Numerical Techniques for Conservation Laws with Source Terms

Transcription

1 Nmercal Techqe or Coerao Law wh Sorce Term by J Hdo Projec Speror Dr. P.K. Sweby Pro. M.J. Bae Abrac h derao we wll dc he e derece mehod or appromag coerao law wh a orce erm pree whch codered o be a kow co o, ad. Fe derece cheme or appromag coerao law who a orce erm pree are dced ad are adaped o appromae coerao law wh a orce erm pree. Fr we coder he orce erm o be a co o ad oly ad he we coder he orce erm o be a co o alo. Some mercal rel o he dere approache are dced hrogho he derao ad a oerall comparo o he dere approache made whe he orce erm. Th derao wa ded by he Egeerg ad Phycal Scece eearch Cocl.

2 Coe SYMBOLS AND NOTATON... NTODUCTON D CONSEVATON LAW D Lear Adeco Eqao Fr Order Scheme Secod Order Scheme mplc Scheme.... -D Coerao Law..... No-Coerae Scheme..... Coerae Scheme Trcao Error ad Sably Trcao Error Sably Dpao, Dpero ad Ocllao Dpao Dpero ad Ocllao Fl-lmer Mehod CONSEVATON LAW WTH SOUCE TEM (X,T) Bac Approach La-Wedro Approach MPDATA approach Bac MPDATA MPDATA Approach or Adeco Eqao wh Sorce Term (,) MPDATA Approach or Coerao Law wh Sorce Term (,) Comparo o Scheme Ug Te Problem...48

3 4 CONSEVATON LAW WTH SOUCE TEM (X,T,U) Adapao o he Scheme or he Coerao Law wh Sorce Term (,) Bac Approach La-Wedro Approach MPDATA approach oe Upwd Approach Adeco Eqao wh Sorce Term () Coerao Law wh Sorce Term (,,) Some Nmercal el or he Eplc Upwd Approach mplc Upwd Approach Fr Order mplc Upwd Approach Secod Order mplc Upwd Approach Some Nmercal el or he mplc Upwd Approach LeVeqe ad Yee MacCormack Approach Eplc MacCormack Approach Sem-mplc MacCormack Approach LeVeqe ad Yee Splg Mehod or he MacCormack Approach Some Nmercal el or he MacCormack Approach SOME NUMECAL ESULTS Eplc ad mplc Addg Approach La-Wedro approach MPDATA Approach oe Upwd Approach mplc Upwd Approach MacCormack Approach Oerall Comparo Fr Order Comparo Secod Order Comparo Secod Order wh TVD Comparo Coclo Chagg he Sep-Sze whe he Sorce Term S CONCLUSON Fal Comparo Frher Work...4 EFEENCES... 5 APPENDX A... 6 ACKNOWLEDGEMENTS... 3

4 Fgre CHAPTE : Fgre -: Shallow Waer Eqao...4 CHAPTE : Fgre -: No-coerae cheme...3 Fgre -: The Upwd cheme becomg able...9 Fgre -3: eral o ably or La-Wedro... Fgre -4: Dpao o he r order Upwd cheme... Fgre -5: Dpero leadg o ocllao o he La-Wedro cheme....5 Fgre -6: Sperbee l-lmer mehod appled o he La-Wedro cheme...8 Fgre -7: TVD rego or e derece cheme...3 Fgre -8: Secod order TVD rego or e derece cheme....3 Fgre -9: Sperbee l-lmer or e derece cheme...3 CHAPTE 3: Fgre 3-: The Upwd cheme wh orce erm added o...35 Fgre 3-: The La-Wedro cheme wh orce erm added o...35 Fgre 3-3: The La-Wedro cheme wh Sperbee l-lmer ad orce erm added...36 Fgre 3-4: Comparo o dere cheme wh he orce erm added o...36 Fgre 3-5: The La-Wedro approach or adeco eqao wh orce erm....4 Fgre 3-6: MPDATA approach or adeco rapor eqao Fgre 3-7: Comparo bewee La-Wedro approach ad MPDATA...46 Fgre 3-8: Comparo o he hree approache dced h chaper....5 Fgre 3-9: Comparo o re error o he hree approache dced h chaper....5

5 CHAPTE 4: Fgre 4-: Comparo o dere cheme wh he orce erm added o...55 Fgre 4-: La-Wedro approach wh ad who Sperbee l-lmer...57 Fgre 4-3: MPDATA approach wh ad who Sperbee l-lmer...6 Fgre 4-4: Comparo o cheme baed o oe Upwd approach...66 Fgre 4-5: Comparo o cheme baed o he mplc Upwd approach...73 Fgre 4-6: Comparo o eplc, em-mplc ad plg mehod or MacCormack approach Fgre 4-7: Comparo o eplc, em-mplc ad plg mehod or MacCormack approach wh TVD CHAPTE 5: Fgre 5-: The eac olo (5.)...9 Fgre 5-: La-Wedro approach appled o (5.) wh μ....9 Fgre 5-3: La-Wedro approach appled o (5.) wh μ....9 Fgre 5-4: La-Wedro approach appled o (5.) wh μ....9 Fgre 5-5: La-Wedro approach appled o (5.) wh μ....9 Fgre 5-6: Eplc addg approach wh orce erm...9 Fgre 5-7: Sem-mplc addg approach wh orce erm Fgre 5-8: Comparo o eplc ad em-mplc addg approach...93 Fgre 5-9: Comparo o La-Wedro approach Fgre 5-: Comparo o La-Wedro wh Sperbee l-lmer...94 Fgre 5-: MPDATA approach or orce erm...95 Fgre 5-: oe Upwd approach wh orce erm...95 Fgre 5-3: mplc Upwd approach wh orce erm Fgre 5-4: Eplc MacCormack approach wh orce erm...96 Fgre 5-5: Sem-mplc MacCormack approach wh orce erm Fgre 5-6: Splg mehod (MacCormack approach) wh orce erm...97 Fgre 5-7: Comparo o MacCormack approach wh orce erm Fgre 5-8: Comparo o r order cheme led Table 5- wh orce erm...98 Fgre 5-9: Comparo o ecod order cheme led Table 5- wh orce erm...99 Fgre 5-: Comparo o ecod order cheme wh TVD led Table 5- wh orce erm...99 Fgre 5-: Eplc Upwd approach wh orce erm.... Fgre 5-: Eplc Upwd approach wh orce erm.... CHAPTE 6: Fgre 6-: Comparo o mo accrae approache wh orce erm... Fgre 6-: Comparo o eplc, em-mplc ad mplc ecod order Upwd wh orce erm...

6 Symbol ad Noao The ollowg a l o ymbol ad oao ed hrogho h projec. N (,) (,) ((,)) (,,(,)) Sep-ze -dreco. Sep-ze -dreco. eger deog crre ep mber. eger deog crre ep mber. Toal mber o ep -dreco. Toal mber o ep -dreco. Crre poo pace. Crre poo me. The eac olo. The al daa. The l. The orce erm. ( ), The mercal appromao o he eac olo. (, ( ) ) ( ) The r order mercal appromao o he eac olo. c ( ( ), The mercal appromao o he l. (, ( ),, The mercal appromao o he orce erm. The wae peed or he adeco eqao.

