FUNDAMENTALS OF FINITE DIFFERENCE METHODS

Transcription

1 FUNDAMENTALS OF FINITE DIFFERENCE METHODS By, Varn Khatan 3 rd year Undergradate IIT Kanpr Spervsed by, Professor Gatam Bswas, Mechancal Engneerng IIT Kanpr

2 We wll dscss. Classfcaton of Partal Dfferental eqatons nvolved n fld mechancs Fndamentals of Dscretzaton. Fnte Dfference Qotents, startng from Taylor s seres. Fnte Dfference eqatons from the pont of vew of ther consstency and convergence. Explct and Implct methods of dscretzaton. Errors and stablty analyss. Fndamentals of Fld Flow Modelng: Transportve and Conservatve propertes. Artfcal Vscosty. Upwnd Scheme of Dscretzaton (Frst and Second).

3 Partal Dfferental Eqatons Consder a second order dfferental eqn φ φ φ φ φ φ =, ( ) A B C D E F G x y x x y y x y A,B,C,D,E and F are constants or fnctons of only x and y, then lnear eqaton Qaslnear eqaton Homogenos eqaton Classfcaton: 4 = 0, B AC parabolc B 4AC < 0, ellptc B 4AC > 0, hyperbolc

4 Partal Dfferental Eqatons (examples) Laplace eqaton: Posson eqaton: o φ x φ x φ y + = 0 φ y + + S = 0 Both are ellptc eqatons Heat condcton eqaton s parabolc. φ φ = B t x

5 Contned Fld flow problems have nonlnear terms de to nerta and energy terms known as advecton and convecton respectvely. φ φ φ φ φ + + v = B ( + ) + S t x y x y φ denotes some transported property lke velocty, temperatre etc. and v denote velocty components. B s dffsvty for momentm or heat. S s sorce term. Parabolc n tme and ellptc n space. For very hgh speeds t becomes hyperbolc.

6 Bondary and Intal condtons No. of B.C.s=order of hghest dervatves n dfferental eqatons. Unsteady problems reqre ntal condtons for tme ntegraton. Dffson terms need two BC s. Spatal Bondary Condtons: Drchlet: Φ=Φ (r) A Nemann: Φ n=φ (r) A Mxed:a(r)Φ+b(r) Φ n=φ 3 (r) A 3

7 Fnte Dfference Methods Dscretzaton s a process by whch a partal dfferental eqaton s approxmated by analogos expressons whch prescrbe vales at only a fnte no. of dscrete ponts or volmes n doman. Basc phlosophy of fnte dfference methods s to replace dervatves wth algebrac dfference qotents, resltng n a system of algebrac eqatons whch can then be solved.

8 Grd Generaton x and y need not be same. When they are dfferent n the physcal plane, the plane s transformed. In transformed comptatonal plane the spacng s nform n all drectons.

9 Fnte dfference qotent Taylor seres expanson Frst order Forward dfference

10 Fnte dfference qotent (Contd )

11 Fnte dfference qotent (Contd ) Second order central dfference If mxed dervatves are nvolved the second order dfference looks lke xy φ + +, j+, j +, j, j+, j φ φ φ φ ( ) ( ) = +Ο x, y 4 x y Δ Δ ΔΔ

12 Fnte Dfference Eqatons The basc concept n solvng the PDE sng ths approach s to replace each dervatve term wth ts dfference element. Consder the nsteady -D heat condcton: t = α x Usng the concept of FTCS (Forward Tme Central Space) We transform the above eqaton to n+ Δt n ( n n n ) + + α + TE Δx ( )

13 Consstency and Convergence A fnte dfference representaton s sad to be consstent f ( PDE FDE ) lm = lm ( TE ) = 0 mesh 0 mesh 0 o A necessary condton s that Δ t Δ x 0 As x ->0 and t->0 o A FDE s sad to be convergent f the approxmate solton approaches the exact one as the sze of the mesh tends to zero. n = x, t asδx, Δt 0 ( ) n

14 Explct and mplct methods Explct method: Consder -D nsteady state heat condcton eqaton. o n+ Δt t n = α x The correspondng Fnte Dfference eqaton s: = α n n n ( + + ) ( Δx ) n + The only nknown n the above eqaton s Its vale can be obtaned from the vales of the varables at tme n. Ths the eqaton takes the form of a marchng procedre n steps of tme.

15 Explct and mplct methods (contd ) Implct method: Consderng the earler eqaton wth a dfferent dscretzaton algorthm we get the form n+ Δt n = α ( n+ n n+ n n+ n ) ( Δx) Here each term has been replaced by the tme average of ts neghborng terms. Ths s known as the Crank-Ncholson Implct Scheme.

