Numerical Solutions to Partial Differential Equations

Transcription

1 Nmeral Soltos to Partal Dfferetal Eqatos Dr. Ismal Celk Referee: Celk, I. (00 ) Itrodtory Nmeral Methods for Egeerg Aapplatos, Ararat Books & Pblshg, Morgatow WV araratbp@gmal.om; a

2 Partal Dfferetal Eqato Adveto or Coveto Eqato 0 t Wave eqato t t d t 0 Laplae eqato y 0

3 Geeral Trasport Eqato (oveto dffso eqato) {tme rate of hage sde otrol volme} {et fl throgh the srfaes} y F North ρ t ( ) Fw Fe Fs F V y F w West Cotrol Cotrol Volme Volme East F e y F F e F F w s F F y w s y Soth F s 3

4 Cotd Sbstttg t F ( V ) y w F y The fles are modeled as F m ( -dreto) Aα where m s oveto ad A α s dffso. s Fld flow rate m ρ A Cotrol volme srfae areas: A Aw Ae ; Ay As A y 4

5 5 Sbstttg Dvdg throgh ot by y ad re-arragg Whe α s ostat ad 0, v 0 (Traset heat odto eqato) Cotd ( ) y y v y y y t α ρ α ρ ρ ( ) ( ) ( ) y y v y t α α ρ ρ ρ α t y

6 Speal ases of Trasport Eqato α 0, D ost. : t Takg the dervatve w.r.t. tme ad sbstttg the seod term yelds the wave eqato t 0 0 t 0 v0 heat eqato Whe steady Laplae eqato: y 0 Wth a sore term Poso Eqato: y s 6

7 Well-Posedess Codto PDE s wthot ay restrtos may have fte mber of soltos Ital ad/or bodary odtos shold be mposed for a qe solto Bodary odtos are mposed o bodary B of a spatal doma S -dreto ormal to the bodary Doma S Bodary B Ital odtos are presrbed throghot the doma at the stat whe osderato of physal evet begs 7

8 Types of bodary odtos I geeral, there are three types of bodary odtos: (.) The fto vales are presrbed o the bodary, ƒ(t) o B (Drhlet odto) (.) The dervatve of the fto s presrbed o the bodary, g(t) o B (Nema or fl odto). (.) Med bodary odto, a b h(t) o B (Rob or radato odto). 8

9 Classfato of PDE s Partal dfferetal eqatos are lassfed aordg to ther haraterst dretos alog whh the eqatos oly volve total dfferetals Notato: Geeral d Order PDE: A B y C yy D E y F G Aalogy to aalyt geometry A By Cy D Ey F 0 B -4AC > 0 Hyperbol; 0 Parabol; <0 Ellpt 9

10 where Geeral Trasformato ε ε(,y); η η(,y) a ee b eh hh d e e e f g(ε,η) a Aε Bε ε y Cε y b Aε η B(ε η y ε y η ) Cε y η y Aη Bη η y Cη y d Aε Bε y Cε yy Dε Eε y e Aη Bη y Cη yy Dη Eη y f F g G * b - 4a J J (B -4AC) ε ε y J η η y (Jaoba) The atre of the eqato remas varat der sh a trasformato f the Jaoba J s ot zero, ad ε ad η are twe otosly dfferetable 0

11 Physal terpretato of the Types of PDE s

12 Classfato of Some Smple Eqatos ad Callato of Charaterst Eqatos (a) Wave eqato: tt 0 Comparg ths eqato wth Eq 7.. reveals (here t takes the role of y) A -, B 0, C, D E F 0 B 4AC 0 4 ( )( ) 4 > 0 for all vales of eept zero. Therefore, the wave eqato s hyperbol for ozero vales of the wave speed. The haratersts a be fod from Eq.: A( dy) B( d)( dy) C ( d) 0

