios etc. Fiite differece etods are attractive because of te reative ease of ipeetatio ad feibiity. Te fiite differece etod is otabe for te great variety of scees tat ca be used to approiate a give differetia equatio. Furterore, fiite differecig eds itsef quite easiy to Tayor series aaysis of te trucatio error, ad tis fact is epoited i te ig-order copact etod to costruct rigorous ig-order accurate approiatios. Te tesis cosists of si capters foowed by te scope for te furter researc ad te ist of refereces usefu for te deveopet ad appicatio of te etods discussed i tis tesis. A brief descriptio of te cotet of eac capter is as foows. Capter : Secod Order Partia Differetia Equatio ad Fiite Differece Metods Te uerica soutio of oe space diesioa secod order o-iear yperboic 40 P a g e