Outline. Finite Difference Grids. Numerical Analysis. Finite Difference Grids II. Finite Difference Grids III

Transcription

1 Itrodcto to Nmercl Alyss Mrc, 9 Nmercl Metods or PDEs Lrry Cretto Meccl Egeerg 5B Semr Egeerg Alyss Mrc, 9 Otle Revew mdterm soltos Revew bsc mterl o mercl clcls Expressos or dervtves, error d error order Nmercl metods or te dso eqto Explct d Implct Frst d secod order tme dervtves Nmercl Alyss Wt to express dervtves d tegrls terms o dscrete dt pots Use deret metods Develop terpolto polyoml d tegrte or derette ts reslt Use Tylor seres to get expressos or dervtves Wt expressos d mesre o error wt ter se Fte Derece Grds Sbdvde rego to dscrete pots Spcg betwee te pots my be orm or o-orm Exmple: grd or x m x x mx wt N odes mbered rom zero to N Itl ode vle, x x m Fl grd ode vle, x N x mx Nodl spcg Δx x x - (, N) Uorm spcg, Δx (x m x mx )/N N odes gve N spces Fte Derece Grds II No-orm grd llstrted below Δx ~ Δx Δx Δx Δx ~ N- N x x x x x N- x N- x N Two spce dmesos reqre x d y grds (M y odes) x x m x N x mx x x - Δx y y mj y M y mx y j y j- Δy j Most geerl cse s tree spce dmesos (x, y, z, d tme) 5 Fte Derece Grds III Grd otto or or depedet vrbles: x, y, z d t x x m x N x mx x x - Δx y y mj y M y mx y j y j- Δy j z z m z K z mx z k z k- Δz k t t m t L t mx t t - Δt Depedet vrble (x,y,z,t) t dscrete pots (x, y j, z k, t ) Use otto below or ts vle o jk ( x, y j, zk, t ) ME 5B Egeerg Alyss

2 Itrodcto to Nmercl Alyss Mrc, 9 Dervtve Expressos Obt rom derettg terpolto polyomls or rom Tylor seres Seres expso or (x) bot x ( x) ( ) d ( x ) Note: d / d!! d ( x - ) d! d ( x) ( x - )! Wt s error rom trctg seres? ( x - )... 7 Trcto Error I we trcte seres ter m terms ( x) m d! ( x - ) d! m ( x - ) Terms sed Trcto error, ε m C wrte trcto error s sgle term t kow locto (dervto bsed o te teorem o te me) m d d m ε ( - ) ( - ) m x x m! ( m )! m ξ ξ kow (betwee x d ) 8 Dervtve Expressos Look t te-derece grd wt eql spcg: Δx so x x Wt Tylor seres or k (x k ) terms o (x ) d dervtves t x x x k x [x ( k)] [x ] k ( x k) ( x ) d x x k! d x x ( k) ' d '' d... d! x d ( k)... x 9 Dervtve Expressos II Combe ll detos or compct seres otto ( x k) ( x ) d x k! d x ( k) d! x '' ' ( k) ( k) k k...!! Use ts orml to get expsos or vros grd loctos bot x x d se reslts to get dervtve expressos ( k)... Dervtve Expressos III Apply geerl eqto or k d k k ' '' ' ( k) k! ( k)! '' '... ''!! '...!! ''...!! A Forwrd ''...!! A Bckwrd '. Dervtve Expressos IV Sbtrct d - expressos '' '!!!... 5! '' '!! '' '... '! 5! 5... Reslt clled cetrl derece expresso A... ME 5B Egeerg Alyss

3 Itrodcto to Nmercl Alyss Mrc, 9 Order o te Error Forwrd d bckwrd dervtve ve error term tt s proportol to Cetrl derece error s proportol to Error proportol to clled t order Redcg step sze by ctor o redces t order error by ε ε Order o te Error Notto Wrte te error term or t error term s O( ) Bg o otto, O, deotes order Recogzes tt ctor mltplyg my cge slgtly wt Frst order orwrd Frst order bckwrd ' ' O( ) O( ) Secod order cetrl ' O( ) Oter Dervtves Secod-order, cetrl-derece, secod dervtve! 5! '' ''... Secod-order, orwrd d bckwrd derece, rst dervtves ' ' ( ) O Oter Dervtve Expressos C derve vros te-derece expressos or dervtves Dervtve order, rst, secod, etc. Order o te error (typclly secod ltog ger orders sed) Forwrd, bckwrds d cetrl derece expressos (typclly se cetrl except t bodres) Derve by Tylor seres mpltos See reslts o pge 7 o Hom Order o Error Exmples Tble trodcto otes sows rst dervtve error or e x rod x Usg rst- d secod-order orwrd d secod-order cetrl dereces Step.,., d. Error rto or doblg step sze. to. or cetrl dereces.7 to.5 or rst-order orwrd dereces. to.9 or secod-order orwrd log ε log( ) log( ) ε ε ε log log( ) log( ) 7 Rodo Error Possble dervtve expressos rom sbtrctg close dereces Exmple (x) e x : (x) (e x e x- )/() d error t x s (e e - )/() - e E.788.5x (.)..785 Secod order error E x (.) E x (.) 9 8 ME 5B Egeerg Alyss

