CE 504 Computational Hydrology Introduction to Finite Difference Methods Fritz R. Fiedler

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1 CE 504 Comptatoal Hydology Itodcto to Fte Dffeece Methods Ftz R. Fedle Itodcto a Taylo Sees Cosstecy, Covegece ad Stablty 3 Ital ad Boday Codtos 4 Methods ad Popetes a Classc Methods paabolc eqatos: fte dffeece eqatos paabolc eqatos: algothms hypebolc eqatos: fte dffeece eqatos v hypebolc eqatos: algothms 5 Mscellaeos Note: I apologze fo mg symbols ths veso of the otes. Ths s doe oly de to tme costas. Howeve, yo shold get sed to eadg/tepetg eqatos that se dffeet symbols ow, as eveyoe has the ow pefeece. Yo may wat to ty e-wtg the eqatos a cosstet symbol set that yo lke ode to avod cofso. Itodcto I the fte dffeece method, appomate dffeece qotets ae sed to appomate the devatves a dffeetal eqato, edcg the dffeetal eqato to a system of algebac eqatos. The defto of the patal devatves of a vaable =,t wth espect to space ad tme, espectvely, ae, t, t = lm 0, t t, t = lm t t 0 t As the ame fte dffeece mples, ad t ae take to be fte qattes the dffeece qotets athe tha allowg them to appoach zeo whe appomatg the dffeetals, t, t 3, t t, t 4 t t Eqatos 3 ad 4 ae appomatos to Eqatos ad, espectvely, ad the dffeece betwee the eact devatve ad the appomato s kow as the tcato eo. The Taylo sees s sed to assess the magtde of the tcato eo ad deve othe foms of fte dffeece appomatos. Gve a smooth fcto.e., a fcto that s cotos ad has cotos devatves, the Taylo sees appomates the vale of a fcto at oe pot based o the vale of the fcto ad ts devatves at aothe, eaby pot. I the dmeso, a Taylo sees s wtte

2 R q =! 3!! L 5 whee R s a emade tem that accots fo all tems to fty that s detemed sg! = R ξ 6 whee ξ s some vale betwee ad. Ths s based o the devatve mea vale theoem, whch essece states that thee mst at least oe pot betwee ad whee the fcto slope s eqal to the slope of a le og the vales of the fcto at ad. The vale of ξ s ot mpotat at ths tme. A fst-ode Taylo sees ca be wtte = ξ 7 ad solved fo the devatve to obta ξ = 8 whch coespods to Eqato 3 above. Theefoe, the tcato eo assocated wth the appomato s detemed by the last tem o the ght sde of Eqato 8. It shold be appaet that the tcato eo gets smalle as s edced, popoto to the fst powe of. Ths fte dffeece appomato s kow as the fowad dffeece fomla fo the fst devatve, ad s fst ode accate. Othe foms of fte dffeece appomatos ca be smlaly deved; fo eample, by epadg abot -, a fst-ode accate backwad dffeece fomla s obtaed. I ode to apply fte dffeece methods, t s ecessay to dscetze the doma to be modeled,.e., costct a gd the space-tme plae composed of dscete elemets. The eqatos ae the solved at the gd pots o odes the tesecto of gd les, o at the cetes of the cells fomed by the gd les. Methods based o the fome ae kow as odeceteed techqes, ad the latte as block-ceteed techqes. The focs hee s o the odeceteed method. A spescpt-sbscpt otato s ofte sed to de the vaables; fo eample, the tege sbscpts ad may efe to the ad y decto odes, ad the tege spescpt cold be sed as a tme de. Fo eample,,,, t y q = 9 ad,, O t y = 0 Hee the commoly sed otato O s sed to dcate that ths s a fst ode method. A patcla gd ode ca be efeeced, assmg a og at zeo, as,,,, t y t y = Gds ca be fom o o-fom both the space ad tme dmesos. The fowad ad backwad fte dffeece appomatos fo the tme devatve ca be wtte, espectvely, t O t t t = ad