7 a((,)) The wae peed or he coerao law. c The Cora mber or he adeco eqao. e (, ) The re error o a cheme a he ode. Τ The rcao error o a cheme. ( ) φ φ θ The l-lmer o a ecod order cheme. / a( ) The local Cora mber or he coerao law. A orward derece appromao. A ceral derece appromao. A backward derece appromao. Alo, we wll be g a ed meh,.e. N N N- N (,)

8 rodco ecely, he mercal olo o coerao law wh a orce erm,.e. () (,, ) (.) where () he l ad (,,) he orce erm, ha bee grea demad. Th de o he reqecy whch coerao law wh orce erm are mahemacal model o phycal ao. For eample, he -D Shallow Waer Eqao model low rer or a chael o e deph ad reqre he mercal olo o a yem o eqao o he orm (.). Coder he Shallow Waer Eqao dced by Bermdez ad Vazqe[4] where h q h h w (, ), ( w) w F q h ( w) q gh F (, w) ad ( w) (),. ghh (.) Here, h(,) ad (,) repree he oal hegh aboe he boom o he chael ad he ld elocy, repecely, ad H() he deph o he ame po b rom a ed reerece leel (ee Fgre -). The aalycal olo o (.) ca be eremely dcl o d ad omeme mpoble. Th, mercal mehod are reqred o appromae he olo o (.). 3

9 A H() h(,) er Seabed L Fgre -: Shallow Waer Eqao. The olo o (.) ca be dcl o mercally appromae accraely ee whe he orce erm o pree,.e. (). (.3) Throgho Chaper, we wll e e derece o appromae (.) ad dc he accracy ad ably o he cheme dered. We wll look a he rcao error ad how ha r order e derece cheme are dpae ad ecod order e derece cheme are dpere. Fl-lmer mehod wll alo be dced o ha we ca mme he dpero pree ecod order e derece cheme. Chaper, we wll ee ha he majory o dcle ecoered whe appromag (.) ca be oercome b we ow eed o coder how o appromae (.), where he orce erm ow pree. A grea deal o reearch ha bee carred o coerao law wh orce erm b how o hadle orce erm, epecally whe hey are, ll a ope e. Chaper 3, we wll 4

10 dc aro approache or appromag (.) b wh he orce erm beg oly a co o ad,.e. d d () d (, ). d We wll coder addg he orce erm, he La-Wedro approach ad he MPDATA approach ad we wll compare he hree approache or a e problem. Chaper 4 we eed he work o coder (.) where he orce erm alo a co o. Th reqre a appromao o he orce erm ce we do o kow. Chaper 4, we wll dc a arey o approache or mercally appromag (.) cldg he hree dced Chaper 3. A mple e problem wll be ed o aalye he dere approache ad Chaper 5, we wll compare he dere approache wh a e problem whoe orce erm. 5

11 -D Coerao Law h chaper, we wll look a ome mercal cheme or appromag he -D calar coerao law ( ) (.) where (,) he coered qay ad () he l. We ca alo rearrage (.) o oba he qa-lear orm a( ) (.) where a() (), whch called he wae-peed. a() c, where c a coa, he (.) become he lear adeco eqao.. -D Lear Adeco Eqao The mo bac orm o he coerao law he lear adeco eqao c (.3) where c a coa ad () c. Here, he coa c kow a he wae peed ce a() c. There are a arey o mercal echqe or appromag he lear adeco eqao, ch a e eleme mehod ad e olme mehod. Aoher cla o mercal echqe ed or appromag he lear adeco eqao are e derece mehod. Fe derece mehod ole replacg he derae o (.3) wh e derece appromao. e.g. 6

12 7 whch called he orward derece appromao me, whch called he ceral derece appromao me ad whch called he backward derece appromao pace. The hree e derece ca be obaed by g Taylor heorem,.e.... ad by re-arragg we may oba he orward derece appromao. Th, by g e derece, we ca oba a e derece cheme ha appromae he lear adeco eqao. For eample, we e a orward derece appromao pace ad a ceral derece appromao me ad ame boh o hee e derece o be appromao a (,), we may oba c, ad by re-arragg, we oba ( ), whch a e derece cheme whch appromae he lear adeco eqao. Uoraely, h e derece cheme codoally able a we wll ee laer.

13 8.. Fr Order Scheme order o oba a r order cheme, we e a orward derece appromao me ad a backward derece appromao pace ad ame boh o hee e derece o be appromao a (,),.e.. ad Sbg hee o (.3) ge, c ad hece, ( ) where c ad kow a he Cora mber. Th cheme oe o he mo bac mercal appromao o he adeco eqao. Howeer, ca be how ha h cheme mercally able c <, whch cae we e a orward derece appromao pace ad me ad ame ha boh are appromao a (,),.e. ad, he bg o (.) ge. c Whece, ( ). Th cheme mercally able c >. Separaely, hee cheme ca become mercally able, b we combe hem ( ) ( ) < > (.4)

14 9 we oba he Upwd mehod wh wchg hrogh. Th cheme ca ll become able b oly or >. Th wll be dced laer. Aleraely, we cold oba aoher r order cheme we e a orward derece appromao me ad a ceral derece appromao pace ad ame ha boh are appromao a (,),.e. ad he bg o (.) ge c ( ). Uoraely h ceral cheme codoally able, b by replacg by he aerage ( ) we oba he La-Fredrch cheme ( ) ( ) (.5) whch able or. (See laer).. Secod Order Scheme Oe o he mo well kow ecod order cheme or appromag he adeco eqao he La-Wedro cheme ad dered a ollow: Ug Taylor heorem... (.6) ad ce

15 c (.7) c c c c c o,. c (.8) Sbg (.7) ad (.8) o (.6) ge... c c ad by g ceral derece appromao pace ad amg ha boh are appromao are a (,),.e. ad we oba. c c Hece, he ecod order La-Wedro cheme ( ) [ ]. (.9)..3 mplc Scheme So ar, all he cheme we hae looked a hae bee eplc cheme. Th becae oe o he cheme we hae looked a hae erm olg me leel o he rgh had de o he cheme. For eample, he La-Wedro cheme eplc ( ) [ ]

16 b we e ceral derece appromao pace ad ame ha boh are appromao a (,) ead o appromao a (,) ( ) [ ] (.) we oba he mplc La-Wedro cheme. Th cheme mplc ce erm olg appear o he rgh had de o he eqao. mplc cheme cae dcle ce we ow hae o ole a r-dagoal yem a each me ep. earragg (.) ( ) ( ) ( ) hece c a b a c b a c b a c b a c b a c b he r-dagoal yem, whch eed o be oled a each me ep, or he mplc La-Wedro cheme where ( ) ( ) c b a ad,. All mplc cheme ake he orm G A where A a () () mar ad G a () colm ecor. geeral, mplc cheme ca be more accrae ha eplc cheme b mplc cheme are harder o mpleme ad reqre a lo more calclao ha eplc mehod. So ar we hae looked a a ew e derece cheme, o r or ecod order, whch mercally appromae he olo o he adeco eqao b here are a

17 grea deal more ad deely oo may o look a h eco. For a more deph dco o e derece cheme or he adeco eqao, look Kroer[8], LeVeqe[7] ad Ame[4].. -D Coerao Law Seco., we dced ome e derece cheme or appromag he lear adeco eqao, whch a orm o he calar coerao law ( ) where () a(). Howeer, we ca adap he echqe dced Seco. o ha we ca mercally appromae he olo o he calar coerao law b we m be carel how we appromae (.) ce we wh o ere coerao... No-Coerae Scheme a cheme o-coerae, he he cheme wll moe dcoe a he correc wae peed. For eample, we appromaed he qa-lear orm o eqao (.) by g he e derece mehod he we wold oba a ocoerae cheme. Coder cd Brger eqao,.e., re-wrg qa-lear orm ge ad by g a orward derece appromao me ad a backward derece appromao pace ad amg ha boh are appromao are a (,),.e.

18 ad we oba [ ], amg >. Th cheme coerae or mooh daa oly ad ed o mercally appromae dcoe, he cheme become o-coerae mog he dcoy a he wrog peed. No-coerae cheme wh d., d. ad (,.5) Eac Appromao Fgre -: No-coerae cheme. we e he o-coerae cheme, whch appromae cd Brger eqao, wh al daa. <.3 (,),.4.3 we may oba he rel Fgre -. Here, we ca ee ha he cheme ha moed he dcoy oo lowly whch mea ha he cheme o coerae. 3

19 .. Coerae Scheme To ere coerao, we reqre ha he mehod be coerao orm,.e. [ F(,..., ) F(, )], q p p q p,..., p where F called he mercal l co ad o p q argme. We ca ere coerao by mercally appromag (.) ad g a mlar approach a we dd he preo b-eco. For eample, whe we dered he Upwd cheme, we ed a orward derece me ad eher a orward or a backward derece pace depedg o he ale o. Here, we ake a ame approach b we wll apply e derece o ead o,.e. ad eher / > or / < where / / a( ). Hece, ( ) ( ) / / > < he Upwd cheme or he calar coerao law where ad ( ). Howeer, dcle are whe appromag /. Th becae a( / / / kow. Oe approach ed o oercome h problem cold be o appromae / by / ) ( ). 4

20 5 Aoher mehod, whch ere coerao, o appromae / by replacg a() by a local deed a each grd po by. oherwe ) ( / a Problem alo occr whe adapg he La-Wedro cheme o he o-lear cae. Th becae (.8) o loger hold. Howeer, we ca oercome h problem by rewrg (.8) a a ) ( ) ( ) ( ) ( (.) ad by g Taylor heorem, d a ) ( ) ( whece we may oba ( ) ( ) ( ) [ ] / / he La-Wedro cheme or he coerao law. Table - l a arey o e derece cheme or he coerao law.

