Finite difference methods An introduction. Jean Virieux Professeur UJF with the help of Virginie Durand

Transcription

1 Fte dfferece methods A trodcto Jea Vre Professer JF wth the help of Vrge Drad

2 A global vso Dfferetal Calcls (Newto, 1687 & Lebz 1684) Fd soltos of a dfferetal eqato (DE) of a dyamc system. Chaos Systems (Pocaré, 1881) Fd propertes of soltos of the DE of a dyamc system. Chaos & Stablty (Smale, 1960) Fd propertes of soltos of a physcal system wthot kowg ts DE After the presetato of Etee Ghys, 13 october 009 Ths corse s abot the dfferetal calcls sg the fte dfferece approach famlar to Newto & Lebz

3 Bblography o Fte Dfferece Methods : A. Taflove ad S. C. Hagess: Comptatoal Electrodyamcs: The Fte- Dfferece Tme-Doma Method, Thrd Edto, Artech Hose Pblshers, 005 O.C. Zekewcz ad K. Morga: Fte elemets ad apprommato, Wley, New York, 198 W.H. Press et al, Nmercal recpes FORTRAN/C Cambrdge versty Press, SA, 0XX Spce grop Erope : FDTD trodcto : ftp://ftp.sesmology.sk/pb/papers/fdm-itro-spice.pdf By P. Moczo, J. Krstek ad L. Halada

4 What s the ma dea? Coverage of the comptatoal doma by a space-tme grd Appromate dervatves ad tal vales at all grd pots Defe bodary codtos at ed pots Costrct the lear algebrac system to be solved by the compter

5 The heat eqato t t (, ) t (, ) k k s the thermal dffso coeffcet (, t) Grate Basalte Calcare Ea Glace Sol sec Sol hmde (8%) k e m /s 1, , , , , , k Replace partal dervatves by fte dfferece appromatos leadg to a algebrac system (,t) ~ where the de s for the dscrete spatal posto ad for the dscrete tme level

6 Other PDE physcs The scalar wave eqato s a partal dfferetal eqato whch belogs to secod-order hyperbolc system. Tme s volved all physcal processes ecept for the Laplace eqato related to Newto law ad mass dstrbto. Posso eqato cold be cosdered as well whe mass s dstrbted sde the vestgated volme t (, ) t (, ) a (, t) t Wave Eqato Fld Eqato Dffso Eqato Laplace Eqato Fractoal dervatve Eqato t (, ) t (, ) c ( ) t t (, ) t (, ) ( ) t t t (, ) t (, ) ( ) 0 t t t (, ) (, ) t (, ) ( ) Advecto Eqato Posso Eqato f( ) ( )

7 Fte Dfferece Stecl (Leveqe 199) cetered h D backward h D forward h D Trcatos errors : 0 h Secod dervatve D D D D D 0 0 ) ( h D Hgher-order terms : same procedre bt yo eed more ad more pots h

8 Fte dfferece appromato D 1 h forward D h 1 backward Forward/backward frst-order appromato D o 1 1 h cetered Cetered secod-order appromato 1 D ( D D ) o Hgher-order appromatos cold be cosdered as well : more vales!!!

9 Dalty betwee a fcto ad ts dervatve Secod-order accrate cetral-dfferece appromato Leapfrog secod-order accrate cetral-dfferece appromato Leapfrog 4 th -order accrate cetral-dfferece appromato Stecl legth

10 4 4 4, 3 3 3,,, 1, 4 6 ) ( 4 4 4, 3 3 3,,, 1, 4 6 ) ( 4 4 4,, 1, 1, 1 Dscretsato ad Taylor epaso ) (, 1, 1,, Assmg a form dscretsato,t o the doma, we cosder terpolato pto power 4 by smmg, we cacel ot odd terms eglectg power 4 terms of the dscretsato steps. We are left wth qadratc terpolatos, althogh cbc terms cacel ot for precso.