16 Explct and mplct methods (contd ) Implct Method (cont.): The eqaton can be wrtten as, n+ n r ( n+ n n+ n n+ n ) = α where r= ( Δt) ( Δx) or, + r r + = + + r r r n+ n+ n+ n n n + + These eqatons when appled at all grd pts. from = to =k+ reslt n a trdagonal matrx whch can be solved to obtan the vales at tme n+. n+ n B() 0 0 K 0 T ( C() + A) n n B() K T3 C() n+ n 0 B(3) K 0 T4 = C(3) M M M M M M M n+ n K Bk ( ) T k ( Ck ( ) + D) Here A and D come from the bondary condtons and C(k) s the vale of the RHS at tme k+.

17 Explct and mplct methods (contd ) o o o Applyng the Crank Ncholson dscretzaton scheme to the -D heat eqaton gven by = α + t x y where we get δ δ Ths system has fve nknowns n+ n, j, j n n Δt α = δ + δ + + ( )( + ) x y, j, j n n n n +, j, j, j x T, j = ( Δ x ) + n n n n, j +, j, j y T, j = ( Δ y ),,, & n+ n+ n+ n+ n+, j +, j, j, j+, j On smplfcaton ths gves a matrx eqaton whch s mch more complcated to solve and consmes more tme.

18 Explct and mplct methods (contd ) Usng alternate drecton mplct method (e) splttng the eqaton nto two steps we obtan the followng new set of eqatons: Frst step: Second step: + / + / + / ( ) ( ) n+ / n, j + +, j = α + Δt / ( Δx) ( Δy) n n n n n n +, j, j, j, j+, j, j ( ) ( ) n+ n+ /, j + +, j = α + Δt / ( Δx) ( Δy) n+ / n+ / n+ / n+ n+ n+ +, j, j, j, j+, j, j Ths system gves two trdagonal matrx eqatons, one for each step. These can be solved to get the solton.

19 Explct and mplct methods (contd ) Implct Method Advantage: Takes lesser tme for calclatons over a tme nterval. Dsadvantage: - Complcated program. -Large comptaton tme n each step -For larger Δt, trncaton error s hgh. Explct method Advantage: The algorthm s smpler Dsadvantage: Reqres many tme steps to carry ot the calclatons over a gven tme nterval, de to restrctons on Δt mposed by stablty constrants.

20 Errors and stablty analyss Errors: A=analytcal sol of P.D.E. D=exact sol. of F.D.E. N=nmercal sol. from a real compter wth fnte accracy. Then Dscretzaton Error =A-D DE=Trncaton Error+ error ntrodced de to treatment of bondary condton. Rond-off Error = ε =N-D Nmercal error ntrodced by the compter as t ronds off the nmbers to some fxed decmal ponts. Ths, N=D+ε

21 Errors and stablty analyss (contd..) Consder the FDE of the one dmensonal heat eqaton: n + n n n n ( ) α + + = Δ t Δ x The Nmercal solton (N) and the exact solton (D) both satsfy the above eqaton. Owng to the lnearty of the eqns ε=n-d wll also satsfy t. Hence we get ε n+ ε Δt n = ( n n n ) + + α ε ε ε ( Δx ) We say the solton s stable f the error shrnks or remans same as we progress n tme. If the error magnfes wth tme then the solton s nstable. Ths the condton for stablty s: ( ) ε ε n + n

22 Errors and stablty analyss (contd..) Expandng the error expresson as a Forer seres, we get Snce the FDE s lnear t wold sffce to consder the term Sbstttng ths n the FDE we get: ε n + n ε 4αΔt e = sn k Δx/ Δx aδt aδt = e where ( ) m

23 Errors and stablty analyss (contd..) Ths we reqre: 4αΔt Δx ( k x ) sn / mδ Ths gves αδt Δx αδt Δx sn ( k / ) mδx / / Ths s called Von Nemann stablty analyss.

24 Stablty of hyperbolc eqatons Applyng Von Nemann stablty analyss to the frst order eqaton = t t + c = x We replace tme dervatve wth frst order dfference and spatal dervatve by central dfference to. Also, we replace (t) by the average of neghborng terms. Ths + Δt ( ) + n+ n n 0 ( n n ) n n n+ + + Δt + = c Δx e Δ = cos k Δx Csn k Δx The amplfcaton factor s at ( ) ( ) m m

25 Stablty of hyperbolc eqatons Ths the reqred condton for stablty s: Δt C = c Δx Here C s called the Corant nmber and the condton s CFL (Corant - Fredrchs - Lewy) condton. Applyng the same procedre to the second order eqaton also gves the same condton for stablty. In actal problems stablty s checked locally. For varable coeffcents Von Nemann analyss s necessary bt not sffcent. To assre stablty we need to carry ot expermentaton and nmercal checks. Rotne stablty analyss only gves gdance.