3 , t = O t 3 t t Note that Eqato 3, the dffeece qotet ca be compted eplctly, as we pesmably kow the vale of fom the pevos tme step at all gd odes. Howeve, Eqato, t s ecessay to solve a system of eqatos fo at the tme level; ths s kow as a mplct method. Ths wll be dscssed moe detal below. Based o the above aalyss of tcato eo sg the Taylo sees, oe mght assme that a smple way to cease the accacy of the -decto fte dffeece appomato s to smply make as small as possble. Ths appoach, howeve, does ot always wok. By makg the space steps small, the comptatoal bde ceases, as the eqatos eed to be solved at evey gd ode. The allowable tme step s also ted to the space step eplct methods. Fally, od-off eo the eo assocated wth sg fte pecso compte epesetatos of eal mbes may become sgfcat fo vey small space ad tme steps. Table shows commo fte dffeece appomatos, sg a depedet vaable, a sbscpt fo the spatal de, ad a spescpt to epeset the tempoal de t s vey commo fo sbscpts to epeset spatal dees, ad spescpts to epeset tempoal dees, o matte what the symbols ae. Also, h s sed stead of, ad k stead of t. The otato Oh dcates a fst-ode accate spatal method, ad Oh dcates a secod-ode accate method; tempoal deftos ae aalagos. Also ote the defto of the opeato δ.

4 Table. Commo fte dffeece appomatos DChatea ad Zachma, 989 Cosstecy, Covegece ad Stablty I sg fte dffeece methods, we ae appomatg cotos patal dffeetal eqatos PDEs wth dscetzed, fte dffeece eqatos FDEs. Cosstecy efes to how well the FDE appomates the PDE as the gd spacg appoaches zeo. If the local tcato eo appoaches zeo wth the gd spacg, the fte dffeece scheme de cosdeato s sad to be cosstet. A patcla FDE s coveget f the solto to the FDE coveges to the solto of the PDE ove the ete doma. Ths s a mpotat dstcto a patcla FDE may be cosstet bt ot coveget. A FDE s stable f odg eos, todced each calclato step, ema boded. I a stable FDE scheme, small eos the tal codto eslt coespodgly small eos the solto. Ths, stablty efes to how eos ae popagated. Cosstecy, covegece, ad stablty ae elated to oe aothe thogh the La Eqvalece Theoem: Gve a well-posed lea tal-vale poblem o talboday-vale poblem ad a fte dffeece scheme cosstet wth t, stablty s both ecessay ad sffcet fo covegece.

5 A well-posed poblem essetally meas that a solto ests, the solto s qe, ad the solto depeds cotosly o the data. The data hee efes to the tal ad boday codtos, pls the coeffcets ad homogeeos soce tems of the PDE. Mathematcally povg cosstecy, covegece, ad stablty s beyod the scope of ths cose. The popetes of commoly sed fte dffeece schemes ae well-docmeted. Howeve, t s mpotat to have a sold destadg of stablty, as t s a ecessay ad sffcet codto fo covegece gve a cosstet scheme, whch we shold be sg. Let s cosde the geeal fte dffeece poblem defed by D[ ] = 0 =,,, I, =,,, N 4a 0 = f =,,, I 4b 0 = I = 0 =,,, N 4c whee D[ ] epesets a fte dffeece opeato o scheme fowad dffeece, backwad dffeece, etc. sed to appomate some PDE, the secod le s the tal codto, ad the thd le shows the boday codtos. Lettg ad f deote colm vectos of the depedet vaables ad tal codtos ove the spatal doma, the fte dffeece method s codtoally stable f fo ay t ad thee ests a depedet costat C sch that C f 0 t T 5 If t mst be fctoally elated to fo ths elatoshp to hold, the fte dffeece method s sad to be codtoally stable. The otato epesets the om of the vecto. Thee ae seveal types of oms, the smplest oe beg, fo eample f = ma f 6 I Eqato 5 dcates that as log as eos stay boded wth espect to the tal codto, the scheme s stable. Rodg ad tcato eos ae also todced evey comptatoal step. To ema stable, these eos mst ot accmlate ay faste tha f they wee smply added togethe. Thee ae seveal ways to aalyze a patcla fte dffeece scheme s stablty, bt we wll ot go to sch detal hee. Istead, we wll keep md the above dscsso, make some geealzatos abot stablty wth espect to eplct ad mplct methods, ad deal wth stablty of commoly sed schemes hydology o a case-by-case bass. Eplct schemes ae sally at best codtoally stable; some ae kow to be codtoally stable. The codtos elatoshps betwee allowable tme ad space steps deped ot oly o the fte dffeece scheme sed, bt also the eqato beg appomated ad soce tems. No-lea eqatos geeally have tghte stablty estctos. If the soce tems chage apdly wth espect to the depedet vaables o foce the depedet vaables themselves to chage apdly, the the soce tems wll dctate stablty eqemets. Eve f we wee to cove the detals of stablty aalyss, the methods sed ae typcally oly tactable fo lea poblems oe dmeso wth o soce tems, ths detemg the lmts of stablty fo a patcla poblem s ofte doe by tal ad eo. I geeal, eplct methods, as s edced say, fo eample, to bette esolve apdly chagg poblem geomety o depedet vaables, t mst also be edced. Implct fte dffeece methods ae geeally codtoally stable. So why bothe wth eplct methods at all? Fst, mplct methods eqe the solto of a system of eqatos at each tme step. The solto of systems of eqatos s ofte comptatoally tesve, ad