13 Classfato of Some Smple Eqatos ad Callato of Charaterst Eqatos ( dt) 0 ( d) 0 Dvdg by dt, we obta: d dt or d dt ± The solto to whh s, Hee, ± t, t t ostat ostat C C ε η are the haraterst of the wave eqato. 3

14 Classfato of Some Smple Eqatos ad Callato of Charaterst Eqatos (a) Laplae Eqato: 0 yy A, B 0, C, D E F 0 B 4AC 0 4 < 0 Hee, the Laplae eqato s always ellpt. Charaterst eqato: dy ( dy ) ( d) 0 ; d dy ± ± (omple roots) d y ± ; y ± ostat I ths ase, the haraterst rves are omple ad are gve by: ε y ; η y - 4

15 Classfato of Some Smple Eqatos ad Callato of Charaterst Eqatos (a) Oe-dmesoal heat eqato: α t 0 ; α > 0 (thermal dffsvty) A -α, B C D E F 0 0. Therefore, the heat eqato s a parabol eqato. Charaterst Eqato: a(d) 0 0(dt) 0 d d 0 ; 0 dt dt It follows that ε ostat ad η t ostat are the haraterst rves. 5

16 Oe-dmesoal salar trasport eqato Oe Dmesoal salar trasport eqato wth ostat dffso α oeffet ad veloty α t (Term I) (Term II) (Term III) Term I s the tme rate of hage, Term II s the oveto (or adveto), ad Term III s the dffso Ths eqato s sed to llstrate the propertes of some ommoly sed fte dfferee methods 6

17 Commoly-Used Dsretzato Shemes Short-Had Notato: FD Fte dfferee LW La-Wedroff CD Cetral dffereg CN Crak-Nolso UW Upwd CS Cetral Spae FT Forward tme US Upwd spae Cetral dffereg s mostly sed for the dffso terms se the dffso proess s ellpt α α ( ) ( ) Where δ represets the etral dfferee operato for αδ methods a be amed depedg o the dsretzato sheme appled to the tme dervatve ad the oveto/adveto terms 7

18 8 FTUS: Forward Tme Upwd Spae Eplt for > 0: f < 0 the seod term s replaed by a forward dfferee. That s, rearragg Where (Corat mber), (dffso mber) ( ) t α ( ) ( ) ( ) d δ t t d α

19 Cotd.. Stablty restrto for the FTUS method s d The leadg terms E T are gve by: E T 6 ( ) α ( 3 ) The frst term o the RHS s alled artfal (or meral) dffso I geeral, eve dervatves the epaso of E T are assoated wth dffso ad odd dervatves wth dsperso or phase errors. The artfal dffsvty α * Eq s: α ( ) * 9

20 0 Cotd. For aray, α * shold be mh less the the real (physal) dffsvty. That s, I terms of pelet mber, Pe /d /α. Pe << /(-) Correspodg flly mplt sheme rearragg ( ) α << ( ) t α ( ) ( ) d d d

21 FTCS (Forward Tme Cetral Spae) t (dstae) (,) (-,) (,) (,) ( ) ( ) C t αδ ( ) ( ) d ( ) T 6 Pe E α 0 d Trato error (leadg terms) Stablty ad aray odtos are ; Pe<

22 La-Wedroff (LW) Stablty ad aray odtos are: ; Pe The odto for Pe s (to avod spatal osllatos) or the so-alled "wggles." d 0 * ( ) ( ) T 4 Pe 6 Pe 6 Pe E α α ( ) ( ) d * * * t d α * α α ( ) ( ) t αδ

23 3 QUICK (Qadrat Upwd Iterpolato for Covetve Kemats) where ad are the frst-order forward ad bakward fte dfferee operator, respetvely, for the thrd dervatve. These are: Neessary odtos for stablty: ; ( ) ( ) q t αδ δ ( ) ( ) ( ) < δ > δ δ b f q ( ) ( ) ( ) ( ) 3 b 3 f δ δ 4 d d