21 6 Name o Scheme Scheme Order Upwd (r order) ( ) ( ) < > / / La-Fredrch ( ) [ ] Secod Order Upwd (Warmg ad Beam) ( )( ) ( )( ) ( )( ) ( )( ) < > 3 3 / / 3/ / 3/ / Leaprog ( ) La-Wedro ( ) ( ) ( ) [ ] / / MacCormack Predcor- Correcor ( ) ( ) [ ] * * * * Table -: Fe derece cheme or he -D coerao law. For all cheme Table -, oherwe ) ( / a ad. Here, we ca ee ha adapg he e derece mehod o he calar coerao law ca cae mor problem..3 Trcao Error ad Sably.3. Trcao Error The rcao error o a cheme ery el, ce ell wheher he cheme coe ad he order o accracy o he cheme. To dere he rcao error o a cheme, we ame ha he ale a he grd po are eac,.e. ), (, ad he e Taylor ere epao. The rcao error alo kow a he

22 7 dcreao error, whch he error caed by g e derece appromao o appromae he derae o (.3). A a eample, coder he La-Fredrch cheme (.5) or he calar coerao law ( ) [ ]. Now, by amg ha he ale a he grd po are eac,.e. ), (, ad by g Taylor heorem he by bg o (.5) ge ) ( ) ( O O Τ where Τ deoe he rcao error. Hece, ) ( ) ( O O Τ he rcao error o he La-Fredrch cheme. The La-Fredrch cheme ecod order pace b oly r order me, whch make he La-Fredrch cheme r order ad coe, ce a ad, he rcao error ed o zero, Τ. Smlarly, we coder he La-Wedro cheme or he adeco eqao ( ) [ ], we ca how ha he La-Wedro cheme ha a rcao error o

23 3 3 3 c Τ O( ) O( 3 ) The La-Wedro cheme ecod order ad coe ce a ad, he rcao error ed o zero, Τ. p q geeral, a cheme ha a rcao error o order O( ) O( ), he he cheme o order p pace, q me ad o oerall order m(p,q). Alo, p ad q are greaer ha or eqal o, he he cheme coe..3. Sably We alo eed o kow he eral o abole ably o a e derece cheme ce, we chooe or ep-ze ch ha he eral o abole ably breached, he he e derece cheme wll become able gg ery accrae rel. Now, a mercal cheme able proded he error a he ode e (, ) doe o blow p..e. he mercal ale a he ode are o eac, he error beg o creep he mercal appromao. hoe error blow p, he he cheme become mercally able. Fgre - how he Upwd cheme becomg mercally able wh al daa <.5 (,)..5 8

24 Upwd cheme or adeco eqao wh d., d.5 ad c U(,) Fgre -: The Upwd cheme becomg able. There are eeral aalycal echqe ha ca be ed o ee a cheme able, oe o whch he Forer mehod. The Forer mehod co o bg a Forer mode k ξ e o he cheme o oba a epreo or he amplcao acor ξ. The cheme wll he be able proded ξ. For eample, coder he La-Wedro cheme or he adeco eqao ( ) [ ]. By bg k ξ e, we oba ξ e k ξ e k k ( ) k ( ) ( ξ ξ ) e e k ( ) k k ( ) [ ξ ξ e ξ e ]. we ow dde by e jk we oba 9

25 ad by re-arragg ( k k) k e e [ k ξ ξ ξ ξ e ] ξ ξ ξ ξ Ug he dee ad we may oba So, or ably we reqre k k k k ( ) ( ) ξ e e e e. e e jk jk jk e e jk cok k ( cok) ( k) ξ ξ. cok k. Here, we ca ee ha he amplcao acor le o a ellpe: we le ad ad by g he dey whece ξ cok k cok y k co k k ( ) y. So, he eral o abole ably a ellpe wh cere (- ) ad croe he - a a ad -. Fgre -3 how he crcle wh he ellpe o he amplcao acor de he crcle. Here, we ca ee ha or he ellpe o ay de he crcle, ad. Hece, or he La-Wedro cheme o be able,. Th codo o called he eral o abole ably. Noce ha, he ellpe ac he crcle. Table - l he ably eral o a ew cheme.

26 y - y -* -* Fgre -3: eral o ably or La-Wedro Name o Scheme Order (pace me) Oerall Order o Scheme eral O Abole Sably Upwd (r order) La-Fredrch Upwd (ecod order) Leaprog La-Wedro MacCormack Predcor-Correcor Table -: The eral o abole ably ad he order o ome cheme. Earler, Fgre - howed he Upwd cheme becomg able or.5. Th becae he Upwd cheme able or, whe c >, ad ce le ode he eral o abole ably, he cheme wll become able.

27 .4 Dpao, Dpero ad Ocllao.4. Dpao ca be how ha all r order cheme er rom dpao whch ca rel a ery accrae mercal olo. Dpao occr whe he raellg wae amplde decreae. Fgre -4 how ome mercal rel o he Upwd cheme appled o he adeco eqao wh al daa <.5 (,). >.5 Fgre -4 how ha he Upwd cheme dpae ce he mercal olo ha ared o decreae amplde. Comparo Bewee Eac olo ad Nmercal Appromao a Upwd chem e or adeco eqao w h d., d., c ad o.3. U(,.3).6.4. U(,) Eac Upwd Fgre -4: Dpao o he r order Upwd cheme order lly derad why dpao occr, we wll e he aaly o he moded eqao, whch dced by Sweby[3] ad LeVeqe[7], o he La- Fredrch cheme or he adeco eqao ( ) [ ].

28 3 Earler, we aw ha h cheme had a rcao error o ) ( ) ( O O Τ ad by g (.8) c we may oba ) ( ) ( O O c Τ. So, he La-Fredrch cheme a ecod order appromao o D c (.) [ ] D where. Eqao (.) kow a he lear adeco-do eqao ad ll-poed D <. h cae, eqao (.) well poed ce o, or (.) o be well poed [ ]. Hece, ce or ably,, eqao (.) well poed a log a he cheme able. So, he La-Fredrch cheme qalaely behae lke he olo o (.). Now, by g he Forer Traorm o wh repec o ( ) ( ) π ξ ξ d e,, ˆ ad bg o (.), we may oba ha (.) a ODE wh olo ( ) ( ) e e c D ξ ξ ξ ξ, ˆ, ˆ ad by g a ere raorm, we may oba ( ) ( ) () ( ) ξ π ξ ω ξ ξ d e e D, ˆ,.