6 sometmes t s dffclt to obta accate, stable soltos to these systems. Secod, fo steady poblems the tme step eqed to obta tme-accate soltos s lmted by how fast the depedet vaables chage; ths tme step s ofte smla to the tme steps eqed fo stable soltos sg eplct methods. Becase of the poblems assocated wth smltaeosly solvg lage systems of eqatos, eplct methods ae eve feqetly sed to obta steady state soltos to comptatoal fld dyamcs poblems. Theefoe, mplct methods ae ot the paacea they fst appea to be, bt ae vey sefl may cases. Ital ad Boday Codtos Ital ad boday codtos ae collectvely kow as alay codtos. These deteme, pat, f a patcla poblem s well- o ll-posed. If thee ae too few alay codtos, the solto wll ot be qe; f thee ae too may, the solto wll ot est; ad f they ae ot cosstet wth the PDE, the solto wll ot deped cotosly po the data. Ital codtos compse the vales of the depedet vaables sed to talze the solto at t = 0. These ae sometmes abtay e.g., costat moste cotet ad pesse head wth sol depth befoe a fltato evet, may epeset physcal ealty e.g., zeo ovelad flow depth po to afall, o they mght be deved fom edg codtos of a pevos smlato cold be sed to talze a ew smlato. I ay case, the tal codtos shold epeset a easoable solto to the eqatos beg solved, othewse the poblem may become ll-posed. Thee ae two boad types of boday codtos: Dchlet, whee solto vales ae specfed o the doma bodaes; ad Nema, whee a vale fo the dectoal devatve, omal to the boday, s specfed o the boday. Dchlet codtos ae faly easy to mplemet, ad may be steady o steady. Fo eample, Dchlet codtos may be wtte fo a oe-dmesoal poblem wth a doma legth of L 0, t = g t, L, t = h t, t > 0 7 Nema boday codtos ae typcally mplemeted sg a fte dffeece appomato. Fo eample, geealzed Nema codtos may be wtte 0, t = g t, L, t, t > 0 8 The fl of sol moste at the sol sface s a physcal eample of ths type of boday codto. We se fte dffeece appomatos to mplemet Nema boday codtos. Cosde a fom gd 0,,., I, I whee = 0 ad N = L. Fo the lowe boday, the followg fst-ode appomato to the devatve boday codto cold be sed 0 = g 9 ad the vale of at 0 ca be compted sce the fcto gt s kow at all tme levels. The choce of dffeece appomato depeds o whch dffeece method s beg sed fo the teo odes. Ths wll be dscssed moe detal sbseqetly. Thee ae othe types of boday codtos that cold be mplemeted. A feqetly sed boday codto hydology s specfcato of ctcal depth at a dowsteam boday. We wll eploe these o a case-by-case bass.

7 Methods ad Popetes I ths secto, we wll look at a few caocal foms of eqatos ad fte dffeece methods to llstate geeal techqes. Oly paabolc ad hypebolc eqatos ae coveed. Paabolc Eqatos: Fte Dffeece Eqatos The caocal paabolc eqato s wtte a = S, t 0 t whee a s a coeffcet ad S,t s the soce tem. A eplct fte dffeece method fo ths eqato s a = S t whee a fst-ode accate fowad dffeece s sed fo the tempoal devatve, ad a secodode accate cetal dffeece s sed fo the spatal devatve. Table lsts commo fte dffeece appomatos fo Eqato 0 ote the dffeeces vaables, as descbed pevosly; the ppe case s sed by these athos to dcate that the vale s appomate. Table. Fte dffeece appomatos fo the caocal paabolc eqato DChatea ad Zachma, 989. The fowad dffeece method show Table s eplct, ad stable oly whe s less tha o eqal to 0.5. The backwad dffeece ad Cak-Ncolso methods ae mplct, ad

