Outline. Fully Implicit (BTCS) Results. Fully Implicit Method. Fully Implicit Results II. Fully Implicit Results III

Transcription

1 ore mercal dso opcs arch 3, 9 Addoal Topcs Nmercal Solos o Parabolc Eqaos arr Careo echacal Egeerg 5AB Semar Egeerg Aalss arch 3, 9 Ole Complee dscsso rom las class Fll mplc ad DFor-Frael mehods Revew mercal dso solos Eplc ad Cra-Ncholso mehods Thomas algorhm Geeral properes o mercal algorhms or PDEs Nmercal dso eqao solos wo or more space dmesos Fll Implc ehod Dscree dso eqao a O ad O[ T O[, Trdagoal ssem o eqaos Almos same wor as CN ad o spros oscllaos, b less accrac or all mes. 3 Fll Implc BTCS Resls Same as CN ps:,,.,.5, / Fll Implc Resls II Fll Implc Resls III Eac Error E 5B Egeerg Aalss

2 ore mercal dso opcs arch 3, Eac Error Compare Cra Ncholso Rchardso/eaprog Use wo me sep ceral dereces O[ O[ Resl s eplc wh secod order accrac me However resl s sable or a ad cao be sed 7 8 DFor Frael odcao o Rchardso mehod o provde sabl Replace secod dervave b average a me seps ad - Irodces aoher O[ error O[ O[ O[ T O,, 9 DFor Frael Rearrage ad rodce / Resl s eplc or vales a me Eplc sar reqred o ge rs se o vales a me - Ca show ha hs s codoall sable Trcao Error Trcao error or DFor-Frael Dervao apped o oes o mercal solos or PDEs Se / or mmm rcao error T TE! T 3! T! T Revew Dso Algorhms Eplc - Fll mplc Cra- Ncholso DFor-Frael New me Old me R [ E 5B Egeerg Aalss

3 ore mercal dso opcs arch 3, 9 Revew Implc Eqaos Cra Ncholso ad ll mplc [ R have hree vales a ew me sep Trdagoal ssem o eqaos easl solved b specal applcao o Gass elmao called Thomas algorhm Geeral rdagoal orm s A B C D 3 Revew Thomas Algorhm Solo procedre or rdagoal eqaos A - B C D Specal applcao o coveoal Gassa elmao Eqaos or E ad F,, N Ge N F N Ge F E or N o B A C B A C B A 3 C B 3 O B N A N D D D CN N DN B N N DN 4 Revew Sabl ms Saw epeced oscllaos whe me sep s oo large Sable algorhm: oscllaos damp o Usable: oscllaos grow who bod Codoal sabl: lm o me sep Eplc mehod reqres / < ½ Fll mplc, Cra-Ncholso, ad DFor-Frael are codoall sable Have large errors wh oo large Keep abo 5 RS Temperare Erro.E-.E-3.E-4.E-5.E-6.E-7.E-8 Error Eplc Solo o Codco Eqao Thermal Dsv < < T, T, T, Ed Tme mm error or /6... / N 5 N N N 5 N N N 5 N.E- Error Cra-Ncholso Solo o Codco Eqao Error Fll Implc Solo o Codco Eqao.E-. RS Temperare Erro.E-3.E-4.E-5.E-6.E-7.E-8 Thermal Dsv < < T, T, T, Ed Tme.... / N 5 N N N 5 N N N 5 N RS Temperare Error / Thermal Dsv < < T, T, T, Ed Tme N 5 N N N 5 N N N 5 N E 5B Egeerg Aalss 3

4 ore mercal dso opcs arch 3, 9.E- Error DFor Frael Solo o Codco Eqao.E3 RS Temperare Error b Eeco Tme.E- mm error or /.E.E RS Temperare Erro.E-3.E-4.E-5.E-6.E-7 Thermal Dsv < < T, T, T, Ed Tme N 5 N N N 5 N N N 5 N RS Temerare Error.E.E-.E-.E-3.E-4.E-5.E-6.E-7 Cra-Ncholso Eplc Fll Implc DFor-Frael.E /.E-8.E-.E-.E.E.E.E3.E4.E5 Eeco Tme secods Properes o Solos Cossec rcao error becomes ero as sep ses approach ero Sabl errors rema boded Coverge eds o he eac solo o he dereal eqao as he grd se eds owards ero Phscal real Solos prodce phscall realsc resls Accrac a sorces o error Cossec Wre e derece epresso The wre Talor seres or each algebrac erm e derece epresso Resl wll be a eqao wh dervaves ad,,,, ec. I we le,,,, ec.approch ero are we le wh he dereal eqao ha we are rg o solve? mercal solos. Sabl Solo o derece eqaos wh e seps gves dere resls Absole sabl: solo remas boded or a choce o sep se Codoal sabl: solo remas boded or lmed sep ses Ucodoal sabl: Solo s o boded or a choce o sep se Preer absole sabl b ca wor wh codoal sabl Covergece A algorhm ha s cosse ad sable s coverge a s heorem No sall a praccal cocer o a ser o algorhms Ca r a code wh dere sep ses ad see ha solo approaches a cosse vale as sep se decreases DFor Frael s eher cosse or coverge, b gves good resls or some choces o ad 3 4 E 5B Egeerg Aalss 4