24 Predtor-Corretor Method Ths method ses the bas dea of Rge-Ktta sheme ad s seod order arate tme Let s wrte the geeral trasport Eq. t 0 ( t ) We a se eplt methods to allate a appromate solto, *, at a ew tme level. That s, 0 ts α S f ( t, () t ) f ( t, ) Where S s the rght had sde (the slope) of the above eqato at the old tme level. Ths s kow as the predtor step. 4

25 Cotd.. Ths appromate solto a the be sed to allate the rght had sde at the ew tme level. S * ( t, ) * f whh tr a be sed to allate a better appromato to the slope at the rret tme level. [ S ] S * S avg Hee, the orretor step follows: ts avg 5

26 6 CN (Crak-Nolso) or Sem-Implt Method CN-method s a wdely sed strtly d order method. Ths method ses the average of the r.h.s. at the old ad ew tme levels, ad, respetvely. Stablty restrto: oe Aray restrto: (to avod spatal osllatos) ( ) ( ) [ ] C C t δ δ α ( ) ( ) ( ) ( ) ( ) ( ) 0.5 d d 0.5 d 0.5 d d 0.5 d ( ) ( ) T E α 3 Pe

27 Eample I ths eample we preset reslts of some methods seleted from the oes desrbed ths seto whe they are appled to the geeral oe dmesoal trasport eqato. The FORTRAN program Trasd_odf.for (traset, oe-dmesoal oveto dffso) was sed to obta the meral soltos. For brevty, we re-wrte the trasport eqato aga ad alog wth the bodary ad tal odtos: t α Ital odto: at t 0; for a < < b, otherwse 0. Bodary odtos: at -, 0; at (large!); 0. 7

28 The aalytal solto s gve by Solto (, t) 0.5[erf(η)erf(η')] where η [ - ( * - * * t)]/( ), η' [ ( * - * * α t t)]/( α) t * [ -(b-a)]/(b-a); * /(b-a); α * 4α/(b-a) erf ( η) π η 0 e w dw s the error fto. 8

29 Nmeral Solto The meral soltos are preseted the followg sldes for varos methods ad parameters. The sze of the doma s 0 m. The tal sqare wave s loated 4 < < 6 ad all soltos are preseted after 5 seods. The mportat dmesoless parameters are: Corat mber, t/ Dffso mber, d αt/ Cell (or grd) Peklet mber, Pe /d /α. The Peklet mber shows the relatve mportae of oveto vs. dffso the dfferee eqato. 9

30 30

31 3

32 3

33 33

34 Geeralzed Method (Smart I.C. Shemes) Upwd ( ) La-Wedroff: Geeral: ( ) ( ) a b Selet a, b, ad sh that σ m 0 γ 0 m oef. of oef.of Also for large For small Pe t reovers Ceteral Dffereg! et. 34

35 Mlt-Dmesoal Dsretzato The stablty odto for eplt methods whe appled to mltdmesoal problems beomes too restrtve. Oe way to mprove stablty odto s to treat oe dreto mpltly ad the other dretos epltly. However, ths prate does ot provde mh beeft. Whe Crak-Nholso method s appled to the two-dmesoal heat eqato, t yelds:, j t, j α [ δ ( ) δ ( )], j, j yy,j,j where δ s a etral dfferee operator defed by: δ ( ),j ( ),j,j,j ad δ ( ) yy,j ( ),j,j y,j 35

36 Cotd.. For smplty, we let d αt αt y d y ad apply the C-N method. The smple grd show Fg. s sed llstrato of the method. The system of eqatos to be solved takes the followg form: [A]{U} {R} y The kow vetor s: {}T [-, ; 3, ;, 3; 3, 3] (,3) (3,3) (,) (3,) The rght had sde (kow from prevos tme level or from bodary odtos) s: {R}T [(F,,, ); (F, 3, 3, 4); (F3, 4, 3, ); (F3, 3 4, 3 3, 4)] F,j d δ y ( ) δ ( ),j d yy,j kow pots o kow pots 36