8 codtoally stable. The Cak-Ncolso method s secod-ode accate both tme ad space. Both mplct methods eqe the solto of a system of eqatos. It s sefl to place these eqatos a mat fomat to see how the eqatos shold be solved ad how the boday codtos ae copoated. sg the symbols of Table, the fowad dffeece method mat fomat s = N N N q ks ks p ks M M O M Note that Eqato, Dchlet boday codtos ae sed, whee pt s the boday codto fcto at the lowe boday, ad qt s the boday codto mposed at the ppe boday. All of the vales o the ght sde of ths eqato ae kow, so the vales of at the tme level ca be compted eplctly. Compae ths to the backwad dffeece method mat fomat = N N N q ks ks p ks M M M O 3 Hee, a system of eqatos mst be solved. The mat s tdagoal, makg ths system faly easy to solve. Table 3, also take fom DChatea ad Zachma 989, smmazes the methods ad boday codtos fo the caocal paabolc eqato.

9 Table 3. Smmay of fte dffeece methods ad boday codtos sed fo the paabolc eqato DChatea ad Zachma, 989. O Table 3, the followg vectos ad mates ae defed:

10 Paabolc Eqatos: Algothms The easest of the methods descbed above to mplemet o a compte s the fowad dffeece method combed wth Dchlet boday codtos. sg the vaables as defed by DChatea ad Zachma 989, a basc algothm fo ths method s peseted sbseqetly. Pelmaes: defe vaables: eal a, dffsvty eal L, legth of doma eal k, tme step t eal h, space step tege ma, mbe of tme steps tege ma, mbe of gd pots defe fctos may call sbotes S,t, soce tem f, tal codto pt, Dchlet boday codto at =0 qt, Dchlet boday codto at =L ead pt data a, L, Defe gd: h = L / ma = a k/h Italzato V0 = p0 = f0 these shold be cosstet Vma = q0 = fma these shold be cosstet fo =,,, ma V = f edloop Tme loop fo =,,, ma fo =,,, ma = *V--**V*Vk*S,

11 edloop 0 = p ma = q Otpt Pepae fo et tme step fo = 0,,, ma V = edloop edloop I ode to mplemet ethe of the mplct methods, we mst solve a system of eqatos tdagoal fom ths type of system ases feqetly solvg PDEs, patclaly whe sg fte dffeece methods. I geeal tems, a tdagoal system s wtte b c d a b c d O M = M 4 an bn cn N d N a N bn N d N I the Thomas algothm, the mat defed Eqato 4 s coveted fom tdagoal to ppe bdagoal fom makg a coeffcets eqal to 0 ad solved sg backwad elmato. I backwad elmato, sce the last eqato epeseted the mat has oly oe kow, t ca be solved; the eslt s the sed the secod-to-last eqato, ad so o. Note that the ete mat does ot have to be stoed; athe, the mat s stoed as thee vectos: sbdagoal a, dagoal b, ad spedagoal c. The followg s the Thomas algothm: Pelmaes defe vaables Decomposto fo =, 3,, N a = a/b- b = b a*c- edloop Fowad Sbsttto fo =, 3,, N d = d a*d- edloop Backwad Sbsttto N = dn/bn fo = N-, N-,, = d c*/b edloop Ths algothm ca be sed wth a ote, fo eample, to solve the paabolc eqato wth Dchlet boday codtos sg the backwad fte dffeece method, as follows. Pelmaes: defe vaables:

12 eal a, dffsvty eal L, legth of doma eal k, tme step t eal h, space step tege ma, mbe of tme steps tege ma, mbe of gd pots defe fctos may call sbotes S,t, soce tem f, tal codto pt, Dchlet boday codto at =0 qt, Dchlet boday codto at =L ead pt data a, L, Defe gd: h = L / ma = a k/h Italzato 0 = p0 = f0 these shold be cosstet ma = q0 = fma these shold be cosstet fo =,,, ma = f edloop Tme loop fo =,,, ma fo =, ma a = - b = * c = - d = k*s, edloop d = d *p dma = dma *q call tdagoal solve ma, a, b, c, d, 0 = p ma = q edloop So fa, oly algothms employg Dchlet boday codtos have bee show. Eqato 9 shows a fst-ode accate dffeece appomato that cold be sed fo the Nema boday codto. Sce the teo odes ae solved sg a secod-ode accate spatal devatve, t makes sese to se a secod-ode accate spatal devatve to appomate the boday codto as well. Fo eample, the followg ceteed spatal dffeece appomato ca be sed at the lowe boday whth a backwad tempoal dffeece ths s the fomla sed fo the Nema codtos show the Table 3 0 = p 5 h whch ca be e-aaged = hp 6 0