5 ore mercal dso opcs arch 3, 9 Phscal Real Solos o dso eqao show orm decreases Cra-Ncholso, a sable ad coverge algorhm ca gve oscllaos Solos o he wave eqao shold have o dsspao a orgal wave orm shold propagae wh o chage shape some algorhms ma o gve hs resl Problems wh Accrac Trcao ad rod-o errors Ierave solo o algebrac eqaos Poor phscal models or assmpos Blders Programmg errors Icorrec daa eres serpreao o resls Bele ha pre op ms be correc 5 6 Two Space Dmesos Eeso o oe space dmeso Have grd as well as ad me Eplc mehod has almos o derece Implc mehods harder o solve Dereal eqao show below Eplc mehod ses same me ad space dervaves as oe space dmeso 7 A Small Grd N 6, T Oe grd or 4 each me 3 sep Compaoal olecle Bodar odes 8 Two Space Dmesos II Eplc mehod has error ha s secod order space ad rs order me 9 Two Space Dmesos III I h so ha, he eplc eqao becomes 4 4 ms be posve or sabl 4 I geeral ms be posve 3 E 5B Egeerg Aalss 5

6 ore mercal dso opcs arch 3, 9 CN Two Space Dmesos CN Two Space Dmesos II Oe-dmesoal e-derece eqao T Eed hs o wo dmesos 3 Sbse / ad / T T T [ lpl b ad rearrage o oba [ R 3 FI Two Space Dmesos Oe-dmesoal ll mplc Eed hs o wo dmesos Ssem o mplc eqaos s o rdagoal oo a small grd Keep same grd or all me seps Have o sore oe me sep ol 33 Old me - Fe-derece Grd - Dscss erao mehods wh aplace eqao Node relaed o or eares eghbors a boh me seps CN ses all poeal vales a old me sep - - New me 34 A Small Grd N 6, T Oe grd or 4 each me 3 sep Compaoal olecle Bodar odes 35 Alerag Dreco Implc ADI s echqe or allowg se o Thomas algorhm or smple solo A each me sep, rea oe dreco as eplc ad oe as mplc Echage eplc ad mplc drecos a each me sep Keep resls a each me sep, b alwas have a eve mber o me seps Dere approaches clde Peacema- Radord ad Doglas-Radord 36 E 5B Egeerg Aalss 6

7 ore mercal dso opcs arch 3, 9 E 5B Egeerg Aalss 7 37 Peacema-Radord ADI Solve rs or appromae a Solve e or acal resl Each eqao gves a rdagoal ssem 38 Peacema Radord II Irodce / ad / ad ge rdagoal orms Frs me sep Secod me sep R R 39 Three Dmesoal Problem Eed eplc wo-dmesoal ederece orm o hree-dmesos 4 Doglas-Bra ehod A ADI mehod hree seps 4 Doglas-Bra ehod II Each sep solvg or,, ad, has a rdagoal ssem o eqaos Ca show ha hs s codoall sable wh O[,,, rcao error 4 Operaor Splg A mehod smlar o ADI Allows se o Thomas algorhm o solve a rdagoal ssem o eqaos Uses derece operaor oao, φ φ φ - φ ad φ φ φ - φ Uses wo operaor acors ad gores cross prodc whch prodces same error as me dervave, O[

8 ore mercal dso opcs arch 3, 9 E 5B Egeerg Aalss 8 43 Operaor Splg e ADI, b do o se mdpo sep Here se derece operaor oao, 44 Operaor Splg II Use appromae acorao Igorg hs erm gves O[ error Resl wh O[ erm egleced 45 Operaor Splg III Dee ermedae vale Solve wo ses o rdagoal eqaos or each me sep