29 Here, we ca ee ha he olo o he orm ( ω e () ξ ξ), whch repree a raellg wae wh decreag amplde, ˆ ( ξ,) e D ξ. The reqecy ω(ξ) ad depede o he wae mber ξ. h cae he reqecy ω(ξ) cξ, h alo kow a he dpero relao. Alo, () ξ ω ξ kow a he phae elocy ad ge he wae peed o each wae. For he La-Fredrch cheme, he phae elocy () ξ ω ξ c. Hece, he wae all rael a he ame peed ad o, he La-Fredrch cheme o-dpere. Howeer, he La-Fredrch cheme er rom dpao, de o he wae raellg wh decreag amplde. Hece, he La-Fredrch cheme er rom dpao b o dpero. We ca alo how ha he Upwd cheme wh > er rom dpao, ce he rcao error o he cheme c [ c ] Τ O( ) O( ), he cheme a ecod order appromao o (.) wh c D ( ) Hece, he Upwd cheme alo dpae ad ce c c ( ) > ( ) where he le-had de repree he ale o D or he La-Fredrch cheme, we ca ee ha he La-Fredrch cheme more dpae ha he Upwd cheme or >.., 4

30 .4. Dpero ad Ocllao Dpero occr whe wae rael a dere wae peed ad commo all ecod order cheme. Fgre -5 how ome mercal rel o he La-Wedro cheme appled o he adeco eqao wh al daa <.3 (,)..3 Here, we ca ee ha he La-Wedro cheme er rom dpero ce ocllao are occrrg he mercal olo behd he dcoy. Comparo bewee eac olo ad mercal appromao a La-Wedro cheme or adeco eqao wh d., d., c ad o.5.4 U(,.5) U(,) Eac La-W edro Fgre -5: Dpero leadg o ocllao o he La-Wedro cheme. We ca ee why he La-Wedro cheme er rom dpero by akg he ame approach a we dd or he dpao cae. Coder he La-Wedro cheme or he lear adeco eqao whoe rcao error wa ( ) [ ], 5

31 3 3 d d 3 Τ c O( ) O( 3 ) 3 3, 6 d d we ca ee ha he La-Wedro cheme a beer appromao o d d 3 d η d c c where η 3 d d 6 ad by g Forer Traorm, we may oba ( ) (, ) ˆ ( ξ,) ( () ξ ξ ) d where ω () ξ cξ η ξ π e ω 3. Here, we ca ee ha he olo o he orm ( ω e () ξ ξ), whch repree a raellg wae wh coa amplde, ˆ( ξ,). Th mea ha he cheme o loger er rom dpao, howeer, coder he phae elocy () ξ 3 ω ξ cξ η ξ ξ c η ξ Here, we ca ee ha dere wae mber rael a dere peed ad o, he La-Wedro cheme dpere. Alo, η < wh he ably rego o a cheme, he ocllao wll occr behd he dcoy ad η > wh he ably eral o a cheme, he ocllao wll occr ro o he dcoy. Th becae, η <, he hgh wae mber rael wh a lower elocy ha hey hold creag ocllao behd he dcoy, b η >, he hgh wae mber rael wh a aer elocy ha hey hold creag ocllao ro o he dcoy. Fgre -5 how ha he La-Wedro cheme er rom ocllao occrrg behd he dcoe, whch wold mply ha η <, η c 6 ( ).. 6

32 For ably, we reqre, whch mea ha or η <, c 6 ad ce c ad >, ere ha η < creag ocllao behd he dcoy. geeral, all r order cheme er rom dpao b are o-dpere, ad all ecod order cheme er rom dpero b are o-dpae. For a more deph dco o wae heory, ee Whham[9] ad Ame[4]..5 Fl-lmer Mehod So ar we hae ee ha, geeral, all r order cheme er rom dpao ad all ecod order cheme er rom dpero, whch creae ocllao arod he dcoy. Howeer, here a mehod whch wche bewee a ecod order appromao whe he rego mooh ad a r order appromao whe ear a dcoy. Th mehod coderably redce he ze o he ocllao by g a r order appromao ear dcoe ad called he l-lmer mehod. Fgre -6 how ome mercal rel o he La-Wedro cheme wh ad who he Sperbee l-lmer mehod appled o he cheme ad wh he eac olo or al daa <.3 (,)..3 7

33 .4 La-Wedro cheme or adeco eqao wh c, d., d. ad.5.. U(,.5) U(,) Sperbee l-lmer appled o La-Wedro cheme wh o Eac Solo No l-lmer Sperbee l-lmer Fgre -6: Sperbee l-lmer mehod appled o he La-Wedro cheme. Here we ca ee ha he Sperbee l-lmer mehod ha elmaed all ocllao rom he La-Wedro cheme relg a eremely accrae ecod order cheme. To lly derad l-lmer mehod, we hall cloely ollow he work o Sweby[3] ad LeVeqe[7]. Now, we ca re-wre ay ecod order cheme a where [ F( ; ) F( ; ) ] (.3) F ( ; ) ( ; ) ( ) F L F H ;. (.4) Here, F L (;) repree a r order cheme ad F H (;) repree a ecod order correco erm. order o oba he l-lmer mehod or a ecod order cheme, we re-wre (.4) a ( ; ) ( ; ) ( )φ F ; F L F H where φ repree he l-lmer, whch ye o be peced. Beore we dc he choce o he l-lmer, le re-wre he La-Wedro cheme or he calar coerao law he orm o (.3). 8

34 9 ( ) ( ) ( ) [ ] / / We ca re-wre h eqao a he r order Upwd cheme pl a ecod order correco erm. Amg ha / >, he La-Wedro cheme ca be wre a [ ] ( )( ) ( )( ) / / ad we may oba ( ) F L ; ad ( ) ( )[ ] F H / ;. Here, F L (;) repree he Upwd cheme ad F H (;) repree he ecod order correco erm. Smlarly, amg ha / <, we may oba ( ) F L ; ad ( ) ( )[ ] F H / ;. Hece, we may oba ( ) ( ) [ ] ; ; F F where ( ) ( ) ( )φ H L F F F ; ; ; ad ( ) < > ; / / F L ( ) ( )( ) ( )( ) < > ; / / F / / H. We ow eed o meare he moohe o he daa o ha we may chooe he llmer o oba ecod order accracy ad he TVD propery. The TVD propery

35 called he Toal Varaoal Dmhg propery ad wll o be dced ll h he. Howeer, we wll how ome rego o TVD or he l lmer, φ. order o meare he moohe o he daa, we cold look a he rao o coece grade. θ j where j g( / ). Here, θ cloe o he he daa codered o be mooh, b θ ar rom, he here are kk he daa a. We ca ow ake φ o be a co o θ,.e. ( ) φ φ θ where φ a ge co. Now, we reqre he l-lmer o be o ecod order ad o ay he TVD propery. he l-lmer o ay he TVD propery, we m r ame ha φ θ ad we m chooe he l-lmer o le he TVD rego how Fgre -7. B o oba ecod order accracy, he l-lmer m pa hrogh φ() ad le he rego how Fgre -8. oe Sperbee l-lmer j φ(θ ) ma(,m(θ,),m(θ,)) ae he ecod order TVD rego a how Fgre -9 ad hereore ecod order accrae ad Fgre -6 how ha he La-Wedro cheme wh Sperbee l-lmer mehod coderably more accrae ha who he Sperbee llmer mehod. See Sweby[] or a more -deph aaly o l-lmer mehod. 3

36 φ(θ) 3 φ(θ) θ φ(θ) 3 θ Fgre -7: TVD rego or e derece cheme. φ(θ) 3 φ(θ) θ φ(θ) 3 θ Fgre -8: Secod order TVD rego or e derece cheme. 3

37 3 φ(θ) 3 θ Fgre -9: Sperbee l-lmer or e derece cheme. Table -3 l a ew l-lmer, whch ay he TVD propery ad are ecod order accrae. Name o Fl-lmer φ(θ) Mmod φ(θ) ma(,m(,θ)) oe Sperbee φ(θ) ma(,m(θ,),m(θ,)) θ θ a Leer φ() θ θ θ θ a Albada φ() θ θ Table -3: Some Fl-lmer or ecod order cheme. Throgho Chaper, we hae dced he e derece echqe or appromag he calar coerao law ad, parclar, he lear adeco eqao. Howeer, omeme he rgh had de o (.) o eqal o zero b ead, a orce erm pree whch ca cae dcle appromag he olo accraely. he e chaper, we wll coder ch a cae, where a orce erm ow pree o he rgh had de o (.). 3

38 3 Coerao Law wh Sorce Term (,) Chaper, we dced a arey o e derece cheme or mercally appromag coerao law ad he lear adeco eqao. We alo dcoered ha a mber o problem occr whe mercally appromag coerao law ee whe a orce erm o pree. h chaper, we wll dc ome mercal echqe or olg coerao law whe a orce erm pree. Howeer, h chaper we wll oly coder orce erm ha are co o ad oly,.e. ( ) (, ) (3.) where (,) he orce erm. h chaper, we wll alo e he lear adeco eqao wh c ad orce erm.e. ( ) e. 3,, e >.3. 3, (3.) >.3 wh al daa.3 (,) (3.3) >.3 whoe eac olo 33

39 34 ( ) > e.3.3, a a e problem o llrae ome mercal rel. 3. Bac Approach The mo bac e derece approach ed o mercally appromae (3.) o add he orce erm o a cheme ha mercally appromae he coerao law who orce erm (.). For eample, we e a orward derece appromao me, a ceral derece pace ad ame he orce erm o be a appromao a (,) he (3.) become ad by re-arragg we may oba [ ]. Th ceral cheme codoally able, by g he aerage ( ) we may oba ( ) [ ] whch he r order La-Fredrch cheme wh he orce erm added o ( ) [ ] LF where LF.