26 Fld Flow modellng Fld flow eqatons are complex as they form a nonlnear system. To model fld flow we consder the Brger s eqaton gven by: ζ ζ + = υ ζ t x x Here s the velocty, v s the vscosty and T s any property whch can be transported and dffsed. If vscos term on the rght sde s neglected the eqaton redces to: ζ ζ + = 0 t x An FDE s sad to be conservatve f t preserves the ntegral conservatve relatons. ω t = V. ω + υ ω Consder the Vortcty transport eq. ( ) Integratng over fxed regon, fnally get,

27 Fld Flow modellng (contd..) ( ) ( ) ω d R= V ω. nda + υ ω. nda t R A0 A0 Ths mples that the tme rate of accmlaton of ω n R eqals net advectve flx rate of ω across A0 nto Rand net dffsve flx rate of ω across A0 nto. Ths relaton needs to be preserved for conservatness. R Consder the conservatve form of the Brger eqaton: ω = ( ω ) t x Usng FTCS the correspondng FDE s ω ω ω ω = Δt Δx n+ n n n n n + + Evalatng the ntegral over a regon between =I R and =I as = I ω Δ x Δ t = I I I I we get n+ n n n ω Δx ω Δ x = ( ω) ( ω) + Δt = I = I = I

28 Fld Flow modellng (contd..) Smmaton of R.H.S. fnally gves, Integral ( ω) ( ω) = I I + Ths the accmalaton of w n s eqal to the net advectve flx rate across the bondary of R. Ths the FDE preserves the ntegral relaton and s conservatve n natre. If we consder the non-conservatve form of the Brger s eqaton ω ω =, t x then followng the earler procedre the ntegral comes ot to be I n n n n = I ω + Here t fals to preserve the Gass-Dvergence property and hence t s not conservatve. R ω

29 Upwnd scheme and Transportve property Von Nemann s stablty analyss appled to normal dscretzaton of the Brger s eqaton ndcates ncondtonal nstablty. Ths we apply an alternate scheme to the convectve term n the conservatve form: n+ n n n ζ ζ ζ ζ = + Δt Δx n+ n n n ζ ζ ζ + ζ = + Δt Δx vscos terms, >0 vscos terms, <0 Ths s known as the pwnd scheme. Transportve property: An FDE of a flow eqaton s sad to possess ths property f the effect of a pertrbaton s convected or advected only n the drecton of the velocty. Reglar formlaton of the FDE volates ths property

30 Transportve property (contd.) Consder a pertrbaton ε m =δ n ζ at the locaton m at tme=n. For >0, at the downstream locaton m+ we get o o n+ n ζ m+ ζ m+ 0 δ δ = =+ Δt Δx Δx Ths s acceptable. At the pont of pertrbaton,(e) m we get ζ n + n m ζ m δ 0 = = δ Δt Δx Δx Ths ndcates that the pertrbaton s beng transported ot of the affected regon. At the pstream staton, m- we see that ζ n + m n ζ m 0 0 = = Δ t Δ x Ths flow s ndrectonal and property s satsfed. 0

31 Upwnd scheme and artfcal vscosty Usng Taylor seres expanson of the terms n the Upwnd Scheme eqaton, we get or ( Δt) Ths eqaton may be wrtten as ( Δx) ζ ζ 3 ζ ζ 3 Δ t + +Ο ( Δ t) = Δx +Ο ( Δx) +[dffsve terms] Δt t t Δx x x ζ ζ Δt ζ ζ = + Δx + ν +Ο Δx t x x Δ x x ζ = ζ + ν ζ e + ν ζ + hgher-order terms t x x x ( ) Where ν ( ) s the artfcal vscosty and C e = Δx C 3 s the Corant nmber. In dervng ths we have taken ζ t ζ x =

32 Upwnd scheme and artfcal vscosty In steady state,.e., ζ/ t=0,νe= x/. For a two dmensonal convectve-dffsve eqaton, on followng smlar steps the eqaton obtaned s ζ ζ v ζ ( ) ζ = + ν ex + ν + ( ν ) ey + ν ζ t x y x y Where ν ex = ( ) wth C x = t/ x, Δx Cx ν ( ) ey = Δy C y C y =v t/ x

33 Hgher order pwnd dfferencng In order to mantan the transportve property the comptatons are naccrate de to the presence of false dffson. To get rd of ths we se hgher order dfferencng. Accordng to second order dfferencng ( ζ ) Rζ R Lζ L = x Δx, j where, R=(,j++,j)/ ; L=(,j+-,j)/ ; and, ζr=ζ,j for R>0; ζr=ζ+,j for R<0; ζl=ζ-,j for L>0; ζl=ζ,j for L<0; For R>0 and L>0 we get ( ζ ), +,,, j+ j j+ j = ζ, j ζ, j x Δx

34 x Hybrd scheme We can also represent t as a weghted combnaton of central and pwnd dfferencng, (, j +, j) η (, j +, j) = 4Δ + + ( ) ( ), j+, j η, j, j, j x Where 0<η<. The accracy of the scheme can always be ncreased by a stable adjstment of η vale.

35 Acknowledgements I take ths opportnty to thank my spervsor Prof. Gatam Bswas (Mechancal Engneerng,IIT Kanpr) for hs gdance and enthsastc spport, whch has played a pvotal role n the preparaton of ths talk.