37 37 Cotd.. ad the oeffet matr s: Ths matr does ot have a easly detfable strtre ad s qte dfflt to solve ompared to a tr-dagoal matr ts rret state. The omptatoal ost of the C-N method a be avoded by sg the well-kow Alteratg Dreto Implt (ADI) method of Peaema ad Rahford (955). [ ] A

38 ADI Method Appled to the Heat Eqato Frst marh t step mplt -dreto ad eplt y-dreto. That s, The marh aother step eplt the -dreto ad mplt the y- dreto. That s, j, j t, j j t, j ( ) δ δ, α yy,, α δ ( ) δ, j Ths method has a seod-order aray both spae ad tme. It s odtoally stable ad oly reqres the solto of tr-dagoal matres yy j, j 38

39 Two-Dmesoal Heat Codto Here we llstrate the dsretzato of traset heat odto eqato Eq. sg a smple mplt sheme. t α( yy ) y 0 The bodary vales are dated by arrows o respetve bodares t old, j, j t ; bakward dfferee 0 y m m 00 (,j,j -,j )/ ; etral dfferee Fgre Bodary odtos for the heat eqato o a sqare slab. yy (,j,j,j- )/y ; etral dfferee 39

40 Cotd.. Colletg all the terms together, yelds: wth, αt αt y, j, j, j t α αt y, j, j t α αt y, j, j old, j,, 3, 4, j,, 3, 4, j Assmg that the temperatre feld s kow (gve as a tal odto) at the old tme level, the above eqatos effet are a baded-blok, strtred matr ad t a be solved sg a approprate matr solver. The solto mst be repeated at every tme level. 40

41 Steady-State Solto to Heat Eqato For steady state odtos ( t 0), the heat eqato redes to: y 0 yy Applyg etral dfferees, we obta: (,j,j -,j )/ 0 (,3) (3,3) (,) (3,) (,j,j,j- )/y For smplty, let y 4 (see Fg.): The, 4,j,j -,j,j,j- Note that the vales o the bodares ( ad 4; j ad 4) are gve (see Fg.) kow pots o kow pots Fgre: Dsrete 6 pot meral mesh (or grd) sed for dsretzato of the heat eqato. 4

42 Cotd.. Note that the vales o the bodares ( ad 4; j ad 4) are gve (see Fg. ) j,, ( 3,,,3, )/4 j, 3 3, ( 4,, 3,3 3, )/4 j 3,,3 ( 3,3,3,4, )/4 j 3, 3 3,3 ( 4,3,3 3,4 3, )/4 Usg the bodary vales gve Fg. (ote that by symmetry, 3, ad,3 3,3), Eqs. redes to:, ( 3,,3 00)/4 3, (, 3,3 00)/4,3 ( 3,3, 00)/4 3,3 (,3 3, 00)/4 These eqatos a be solved by sg the Gass-Sedel terato method. Start wth zeros (,j 0) at the teror pots as a tal gess. The reslts are show Table 4

43 Table 7.3. Reslts of Gass-Sedel terato Appled to Eq Iteratos, 3,,3 3, Coverged fal solto (to demal plae)

44 Le-by-Le Iteratve Solto Istead of sg pot-by-pot Gass-Sedel terato, we a mprove the overgee rate by sg the so-alled le-by-le G-S method whh follows: Step : Start wth a gessed solto,,j Step : Solve sg TDMA for j, 3, 4,, j-, the followg system of eqatos: - -,j 4,j,j (,j,j- ) Step 3: The solve sg TDMA for, 3, 4,, -, the followg system of eqatos: -,j- 4,j,j (,j -,j ) The qattes sde the parethess o the rght had sde (are assmed to be) kow from prevos terato. Step 4: Repeat steps () ad (3) tl the solto does ot hage more tha a spefed error bod, that s the appromate relatve error ea < 0-3 The system of eqatos represeted by the above eqatos reslt trdagoal matres ad they a be solved by the TDMA (or Thomas) algorthm. 44