40 Upwd (r order) wh d., d. ad o (,) Fgre 3-: The Upwd cheme wh orce erm added o. La-Wedro wh d., d. ad o U(,) Fgre 3-: The La-Wedro cheme wh orce erm added o. 35

41 La-Wedro TVD wh d., d. ad o U(,) Fgre 3-3: The La-Wedro cheme wh Sperbee l-lmer ad orce erm added Comparo o cheme wh orce erm added o eplcly. d., d. ad.5. U(,.5) Eac Solo Upwd (r order) La-Wedro La-Wedro TVD Fgre 3-4: Comparo o dere cheme wh he orce erm added o. 36

42 Th approach wll work wh all cheme dced Chaper ad, geeral SCHEME. (3.4) Here, SCHEME repree a mercal cheme o he coerao law who a orce erm pree. Alo, by amg he orce erm o be a appromao a (,), we ca oba a em-mplc cheme SCHEME. (3.5) Fgre 3-, Fgre 3-, Fgre 3-3 ad Fgre 3-4 are all rel o cheme o he orm (3.4) appled o (3.) wh al daa (3.3). Fgre 3- how he Upwd cheme wh he orce erm added, Fgre 3- how he La-Wedro cheme wh orce erm added ad Fgre 3-3 how he La-Wedro cheme wh Sperbee l-lmer ad orce erm added. Fgre 3-4 how he Upwd cheme, La-Wedro cheme ad La-Wedro cheme wh Sperbee l-lmer, all wh he orce erm eplcly added o. Here, we ca ee ha he Upwd cheme wh orce erm added er badly rom dpao ad ha he La- Wedro cheme wh orce erm added er badly rom dpero relg ery large ocllao beg pree. The mo accrae cheme wa he La- Wedro cheme wh Sperbee l-lmer ad orce erm added. addo, we ca ee all cheme are coerae ce he dcoy wa moed a he correc wae peed. 37

43 38 3. La-Wedro Approach We ca alo e he La-Wedro approach ha we ed Chaper, Seco, o appromae he calar coerao law wh orce erm. Howeer, we m r rewre (.) o clde he orce erm. Now, we ca re-wre (3.) a ) ( ), ( (3.6) a ) ( ad we may oba () ( ) () a a. (3.7) Now, by g Taylor heorem... (3.8) ad bg (3.6) ad (3.7) o (3.8) ge ( )... ) ( ) ( a a ad by g ceral derece appromao pace ad amg ha boh are appromao a (,) he ( ) ( ) ( ) [ ]. ) ) ( ( / / d a d d d Hece, by g a orward derece appromao pace ad me, we may oba

44 39 ( ) ( ) ( ) [ ] a a / / / / / / ad by re-arragg ( ) ( ) ( ) [ ] [ ] ( ) ( ) [ ] / / / / 4 (3.9) we may oba a ecod order appromao o (3.), whch baed o he La- Wedro cheme. We ca alo apply l-lmer mehod o (3.9) by re-wrg (3.9) a ( ) ( ) [ ] [ ] ( ) ( ) [ ] F F / / 4 ; ; where ( ) ( ) ( )φ H L F F F ; ; ; ad ( ) < > ; / / F L ( ) ( )( ) ( )( ) < > ; / / F / / H. where φ deoe he l-lmer mehod decrbed Chaper, Seco 5 ad we cold e ay o he l-lmer Table -3 o oba a ecod order l-lmer mehod. we ow apply he La-Wedro approach, who a l-lmer mehod, (3.9) o he e problem (3.) wh al daa (3.3), we may oba he rel Fgre 3-5.

45 La-Wedro approach wh d., d. ad o U(,) Fgre 3-5: The La-Wedro approach or adeco eqao wh orce erm. Fgre 3-5 how praccally he ame rel a Fgre 3- where he orce erm wa added o he La-Wedro cheme. Th becae he orce erm a kow co o ad o all appromao o he orce erm wll be eremely accrae. Howeer, whe he orce erm alo a co o, he appromao o he orce erm are o a accrae makg he wo cheme accracy chage dramacally, a we wll ee laer. 3.3 MPDATA approach Smolarkewcz ad Margol[3] dered a algorhm o appromae he adeco rapor eqao (3.7) called MPDATA. MPDATA a Mldmeoal Poe Dee Adeco Trapor Algorhm ad appromae he adeco eqao (.3) wh a orce erm pree 4

46 4 ), ( c (3.) whch alo kow a he adeco rapor eqao. The ahor ae ha h algorhm e a mlar approach o ha o he La-Wedro, whch ca be ewed a he Upwd cheme m a error emae, b eplo pecal propere o he Upwd cheme Bac MPDATA Beore we ca dc he MPDATA algorhm or (3.), we m r look a he mo bac MPDATA algorhm, whch baed o he adeco eqao who orce erm, c (3.) we ame ha oegae, he he bac MPDATA algorhm he Upwd cheme (.4) re-wre l orm ( ) ( ) [ ] C F C F,,,, (3.) where ( ) C C C F L L,,, ( ) ( ) C C C C C C c C ad,. Th cheme oly r order ad we reqre a ecod order cheme, b we look a he rcao error o (3.) ( ) ) ( 3 O C C Τ ad ce c > ( ) ) ( ) ( O O c c Τ.

47 Here we ca ee ha (3.) a beer appromao o he adeco-do eqao, K c where K ( C ) ad oly r order pace ad me. Howeer, we ca corc a mercal emae o he error ad brac rom (3.) whch wll make he cheme ecod order. Th approach mlar o ha o he La-Wedro cheme or he adeco eqao, whch e ceral derece o appromae he rgh had de o (3.) wherea MPDATA e pecal propere o he Upwd cheme or appromag ad compeag he error. We ca re-wre he error erm a () ( ) c C, where ( C ) C ( ). a pedo elocy. The by g / ( ) ad () (), where he percrp () deoe he r appromao o he adeco eqao (3.), we may oba he r order accrae appromao V () () ( C ) () / C () ( ) o he pedo elocy. order o oba a ecod order appromao, we brac he error he ecod pa () () () () () () [ F (, ) F(, )] (), V / V /., Hece, we may ow oba he bac MPDATA algorhm () () () () () () [ F (, ) F(, )] (), V / V / (3.3), 4

48 where he pedo elocy () () () ( ) V C / C (3.4) () ( ) ad he r order appromao [ F(, C) F(, C) ],. (), So ar we hae oly codered he adeco eqao wh oegae b he bac MPDATA algorhm ca alo be pdaed or o be o arable g. Th acheed by g he pedo elocy V () () () / ( C C ) () () ead o (3.4). Alo, we ca apply l-lmer o he bac MPDATA algorhm by replacg (3.3) wh () ) () ) () () [ F (, ) φ F(, ) φ ] () ( (, V / V / (3.5), where φ deoe he l-lmer mehod decrbed Chaper, Seco MPDATA Approach or Adeco Eqao wh Sorce Term (,) So ar we hae oly dced he bac MPDATA cheme or he adeco eqao b MPDATA ca alo be adaped o appromae he adeco rapor eqao (3.) c (, ). we e a orward derece appromao me ad ame a appromao a (,). Alo, by amg he orce erm a appromao a (,½) we may oba c / (3.6) 43

