Computational Fluid Dynamics. Lecture 3

Transcription

1 Computatioal Fluid Dyamics Lecture 3 Discretizatio Cotiued. A fourth order approximatio to f x ca be foud usig Taylor Series. ( + ) + ( + ) + + ( ) + ( ) = a f x x b f x x c f x d f x x e f x x f x ( 0 ) ( 0 ) ( 0 ) ( 0 ) v a+ b+ c+ d + e= 0: f x a+ b d e= : f x 4a+ b+ d + 4e= 0: f x 8a+ b d 8e= 0: f x 6a+ b+ d + 6e= 0 : f x 4 th order solutio: 0 ( + ) ( ) ( + ) ( ) 4 f x0 x f x0 x f x0 x f x0 x Aother Example: st derivative ear a boudary st grid poit 3 rd order oe sided Dirichlet B.C.usig Taylor Series expasios at grid poits, 3, ad 4 with h= x. f = [ Af+ Bf + Cf3+ Df4] h 3 h h 4 f = f+ hf + f + f + σ ( h ) f3 = f+ hf + h f + h f + σ ( h ) f4 = f+ 3hf + h f + h f + σ ( h ) 3 h h 4 3 f = Bf+ Bhf + B f + B f + Af+ Cf+ Chf + Ch f + C h f h Df + D3hf + D h f + D h f 0

2 A+ B+ C+ D f = 0 h f = B+ C+ D f ( 3 ) 9 B+ C+ D hf = 0 B C4 9D h f + + = A 0 9 B 0 = C D Matrix iversio gives: A =.8333 = 6 8 B = 3.0 = C =.5 = = 6 D = = 3 6 which agrees exactly with page 6 of ATP. f = [ f+ 8 f 9f ] 3 + f4 z 6 z Cosistecy: The Fiite differece approximatio is cosistet if the trucatio error for the whole equatio goes to zero as the mesh size goes to zero. Accuracy is rate that F.D. approximatio approaches zero as 0. Stability + Cosistecy = Covergece (Lax Equivalece Theorem)

3 Stability: Begi by defiig the N φ = φi i= Norm. L a fiite differece method is coverget of order p ad q if p q ψ (, j) φj = σ ( x ) + σ ( ) ad stable. Vo Neuma s Method for examiig stability of a liear PDE. Assume the solutio cosists of some Fourier wave umbers φ N ij j = ae = N a where is the amplitude of the Kth Fourier mode at time level. Strictly correct for spatially periodic domais (still useful otherwise). Key property so if φ j = e ij d e dx ix = ie ix for a liear fiite differece scheme at some iitial coditios, after oe time iteratio. φ + j where = Ae A ij x is a complex costat called the amplificatio factor for the Kth Fourier compoet a A a A a = = the modulus of the amplificatio factor scheme. o A for each Fourier compoet must be bouded for a stable It is a ecessary ad sufficiet coditio for stability if A + γ t, where γ is a costat idepedet of K,, or. The i the limit as ad 0the scheme will coverge. It is usually advatageous to eforce the more striget criteria A, which will guaratee stability. 3

4 Example: Cosider the upstream or Door cell fiite differece scheme. This scheme is forward i time ad bacward i space ad coditioally stable. φj φj φj φj + C = 0 Solve for φ C C φj = φj + C let µ = φ = µ φ + µφ ( ) ( ) j j j j = + φ j ( ) ij ij i j Ae µ e µ e ij Divide out commo factor e. A = µ + µ e i x magitude (modulus) of criteria. A foud by multiplyig by complex cojugate ad eforcig the stability i i ( µ µ ) i( µ µ ) µ ( µ ) A = + e + e = cos x so A < for µ ( µ ) µ ( µ ) ( ) ( ) 0 cos ad cos x > 0 for all K, except trivial case K = 0 (steady mea flow) the - 4µ µ 0 µ µ t is sufficiet for stability, ad assumig C > 0 the divide through by µ ad ( µ ) 0 gives µ so 0< C for coditioal stability. x π A = cos shows that the wave = λ is most ustable. Looig at µ ( µ ) π i.e. λ= = ad cos x= cosπ = ad for ay ustable C t value of > the x "blows up" fastest, ad domiates the resultig ustable amplificatio. 4

5 A wave is: i- i- i i+ i+ i+3 x ustable solutios usually become cotamiated with high frequecy oise. A table of results from Durra 999 is attached. Courat Friedrichs & Lewy stability aalysis uses the method of characteristics. The domai of depedece ( 0, 0 ) ϕ ( x, t ) 0 0 x t is the set of all poits ( x, t) for which ϕ ( x, t) exerts a ifluece o x ϕ ϕ i.e. t =, + C = 0 c t x ϕ or α x x = C t 5

6 Domai of ifluece is where ϕ ( x 0, t 0 ) will ifluece as t goes forward. The slope of the umerical domai of depedece, must satisfy ; C C 0 C x taig a smaller value of for fixed ad C ca result i a stable umerical scheme. Forward Time - Cetered Space is ustable. φj φj φj+ φj + C = 0 C φj = φj ( φj+ φj ) ij ij µ Ae = e e e µ i i A = ( e e ) = µ isi i( j+ ) i( j ) ( µ si )( µ si ) A = i x + i stable whe A = + µ si which ca ever so ucoditioally ustable. Note: HW # purposefully has this istability preset o oe of the parts, ad as you decrease µ the result blows up faster sice A A time) tha a larger with smaller. i.e <.0 is larger for larger (the umber of time steps to get to a certai.59 <.70 6