11 Secod dervatves Varos methods for evalato of secod dervatves 11 D h 1 D D D D D h h h 1 1 Taylor terpolato formlato related to Lagrade polyomals ad, therefore, vales are ot restrcted to odes of a grd we assme a smooth cotrbto of the mercal solto. The solto of the heat eqato has a mercal appromate solto 1 1 t t 11

12

13 Is the mercal solto correct? Stablty: Errors are boded drg the comptato of the solto. Cosstece: Trcato errors go to zero whe dscretsato gets smaller. Covergece: Nmercal soltos go to the eact solto as dscretsato gets smaller. Solto Accracy

14 Heterogeeos meda Whe cosderg mercal methods, we ofte address the problem of comple meda for whch there s o aalytcal solto,, Spatal dervatves of medm propertes shold be estmated,,,, Decreasg the order of dervatves by addg alary varables whch have ofte physcal meags.

15 Parsmoos approach F RC V K t 1 1 V V Stadard grd 1 1 K 1 K 1 F RC RC t K K ( K K ) t Not eactly waht we wat becase of dces + ad -: we eed to move to a med grd approach (dalty betwee a fcto ad ts dervatve) F

16 Parsmoos approach 1 1 1/ 1/ F RC V K t 1/ 1/ 1 V V Staggered grd or med grd approach K 1/ K 1/ F RC t K K ( K K ) 1 1 1/ 1 1/ 1 1/ 1/ F RC t Same dces tha the homogeeos case: arthmetc average comg from the physcs behd Idetfy how to defe jmps at a terface

17 Tme marchg procedre Eplct cetered scheme t IS THAT ALL? A very smple costrcto: estmato of a ftre vale from preset vales ad oe past vale. Ital vales: Bodary vales: Drchlet codtos & L 0

18 Freqecy approach Fte Dfferece formlato leads to a lear system K K ( K K ) 1/ 1 1/ 1 1/ 1/ F RC WRITE ON THE BLACKBOARD THE LINEAR SYSTEM

19 FIND A SOLTION? I Tme : marchg approach I Freqecy : lear algebra

20 ODE verss PDE formlatos d yt () Ayt ( (,), t,) dt d yt () At (,) yt (,) dt yt (, ) t yt (, ) t No-lear Dyt ( (, ), t, ) D( t, ) y( t, ) Lear O.D.E Ordary dfferetal Eqatos P.D.E Partal Dfferetal Eqatos Symmetry betwee space ad tme? GOAL : fd ways to trasform dfferetal operators to algebrac operators order to se lear algebra at the ed «smple» solto «comple» solto

21 EXISTENCE & NIQENESS The mercal problem s well posed sg Hadamard crtera f the solto ests, s qe ad depeds cotosly to tal/bodary codtos, to bodares ad coeffcets of the dfferetal eqatos Vo Nema stablty aalyss for FD methods: mercal tegrato caot go beyod physcal dstaces of teracto. For eample, Corat-Fredrchs-Lewy CFL codto for wave propagato

22 Frst-order hyperbolc eqato E v t v t c t v E t Let s defe other varables for redcg the dervatve order both tme ad space The d order PDE became a 1st order PDE Ths s tre for ay order dfferetal eqatos: by trodcg addtoal varables, oe ca redce the level of dfferetato. Amog these dfferet systems, oe has a physcal meag whch becomes v E t t v 1 E c wth stress velocty Other choces are possble as dsplacemet-stress stead of velocty-stress.

23

24 Ital ad bodary codtos 1D strg medm Ital codtos (,0) Bodary codtos (0,t) Bodary codtos (L,t) Drchlet codtos o Nema codtos o f(,t)=s(t)r() Ectato codto,, Dffclt to see how to dscretze the velocty c()! Mch better for hadlg heterogeetes,,,,

25

26

27

28 v=0

29 Sorce ectato Implsve sorce: Rcker wavelet for eample 1 Oscllatory sorce: Ss wavelet for eample s

30 Sorce radato Drectoal sorce (hammer) Eplosve sorce (dyamte) Meag for the strg? Applcato of of opposte sg forces o two odes or a fctos force betwee two odes

31

32

33

34

35 Smlato a two-layer medm c1=000 m/s c=4000 m/s - S=1 the hgh-velocty layer Radato bodary codto Code df1d.3.f

36 Sesmc propagato the Agel Bay earby Nce (Frace) Earthqake of magtde 4.9 at a depth of 8 km

37 CONCLSION Effcet mercal methods for propagatg sesmc waves Tme tegrato verss freqecy tegrato Competto betwee FE & FV for modellg FD a effcet tool for magg

38 THANKS YO!