13 The followg algothm ses ths dffeece fomla to appomate Nema boday codtos at both eds of the doma to solve the oe-dmesoal caocal paabolc eqato. Note that the gd s defed a slghtly dffeet mae fo Nema boday codtos see Table 3. Pelmaes: defe vaables: eal a, dffsvty eal L, legth of doma eal k, tme step t eal h, space step tege ma, mbe of tme steps tege ma, mbe of gd pots defe fctos may call sbotes S,t, soce tem f, tal codto pt, Nema boday codto at =0 qt, Nema boday codto at =L ead pt data a, L, Defe gd: h = L / ma - = a k/h Italzato fo =,,, ma = f edloop Tme loop fo =,,, ma fo =,,, ma a = - b = * c = - d = k*s, edloop d = d **h*p dma = dma **h*q c = -* ama = -* call tdagoal solve ma, a, b, c, d, edloop Hypebolc Eqatos: Fte Dffeece Eqatos fo the IVP The caocal hypebolc wave eqato s wtte a = 0 7 t Note that ths s essetally the kematc wave eqato, ad ca be posed as a pe tal vale poblem,0 = f, - < < o as a tal-boday vale poblem,0 = f, 0 < < ad 0,t = gt, t > 0. Commo fte dffeece appomatos to ths eqato whe sed

14 a IVP ae show Table 4. Note that these ae all eplct methods, as mplct methods ae ot applcable to the pe IVP. Table 5 lsts the stablty eqemets of the methods lsted Table 4. Table 4. Commo fte dffeece appomatos fo the caocal hypebolc eqato wth o soce tem whe posed as a IVP DChatea ad Zachma, 989.

15 Table 5. Stablty of methods show Table 4. Method Stablty Accacy FTFS stable f a < 0 ad sa fst ode tme ad space FTBS stable f a > 0 ad sa fst ode tme ad space FTCS codtoally stable N/A La-Fedchs stable f sa fst ode tme ad space Leapfog stable f sa secod ode tme ad space La-Wedoff stable f sa secod ode tme ad space Hypebolc Eqatos: Chaactestcs ad the CFL Codto Eqato 7 descbes the movemet of a wave, whch does ot chage shape, oe dmeso. The method of chaactestcs s sed to edce ths eqato to oday dffeetal eqatos the -t plae. If we let C be a chaactestc cve descbed by = t, o C,t = t, t, ad dffeetatg alog C eslts the eqato d = 8 dt dt t If a s defed as the wave speed o the cve C, o d = a 9 dt the the chage of wth espect to t o C s zeo d = 0 30 dt Eqato 30 shows that the solto s costat o the chaactestc cve. Eqato 9 s a oday dffeetal eqato that ca be solved to obta = at 0 3 So whe a s costat, the chaactestc cve s a staght le. Gve Eqato 7 ad the tal codto,0 = f 3 fom Eqato 3 we ca see that the aalytcal solto s, t = at,0 = f at 33 povded that f s cotosly dffeetable. A ecessay, bt ot sffcet codto fo stablty of eplct fte dffeece methods fo hypebolc eqatos s the Coat-Fedchs-Lewy CFL codto, whch states that the mecal doma of depedece mst cota the aalytcal doma of depedece, whee the aalytcal doma of depedece s gve by the chaactestc cves. Sce a s the wave speed, o mecal method ca ot tasmt fomato the -t plae faste tha ths wave speed ad ema stable. Ths, the tme step mst be less tha the space step dvded by the wave speed, o that the tme step mst be less tha the tme fo the wave to tavel acoss oe space step t / a 34 Eqato 34 ca be e-aaged to gve the stablty codto show evey eplct method Table 5 ote that addtoal codtos eed to be met fo the FTBS ad FTFS methods.