49 44 Now, by g Taylor heorem / ad by bg o (3.6) we may oba ) ( O c, (3.7) ) O( c, (3.8) ad ) ( O c. (3.9) Sbg (3.8) o (3.9) ad re-arragg ge ) ( O c c ad bg h o (3.7) we may oba ) ( O c c c. Whece we may oba ) ( O c c c. (3.) The r wo erm o he rgh had de o (3.) how he error de o he orce erm ad he hrd erm how he error de o he mehod. Here, he ecod erm o he rgh had de o (3.) ca blow p creag ery accrae mercal rel, epecally he orce erm (ee Chaper 5, Seco ), b MPDATA compeae or h erm makg he cheme coderably more accrae. The

50 MPDATA approach whch mercally appromae (3.) dered by amg he orce erm appromao o be a (,½) gg where ( C) MPDATA ( C) /, MPDATA, correpod o he bac MPDATA algorhm dced he preo eco (3.3.). Now, by g he aerage, we may oba [ ] / MPDATA (, C) ( ) ad by adecg we may oba he MPDATA cheme or appromag he adeco rapor eqao MPDATA, C (3.) where MPDATA, w / / correpod o he bac MPDATA algorhm dced he preo eco (3.3.). Here, by adecg he alary eld,, he erm he rcao error (3.) de o he orce erm do o blow p. Now, by applyg (3.) ad (3.9) o he e problem (3.), wh al daa (3.3), we may oba he mercal rel Fgre 3-6 ad Fgre 3-7. Boh Fgre 3-6 ad Fgre 3-7 how ha he MPDATA cheme er rom a lo le ocllao behd he dcoy ha he La-Wedro approach, whch mea ha MPDATA le dpere ha he La-Wedro approach. Alo, Fgre 3-7 how ha, ear he dcoy, he MPDATA approach a lo le accrae ha he La-Wedro approach. 45

51 MPDATA approach wh d., d. ad o (,) Fgre 3-6: MPDATA approach or adeco rapor eqao. Comparo o La-Wedro ad MPDATA approach wh d., d. ad (,) Eac Solo La-Wedro approach MPDATA approach Fgre 3-7: Comparo bewee La-Wedro approach ad MPDATA. 46

52 So, he MPDATA approach coderably le dpere ha he La-Wedro approach b, ear he dcoy, he La-Wedro coderably more accrae. B oerall, he MPDATA approach a lo more accrae ha he La-Wedro approach or appromag (3.). geeral, he MPDATA approach (3.) ery accrae whe mercally appromag he adeco-rapor eqao (3.) MPDATA Approach or Coerao Law wh Sorce Term (,) So ar we hae oly looked a MPDATA algorhm or he adeco-rapor eqao. Le ow coder MPDATA algorhm or he calar coerao law wh orce erm pree (3.),.e. ( ) (, ) MPDATA ca be adaped o appromae (3.) by coderg he elocy c o he adeco-rapor eqao o o loger be a coa b o be a co o ead,.e. ( w( ) ) (, ) where w ( ) or cd brger eqao, ec. The bac MPDATA algorhm or he coerao law who orce erm ow ake he orm where he pedo elocy V () () () () () () [ F (, ) F(, )] (), V / V / (3.), () () / ( ) / / / / / [ 3/ / ] w [ w / / ] () () w w w (3.3) () / ad he r order appromao / / [ F(, ) F(, )] ( ),, w / w /. 47

53 Howeer, w / / kow ce w a co o ad oly kow a he grd po (,). We cold appromae w / / by g he aerage or by g lear erpolao w w w ( ) / / / / w / / 3 ( ) w / w /. we appromae by g lear erpolao, he mehod wold reqre aoher cheme o ally ar he algorhm o, ce we reqre a ale o a (,-), b we e he aerage, he algorhm become mpraccal ce we reqre he ale o a (,). So ar we hae oly codered he mo bac MPDATA algorhm or he coerao law who orce erm ad hae ecoered a lo o dcle. we ow coder a orce erm he he correpodg MPDATA cheme / MPDATA, w / (3.4) where MPDATA, / w / correpod o he bac MPDATA algorhm, or he coerao law who orce erm, dced Seco 3.. Howeer, care m be ake whe g h cheme ce he orce erm a co o he ee more dcle are whe g h algorhm a we wll ee laer. 3.4 Comparo o Scheme Ug Te Problem Now, by g he e problem (3.) wh al daa (3.3), we ca oba he mercal rel Fgre 3-8 ad Fgre 3-9 ad compare he mercal olo o 48

54 he hree approache dced hrogho h chaper wh he eac olo. The hree approache are:. Upwd (r order) wh he orce erm added o.. La-Wedro approach wh or who Sperbee l-lmer. 3. MPDATA approach wh or who Sperbee l-lmer. Fgre 3-8 compare he mercal olo o he hree dere approache wh he eac olo a.5. Fgre 3-9 compare he re error o he mercal olo o he hree dere approache a.5. Fgre 3-8 how ha he Upwd cheme er rom dpao a epeced ad ha he MPDATA approach who TVD ge le ocllao behd he dcoy ha he La-Wedro cheme who TVD. Fgre 3-9 how ha he mo accrae approach oerall he La- Wedro approach wh Sperbee l-lmer ollowed by he MPDATA approach wh Sperbee l-lmer. We ca alo ee ha ear he dcoy, he MPDATA approach le accrae ha he La-Wedro approach, b away rom he dcoy, he MPDATA approach coderably more accrae ha he La- Wedro approach. Howeer, he MPDATA approach wll be o o accrae whe appled o he calar coerao law ce he MPDATA approach wold reqre pecal arg procedre ad appromae appromao. Th de o he MPDATA approach beg peccally dered o appromae he lear adeco eqao wh orce erm pree ad o he calar coerao law. Throgho h chaper, we hae dced hree ma approache whch appromae he calar coerao law wh orce erm, whch a co o ad ad we hae obaed ome ery accrae rel. Howeer, we hae oly codered kow orce erm ad whe he orce erm ha o be appromaed, he dere approache dced h chaper are o o accrae. he e chaper, 49

55 we wll coder ome mehod or appromag he calar coerao law wh he orce erm beg a co o alo. Sce he orce erm ow a co o a well, we wll ow eed o appromae he orce erm a well. 5

56 Comparo o 're' error o he cheme wh d., d. ad U(,) Eac Solo Upwd (r order) La-Wedro MPDATA La-Wedro TVD MPDATA TVD Fgre 3-8: Comparo o he hree approache dced h chaper. Comparo o 're' error o cheme wh d., d. ad 're' error Upwd Error La-Wedro error MPDATA error La-Weddro TVD error MPDATA TVD error Fgre 3-9: Comparo o re error o he hree approache dced h chaper. 5

57 4 Coerao Law wh Sorce Term (,,) Chaper 3, we dced ome e derece cheme ha mercally appromae coerao law wh a orce erm whch a co o ad. h chaper, we wll dc ome e derece cheme ha mercally appromae coerao law wh a orce erm whch ow a co o, ad,.e. ( ) (,, ) (4.) where (,,) he orce erm. We hall ee ha dcle wll are ce he orce erm ow a kow co o a well a ad, relg he mercal appromao o he orce erm o loger beg eac. Throgho h chaper, we wll be g he ollowg e problem codered by LeVeqe ad Yee[]. where wh al daa () () ( ), (,), (4.).3 >.3 ad whoe eac olo.3, (4.3) >.3 ( ) o llrae ome mercal rel. 5

58 4. Adapao o he Scheme or he Coerao Law wh Sorce Term (,) h eco, we wll dc how he dere approache or coerao law wh a orce erm o he orm (,), whch were dced Chaper 3, ca be adaped o mercally appromae (4.). 4.. Bac Approach The addg o he orce erm ca be ealy adaped o mercally appromae eqao (4.). We do o eed o adap cheme (3.4) ce we ca re-wre he orce erm appromao o clde,.e. (, ) whch kow ce he, ale o are kow o, he cheme rema a SCHEME. (4.4) Howeer, cheme (3.5) em-mplc ce (,, ) are o ye kow o we eed o re-wre b he ale o SCHEME Oe approach o e Taylor heorem o oba ad by bg (4.6) o (4.5), we may oba. (4.5) O( ) (4.6) SCHEME. Here, we cold e e derece o appromae,.e., or, 53

59 54 b we wold he ecoer oher dcle ce we oly kow he ale o ad, ecep ally whe we do o kow he ale o. We cold alo calclae he derae aalycally ad he appromae he derae,.e. b he derae o he orce erm may be eremely dcl o d ce he orce erm a co o ad a co o ad. Aoher approach we cold ake o re-arrage (4.6) by g he cha rle,.e. ( ). Sbg o (4.6) ge ( )... ad by bg o (4.5), we may oba SCHEME, (4.7) a em-mplc cheme whch add he orce erm mplcly. Here, ce he orce erm a kow co o, ad, we ca calclae he derae aalycally ad he appromae he derae,.e..