4 Itrodcto to Nmercl Alyss Mrc, 9 Error vs. Step Sze Plot Fgre -. Eect o Step Sze o Error Plot te log o te error verss te log o te step sze For regos were tere s o rodo error ts wll be strgt le wose slope s te order o te error log ε log ε ε ε ( log log ) log ε log ε log log ε log ε log log 9 Error.E.E.E-.E-.E-.E-.E-5.E-.E-7.E-8.E-9.E- Slope o log error log plot s order Δ(log ε) ; Δ(log ) 5 Slope order.e-.e-7.e-5.e-.e-.e-9.e-7.e-5.e-.e- Step Sze Nmercl PDE Soltos Dee te-derece grd te depedet vrbles (x, y, z, t) Plce grd pots o rego bodry wose vles re od rom bodry codtos or te problem At some grd locto covert deretl eqto to te derece eqto Observe trcto error process Neglect trcto error to get set o lgebrc eqtos to solve t Dso Eqto Apply derece ormls derved or ordry dervtves to prtl dervtves Use otto to cosder deret coordte drectos Apply to dso eqto α t x Grds x x Δx d t t Δt Try te derece expressos below to get smple te-derece eqto O( Δt) Δt d x O[( ] Dso Eqto II Sbsttte te derece expressos to deretl eqto α Δt O[ Δt,( ] Igore trcto error, solve or αδt αδt ( ) Obt potetl t x x d t t terms o vles t old tme step Explct (FTCS) Metod Metod jst derved s clled explct metod; c solve oe eqto t tme Δt ( ) ( ) ( ) αδt α x t t αδ does ot deped o oter vles t te ew tme step () ME 5B Egeerg Alyss

5 Itrodcto to Nmercl Alyss Mrc, 9 Explct Metod Exmple Explct Metod Reslts. Pck α, Δx.5, N x, Δt. αδt/( (.)/(.5). Pck tl d bodres, or tme > ( ) Apply ( ) ( ) [ ] ( ).[ ].8[] 8 [ ] ( ).[ ].8[] [ ] ( ).[ ].8[] 8 Repet or sbseqet tme steps t t t. t. t. t. t.5 t. t.7 t.8 x. x x x x. 5 Explct Metod Reslts. x. x.5 x.5 x.75 x. t. 7.5 t t t t t t t t Exct t Error t Explct Reslts. x. x.5 x.5 x.75 x. t t t. 8 8 t t t t t Exct t Error t Explct Reslts. Wt Hppeed? 5 Exct Error t t t. t.8 t. t. t. t. t. x. x x x x. We re seeg eects o stblty Derece eqtos my ot coverge Ustble eqtos grow wtot bod My ve stble eqtos tt prodce correct reslts Codtol stblty reqres step sze less t tt eeded or ccrcy Gol o bsolte stblty ot lwys possble Dscssos o stblty complex, c sometmes se pyscl rgmets 9 ME 5B Egeerg Alyss 5

6 Itrodcto to Nmercl Alyss Mrc, 9 Stblty o Explct Metod I te vles o d - re xed crese sold crese I s greter t.5, crese wll cse decrese We c vod ts correct reslt by keepg αδt/(.5 Ts mposes tme step lmt tt my be less t te lmt reqred or ccrcy te solto TE FTCS Trcto Error Dervto ppe or otes o solvg PDEs gves ts eqto TE α α( k k T x (k)! k! α( k k T k x T x Settg αδt/( / elmtes rst term te trcto error L Altertve Metods Next clss wll sow ow to vod stblty lmt by sg explct metods Solto or depeds o oter vles t ew tme step Reqres solto o smlteos eqtos or ll Crk-Ncolso ses vles or old d ew tme steps spce dervtves Flly mplct ses ew tme vles oly ME 5B Egeerg Alyss