16 Hypebolc Eqatos: Fte Dffeece Eqatos fo the IBVP I hydologc applcatos, we ae typcally coceed wth poblems defed o fte domas, ad mpose boday codtos o those domas. The eplct methods show Table 4 also apply to the IBVP, whee 0,t = gt s mposed at = 0. Fo eample, fo smlatg a hllslope, both the dschage ad wate depth ca be set to zeo at the hllslope dvde. The FTFS method does ot apply f the coeffcet a s take to be postve. Implct methods ae also applcable to the IBVP. Table 5 shows seveal mplct methods applcable whe a s postve ad a soce tem s clded; all ae codtoally stable. Note that sce the BTBS ad Wedoff mplct methods se adacet gd pots at the tme level, thee s o eed to solve a system of eqatos: to compte the methods have avalable both fom the last tme step ad fom the last space step. Table 5. Implct methods fo the hypebolc IBVP DChatea ad Zachma, 989. Hypebolc Eqatos: Algothms A algothm fo the La-Wedoff method to solve the pe IVP

17 a = s, t 0 < t < t ma, p < < q t 34,0 = f s peseted as follows. Note that the solto doma deceases by two gd pots each tme step, as thee ae o boday codtos ad the - ad vales ae sed to compte at the lowe ad ppe eds of the doma, espectvely. Theefoe, we mst beg comptatos o a doma *ma wde tha the desed solto doma. Also ote how the soce tem s compted, sg both the soce tem ad ts devatves wth espect to tme ad space. Pelmaes: Ipt: Itege p, lowe spatal de at t ma Itege q, ppe spatal de at t ma Itege ma, mbe of tme steps eal k, tme step eal h, space step eal a, coeffcet fcto f, tal codto fcto ss,t, soce tem fctos s, t, s, t s,t ad st,t, espectvely t Defe Gd: s = k/h check stablty Italze otpttg tal codto t = 0 m = p ma ma = q ma fo = m, m,, ma = *h V = f edloop otpt t fo = p, p,, q otpt V edloop Tme Loop: fo =,,, ma m = m ma = ma- fo = m, m,, ma = V-0.5*s*a*V-V- s***a**/*v--*vv k*ss*h,tk**/*st*h,t-a*s*h,t edloop t = t k fo = m, m,, ma V = edloop

18 otpt t fo = p, p,, q otpt V edloop edloop To llstate both how boday codtos ae hadled ad mplemetato of a mplct scheme, a algothm fo the Wedoff fte dffeece method appled to a IBVP s show below. The boday codto s 0,t = gt. Notce how the vale of fom the pevos space step at the tme level s sed the tme loop. Pelmaes: Ipt: Itege ma, mamm de Itege ma, mbe of tme steps eal k, tme step eal h, space step eal a, coeffcet fcto f, tal codto fcto gt, boday codto fcto ss,t, soce tem Defe Gd: s = k/h Italze otpttg tal codto t = 0 ma = ma*h fo = 0,,, ma = *h V = f edloop otpt t fo = p, p,, q otpt V edloop Tme Loop: fo =,,, ma t = t k 0 = gt q = -s*a/s*a fo = 0,,, ma- = Vq*V-q* *k*ssmah/, t-k//s*a edloop fo = 0,,, ma V = edloop otpt t fo = 0,,, ma

19 otpt edloop edloop Hypebolc Eqatos: Cosevato Law Fom Fom the geealzed kematc wave theoy, we have pevosly see that C F = 0 35 t whee F s the fl ad geeal s a fcto of both C, the cocetato, ad F = f C, 36 Eqato 35 ca be sbsttted to Eqato 34 to obta C F C = 0 37 t C A eplct fte dffeece method s sad to be cosevato law fom f t ca be cast the followg fom Q / Q / = 0 38 h k whee the Q vales ae mecal appomatos to the cotos fles acoss bodaes located at some pots below ad above the gd ode fo whch the dffeece eqato s wtte, ad the vales ae appomatos to the cocetatos. I geeal, the Q vales ae detemed as some fcto of the eghbog vales Q / = Q,, Q / = Q, 39 ad t s eqed that the mecal fles be cosstet wth the cotos fl. Eqato 37 ca be e-wtte h = h k Q / Q / 40 Wtg the cosevato law as F = 0 4 t the La-Fedchs method ca be wtte = s F F 4 wth the mecal fles defed as Q / = F / s 43 Q / = F / s ths method s a cosevato law dffeece eqato. Whe F = a ad a s costat, the cosevato fom of the La-Fedchs method edces to the fom show pevosly. Table 6. Fte Dffeece methods Cosevato Law fom DChatea ad Zachma, 989.

20 Refeeces DChatea, Pal, ad Davd Zachma, Appled Patal Dffeetal Eqatos, Hape & Row, Pblshes, Ic., New Yok, New Yok, 989.