60 Comparo o cheme wh orce erm added o eplcly. d., d. ad (,.5) Eac Upwd (r order) La-Wedro La-Wedro TVD Fgre 4-: Comparo o dere cheme wh he orce erm added o. we e (4.4) o mercally appromae he e problem (4.), we may oba he rel how Fgre 4-. Fgre 4- how ha he Upwd cheme wh he orce erm added ll gg he lea accrae rel ad ha he La-Wedro wh Sperbee l-lmer gg he mo accrae rel. 4.. La-Wedro Approach The La-Wedro approach ca be adaped o mercally appromae (4.) b wh dcly. Drg he derao o he La-Wedro approach or mercally appromag (3.), we obaed ( ) [ / ( ) / ( )] ( a( ) ). 55

61 56 We he ed a orward derece appromao pace ad me, o oba ( ) ( ) ( ) [ ] a a / / / / / / ad hece, ( ) ( ) ( ) [ ] [ ] ( ) ( ) [ ] / / / / 4. Howeer, he orce erm ow alo a co o, he (4.8) become emmplc ce we o loger kow he ale o. We cold replace wh (4.6) a we dd he preo b-eco, b h wold oly creae more problem. Howeer, we cold replace wh ( )... ad oba ( ) ( ) ( ) [ ] ( ) ( ) [ ] / / / / 4 he La-Wedro approach or appromag (4.). We ca alo apply l-lmer mehod o he La-Wedro approach ad oba ( ) ( ) [ ] ( ) ( ) [ ] F F / / 4 ; ; (4.8) where ( ) ( ) ( )φ H L F F F ; ; ; ad

62 F ( ; ) F L ( ; ) / / ( / )( ) ( / )( ) > < / > < / H. Here φ deoe he l-lmer mehod, whch ca be ay o he l-lmer Table -3. we e (4.4) o mercally appromae he e problem (4.), we may oba he rel how Fgre 4-. Here, we ca ee ha he La-Wedro approach ha mercally appromaed (4.) ery accraely. Alo, he mercal rel Fgre 4- are ery mlar o he mercal rel Fgre 4-, where we added he orce erm. geeral, he La-Wedro approach more accrae ha eplcly addg he orce erm a we hall ee laer. La-Wedro approach wh ad who Sperbee l-lmer, where d., d. ad U(,.5) Eac La-Wedro La-Wedro TVD Fgre 4-: La-Wedro approach wh ad who Sperbee l-lmer. 57

63 4..3 MPDATA approach The MPDATA approach alo creae dcle whe he orce erm alo a co o. Th becae (3.) reqre he ale o ad we m re-wre (3.) o ha mercally appromae he coerao law wh orce erm ead o he adeco-rapor eqao, whch wa dced Chaper 3, Seco 3.3. Alo, he MPDATA approach reqre aoher cheme o ar o ally, ce we eed o re-wre (4.) a b by g lear erpolao w ( w( ) ) (,, ) / / 3 ( ) w / w /. Here, we wold reqre he al ale o w ale o w /. B, we ed he aerage, w / ad w ( ) / / w / w / / b we oly kow he he algorhm wold become mpraccal ce by g we oba w / / w w w w / ( )... ( ) / / w / b he ale o / are kow. We ca oercome he dcly o he ale o beg kow by g ( )... 58

64 ad by bg o (3.4), o oba / MPDATA, w / (4.9) where ( C) MPDATA, correpod o he bac MPDATA algorhm wh llmer () () () () () () [ F (, ) φ F(, ) φ ] () V / V /,, whoe pedo elocy V () () ( ) / / / [ / / / 3/ / ] w [ w / / ] () () w w w, () / he r order appromao / / [ F(, ) F(, )], w / w /, ( ), w / / 3 / / ( ) w ad φ ca be ay o he l-lmer led Table -3. Here, we ca ee ha he MPDATA approach or mercally appromag (4.) becomg ery mpraccal. Th becae we are appromag appromao relg he accracy o he algorhm redcg rapdly ad we alo reqre aoher cheme o ar he algorhm o. Howeer, MPDATA ca be ed o accraely mercally appromae he adeco-rapor eqao wh orce erm, (,,). we e (4.4) o mercally appromae he e problem (4.), we may oba he rel how Fgre 4-3. Here, we ca ee ha he MPDATA approach qe accrae b o a accrae a he rel obaed Fgre 4- ad Fgre 4-. Alo, MPDATA wll o be o accrae or he cd brger cae wh orce erm. w 59

65 MPDATA wh ad who TVD where d., d. ad (,.5) Eac MPDATA MPDATA TVD Fgre 4-3: MPDATA approach wh ad who Sperbee l-lmer. 4. oe Upwd Approach 4.. Adeco Eqao wh Sorce Term () oe[6] dered a e derece cheme whch mercally appromae c (), (4.) where c > ad () he orce erm, wh ecod order accracy. we coder he al-ale problem o (4.) wh al daa ( ) ( ) geeral olo, o c ( ) ( c) ( ) d c, o, we may oba he whch ca be re-wre a ( ), ) ( ) d. (4.) ( ) c 6

66 Here we ca ee ha he r erm o he rgh had de o (4.) ca cae dcle he Cora mber o a eger. Th becae we are g a meh where we oly kow he ale a he grd po (,) ad o a eger, he he ale o reqred o loger le o he meh ad h kow. Howeer, oe[6] dedced ha he oly reaoable way o appromae h erm o e ( ), ) ( ) (, (4.) whch he Upwd approach, ce o oher ormla coe wh (4.). Alo, (4.) ge he malle rcao error o all poble choce where he rcao error ha poe coece ad deped oly o. The egral erm pree (4.) ca be appromaed mero way. Sce we kow (), we cold egrae he orce erm ad e he eac ale b a more geeral approach o e a wo-po appromao [( α) α ] ( ) d ( ). (4.3) c Here, he ale o α arbrary ad m be choe ch ha α. Hece, by bg (4.3) ad (4.) o (4.), we may oba ( ) [ ( α) ] (4.4) α whch oe Upwd approach or mercally appromag (4.). Howeer, h cheme oly able or c >, b c < he we may oba ( ) [ ( α) ] (4.5) α ad by combg (4.4) ad (4.5) ge ( ) [ ( α) α ] ( ) [ ( α) α ] > < whch oe Upwd approach. Moreoer, oe[6] od ha we ake 6

67 6 α he we may oba a cheme ha ecod order accrae he eady ae,.e. ( ) [ ] ( ) [ ] < >. Uoraely, h cheme oly a r order appromao o (4.) b we ca alo oba a ecod order accrae cheme by g a Leer MUSCL approach [] o oba ( ) ( )[ ] ( ) [ ] α α where c >. Ad hece we may oba ( ) ( )[ ] ( ) [ ] ( ) ( )[ ] ( ) [ ] < α α > α α (4.6) whch a ecod order appromao o (4.). Noce ha (4.6) he La- Wedro cheme or mercally appromag he adeco eqao who a orce erm wh a orce erm appromao added. 4.. Coerao Law wh Sorce Term (,,) Bermdez ad Vazqez[4] adaped oe Upwd approach or mercally appromag (4.) o mercally appromae he adeco eqao wh orce erm (,),.e. ), ( c (4.7) where c >. They ed a mlar approach a he preo b-eco o oba

