Computational Fluid Dynamics. Lecture 5

Transcription

1 Time differecig cotiued. Three level schemes. B. Modified L-F schemes. C. Higher order methods. Three level schemes Computatioal Fluid Dyamics Lecture 5 ψ = α α β tf β cosistet schemes if α α = ad β β = tf It will be purely explicit ad will be secod order accurate if: α = α β = ( α 3) β = ( α ) if β = 0, Leap Frog d If α = 0 - order B dams Bashforth ψ ψ bk C = 0: ickb k = 0 x / oe fourier mode With Fourier Series, e.g, N ψ = b k = 0 k () t e ik x Oscillatio equatio: bk = ickb, similar properties to wave equatio. k L-F o Oscillatio equatio = i = = i = 0 Factor out

2 Every step has same amplificatio factor because this is a liear equatio with costat coefficiets. has two roots. = ik ± t ± k t I limit of good resolutio 0 ad The solutio has two differet modes, is the physical mode ad is the computatioal mode. ( ) If the k t is real ad = k t k t = ad there is o amplitude error i leap frog scheme. I the case k t > ± = ik t± k t ± = ik t± i k t = i k t ( k t ) which has magitude greater tha i. i k t k t > i positive sice k t > so > sice > k t > is ustable. Whe k t < similarly > ad whe k t > the = i costat, ad thus each time step produces a 90 shift i the phase of the oscillatio, ad the ustable mode grows at a period of t. Relative Phase errors: ta RLF = k t I limit of good resolutio 0

3 R LF ad = acceleratig phase - lead error. 6 for stability. dams Bashforth d order scheme 3 = i F F applied to Oscillatio equatio. 3 = i 3 i ik t = 0 ± 3i ( 9) = i ± as 0 0 damped computatioal mode is desirable. but there is a weak istability i the physical mode R B -ustable amplifies some waves 5 = phase leadig error Sometimes the weak istability ca be tolerated if legth of itegratio is short or other dampig is preset i the system. 3

4 First ad Secod Order Schemes Implicit Explicit. Backward ( t. Forward σ t σ ) ustable σ ( t) ( most accurate but slow ) σ ( t ). Trapezoidal. Ruge Kutta. ustable 3. Matsuo σ ( t) heavily damped (two step method). Leap Frog σ t. computatioal mode is udamped, ad oliear behavior will blow up. 5. dams Bashforth σ t ustable. Dealig with computatioal mode of Leap Frog scheme. methods. Periodically restart solutios, throwig away oe time level. You could use RK- smoothig step.. Filter the computatioal mode with the sseli time filter. tf ( ) = = γ 3. Use the L.F. predictor, trapezoidal corrector scheme.. Take a Leap Frog, B LF- B scheme, -Magazekov. Decouples odd/eve time levels. Not frequetly doe, but more attractive. sseli time filterig Regular L-F is secod order.

5 = tf ( ) = γ γ - small umber to typically ad meas filtered pply to oscillatio equatio. ψ = ikψ = ik t = i = i ( ) = ( ) γ ( ) ( ) = γ γ γ = ik t γ Which is of the form ax bx c = 0 ( γ ) = γ ik t± γ = 0 goes back to Leap - Frog ufiltered scheme. eutral for k t 0 ( γ ) lim = i σ ( ) i 3 i i k t exact = e = i σ 6 Note that the amplificatio factor is weakly dampig but the disadvatage is that sseli time filterig lowers the global trucatio error to first order, istead of secod order L-F. But i all practical cases ad γ so the filterig effect has egligible effect ad order of.9 accuracy σ. 5

6 R sseli - LF sseli - LF sseli - LF γ γ γ γ 6 γ γ γ 6 ( ) γ Efficiet d order alterative Leap Frog Trapezoidal Method (L-T) Predictor step. * = tf corrector trapezoidal step. t = F F( ) * max stable For example i the equatio u u u = F( u ) = C v x x u u u C = v x x Evaluate fuctios times per time step. k t = 0.7 efficiecy R LT LT = = other good method - Magazekov scheme LF-B-LF-B. 6

7 = tf = t 3 F F max 3 completely explicit scheme. R = = 6 ad a damped computatioal mode. Takes two ustable schemes ad makes them stable. Explicit schemes wat combiatio of accuracy ad low storage ad efficiecy..73 RK3 3 max 0.58 B3 max 0.7 RK max Higher order Explicit schemes. B3 t = 3F 6F 5F Cosistet if coefficiets = 3 rd order accurate. Code is ormally writte to coserve memory, by savig the fluxes at each previous time level. 7

8 Ruge Kutta q = tf q = tf q q3 = tf q ( ) q = tf q 3 6 [ q q q q ] = 3 Variable time steppig t t t 3 same stability properties if t chages slowly. Coeffiets, -6, 5 chage.see other otes. Oe facy scheme uses variable time steps ad variable order time discretizatio to achieve a miimu tolerable error. For example, whe t is small a low order Ruge-Kutta scheme is used, but whe t is larger a higher order R-K scheme (up to 0 th order) ca be selected. 8