68 ( ), ) ( c( ), ( c( ), ) )d c (4.8) ad hece, ( ) [ ( α) ], α whch mercally appromae (4.7) ad decal o (4.4). They alo dced aoher approach, whch wa o appromae he egral erm o (4.8) wh c ( c( ), ( c( ), ) d ( α) α,( α) ) α ead o c ( c( ), ( c( ), ) d ( α) [ ] α Hece, Bermdez ad Vazqez[4] obaed wo approache o mercally appromae (4.7) whch oe Upwd approach ad ( ) [ ( α) α ] ( ) [ ( α) α ] >, (4.9) < ( ) ( α) α, ( α) α) ( ) ( α) α,( α) α ) > (4.) < Here, α ½ alo ge ecod order accracy he eady ae or boh cheme. Vazqez ad Bermdez[4] alo dc aro choce o α ad ge ome eral o abole ably ad poy, where c >, or he dere ale o α, whch are led Table

69 eral O Abole eral O Abole Scheme Sably ( c > ) Poy ( c > ) λ α λ λ α ad ad λ λ λ α (or () oly) ad λ ad λ λ α ad λ λ ad Table 4-: eral o abole ably ad poy or (,) -λ. Noce ha whe α, boh (4.9) ad (4.) become he Upwd cheme wh orce erm added a dced Chaper 3, Seco. Th approach ca be ealy adaped o mercally appromae (4.) by re-wrg (4.9) ad (4.) a ( ) ( ) α ( ) ( α) [ α ] [ α ] > /, (4.) < / whch oe Upwd approach, ad ( ) ( α) α ( α), ( ) ( α) α, ( α) ( α) ( α ) / / >. (4.) < Here, boh (4.) ad (4.) are r order accrae cheme b we ca oba ecod order accrae cheme by g ad ( ) [( / )( ) ( / )( )] [( α) α ] ( ) [( / )( ) ( / )( )] [( α) α ] / / > < 64

70 65 ( ) ( )( ) ( )( ) [ ] ( ) ( ) ( ) ( ) ( )( ) ( )( ) [ ] ( ) ( ) ( ) < α α α α > α α α α,, / / / / / / Alo, we ca apply he l-lmer mehod o oba ( ) ( ) [ ] ( ) ( ) < α α > α α ; ; / / F F (4.3) ad ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) < α α α α > α α α α,, ; ; / / F F (4.4) where ( ) ( ) ( )φ H L F F F ; ; ; ad ( ) < > ; / / F L ( ) ( )( ) ( )( ) < > ; / / F / / H. Here, φ repree he l-lmer, whch ca be ay o he ecod order l-lmer Table -3.

71 4..3 Some Nmercal el or he Eplc Upwd Approach Now, by g (4.4) o mercally appromae he e problem (4.), we may oba he mercal rel Fgre 4-4. Comparo o cheme baed o oe' Upwd approach wh d., d. ad (,) Eac Upwd (r order) La-Wedro La-Wedro TVD Fgre 4-4: Comparo o cheme baed o oe Upwd approach. Here, we ca ee ha oe pwd approach gg ome ery accrae rel, epecally or he ecod order La-Wedro pl Sperbee l-lmer, b he rel are o a accrae a Fgre 4-, where we added he orce erm, ad Fgre 4-, where we ed he La-Wedro approach. Howeer, we wll ee laer ha, geeral, oe Upwd approach a lo more accrae a mercally appromag (4.) ha addg he orce erm ad he La-Wedro approach, epecally whe he orce erm. 66

72 4.3 mplc Upwd Approach Embd, Goodma ad Majda[] dced ome dere approache or mercally appromag ( ) (, ) (4.5) where he orce erm m be o he orm ( ) e( ) g( ),. They dced he r order Egq-Oher cheme, wh wchg hrogh zero, ad a ecod order Upwd approach baed o he Egq-Oher approach. Here, we wll e he aaly o Embd, Goodma ad Majda[] o dere a r ad ecod order mplc Upwd cheme wh he orce erm added mplcly Fr Order mplc Upwd Approach The r cheme ha we wll dc he mplc r order Upwd approach wh he orce erm added mplcly,.e. ( ) > /. (4.6) ( ) / < Here, we wll eed o re-arrage (4.6) o he yem A G, where A a () () mar ad G a () colm ecor, ad ole h yem a eery me ep. Howeer, dcle ca are whe re-arragg (4.5) o yem orm. Coder (4.6) whe / > ( ) 67

73 68 ad, by re-arragg we may oba ( ) ad ce ( ) ( ) ( ) g e, ( ) g e. (4.7) Howeer, he ecod ad hrd erm o he le-had de o (4.7) creae dcle ce hey are co o ad o we cao re-arrage (4.7) o yem orm. We ca oercome h problem by g Taylor heorem,.e. ( )... ad ( )... g g g Now by bg o (4.7) ( ) ( ) ( ) g g e ad by re-arragg we may oba ( ) ( ) ( ) g e g e (4.8) where / >. Smlarly, we may oba ( ) ( ) ( ) g e g e (4.9) where / <. Hece, by combg (4.8) ad (4.9), we may ow oba he yem orm

74 69 ( ) ( ) d G G G G G a G b a d b a d b a d b a d b a d b (4.3) where ( ) ( ) < > / / g e G, < > / / a, < > / / d ad g e b ) g( /. Here, A a r-dagoal mar ad o, h yem doe o reqre oo may calclao ad ce ad g are kow co o, we ca appromae ad g Secod Order mplc Upwd Approach We ca alo oba a ecod order appromao by g he mplc ecod order Upwd approach wh he orce erm added mplcly,.e. ( )( ) ( )( ) ( )( ) ( )( ) < > 3 3 / / 3/ / 3/ / (4.3) We wll eed o re-arrage (4.3) o yem orm ad ole or each me ep he ame way we dd he preo b-eco o oba

75 7 ( ) ( ) ( )[ ] ( )[ ] ( )( ) ( )( ) [ ] g e g e 3/ / 3/ 3/ / / (4.3) where / > ad ( ) ( ) ( )[ ] ( )( ) ( )( ) ( )( ) [ ] g e g e / 3/ 3/ 3/ / / (4.33) where / <. Hece, by combg (4.3) ad (4.33), we may ow oba he yem orm ( ) ( ) ( ) ( ) ( ) ( ) k d G k G G G l G l a G b a l d b a l k d b a l k d b a l k d b a k d b where ( )( ) ( )( ) ( )( ) ( )( ) < > 3 3 / / 3/ / 3/ / g e G, [ ] < > / / 3/ l,

76 7 [ ] < > 4 / / 3/ / a, < > 3 3 / / - g e b / /, [ ] < > 4 / 3/ / / d ad [ ] < > / 3/ / k. Here, A a pea-dagoal mar ad oraely reqre a lo more calclao ha beore relg he eral o abole ably ad he accracy o he cheme beg redced. Howeer, Embd, Goodma ad Majda[] dced g he r order r-dagoal mar or he ecod order Upwd approach baed o he Egq-Ocher cheme o creae he eral o abole ably. Ug he ame approach, we ca oba he ecod order mplc Upwd approach,.e. ( ) ( ) d G G G G G a G b a d b a d b a d b a d b a d b (4.34) where ( )( ) ( )( ) ( )( ) ( )( ) < > 3 3 / / 3/ / 3/ / g e G,

77 ad a / > /, b g( < d g / ) e. / / > < They alo ae ha by g he r order mar, he eral o abole ably creae relg a more rob cheme. We ca alo apply l-lmer mehod o (4.34) o mme ay ocllao pree he mercal olo. Th obaed by replacg G (4.34) wh [ F( ; ) F( ; ) ] G e g where ad F ( ; ) ( ; ) ( ; ) ( )φ F L F H F ; F L ( ; ) / / ( / )( ) ( 3/ )( ) > < / > < / H. where φ repree he l-lmer, whch decrbed more deal Chaper, Seco Some Nmercal el or he mplc Upwd Approach we apply (4.3) ad (4.34) wh ad who l-lmer o he e problem (4.), we may oba he mercal rel Fgre 4-5. Here, we ca ee ha he rel o he r order mplc Upwd approach are qe accrae b he mehod er rom dpao. Alo, oce ha he ecod order mplc Upwd approach prodced he mo accrae rel. 7