Fourier Techniques lecture by Håkon Hoel

Transcription

1 CSC Nada DN55 Sprig - Differetial Equatios II JOp p (7) Fourier Teciques lecture by Håo Hoel Fourier series... Applicatios to liear PDE... 3 Numerical metods Vo Neuma aalysis Spectral metods Modified Equatios...7 Fourier series Ay fuctio f L [ A, A] ca be represeted as a Fourier series A f ( x) = f e x, f = f ( x) e x π, ω =, =...,,0,,... = A A A Te set {. e } Z is a liearly idepedet basis for te ifiite dimesioal fuctio space L [-A,A] just lie ay vector z i R ca be expressed as z = < z, l > l were <.,.> is a = scalar product, for example <a,b> = a T b for colum vectors a,b. Example ix ix f ( x) = cos x = / ( e + e ), f /, = ± = 0, else Importat properties I. Parseval s teorem = f f A II. Liearity: Let te Fourier coefficiets of f ad g be f, g. Te te Fourier coefficiets of te liear combiatio af + bg are af + bg. III. Differetiatio: ( m) ( m) If te ed derivatives of f matc up to :t order, f ( A) = f ( A), m = 0,,,..., (m) m te te Fourier coefficiets of f are ( ) f,m=0,,,. Equivaletly, te fuctio correspodig to IV. Traslatio: Te Fourier coefficiets of f(.+) are e f. Applicatios to liear PDE Example advectio equatio Advectio equatio wit periodic boudary coditios Ut + au x = 0, x [ A, A], a R U ( x,0) = U 0 ( x) (ADVECT)

2 CSC Nada DN55 Sprig - Differetial Equatios II JOp p (7) Fourier trasformatio of te equatio, usig III, gives U ( ) t = au ( ) wit solutio a U ( ) e t = U 0 ( ) wic, by te traslatio relatio IV are te F. coefficiets of U 0 (x-a. Let us sow te full computatio ( ) U ( x, U ( ) e x a e t + x x at U ( ) 0 e = = = U0 ( ) = U 0 ( x a Example - eat equatio Te eat equatio wit periodic boudary coditios o [-A,A] Ut = DU xx, U ( x,0) = U0 ( x) (HEAT) We obtai te ODE for te Fourier coefficiets d U (, = D( ) U (,, dt U Dω t (, = e U 0( ) We ca use tis, ad te Parseval relatio, to sow a boud for te solutio growt U ( t,.) U (, e D ω t = = U0 ( ) U0 ( ) = U0 A A Tis ca also be obtaied by te eergy metod, lie i Lect. Te secod derivative term is diffusive or dissipative ad acts to dissipate te eergy, faster for iger wave umbers ω. (Sow solutio wit differet wave umbers). Te same olds for all eve order space derivatives wit te correct sig, i U, =,,... x Example 3 - dispersive U t = β U xxx, U ( x,0) = U 0( x ) We get d 3 U (, = β ( ) U (,, dt 3 U iβω t (, = e U 0 ( ) ad 3 + = x iβω t x (, ) (, ) = ( x βω U x t U t e e U ( ) = 0 e U 0 ( ) For a give wave umber ω, tis is a traslatio wit pase speed βω, so waves of differet wavelegt travel at differet speeds: dispersio. Te growt estimate becomes U (., = U 0 because all te e i( ) factors ave purely imagiary expoets, ece absolute value. It follows tat odd order spatial derivatives are dispersive, a rigt ad side wic is a liear combiatio of odd-order spatial derivatives gives solutios wit L -orms costat i time.

3 CSC Nada DN55 Sprig - Differetial Equatios II JOp p 3 (7) 3 Numerical metods N / Give N fuctio values { } f ( x j ) (N eve) o te grid x j = N / j = j, = A/N, we defie te Discrete Fourier Trasform by N / x j f = f ( x j ) e A j= N / wit te iverse A / i x j f x j = f ω ( ) e A = A / + Te limits o wave umber are +- A/ because a wave wit sorter wavelegt caot be distiguised from oe wit a wavelegt i tis iterval. Tis peomeo is called aliasig, resposible for te illusio of stage coac weels turig te wrog way because te motio is sampled at movie-frame rates 5 fps. Here is a picture of si π/0 x, si 0.x ad si 6.48x o a grid wit =. Te log wave is clearly idetifiable, te oter two are almost idetical, but ot quite because π + 0. = , ot (6.78 i te leged is a mispri Aliasig Te sortest sie-wave represetable as a pase sift of π per grid iterval ad is a perfect saw-toot, ad we do ot plot it because it would drow out te oter curves. Wit te orms N / f = f ( x j ), j = N / Parseval s relatio still olds, f = f A f A / = = A / f Exercise Ti of te trasformatio as a liear mappig of C N to C N f = Ff Te Parseval relatio sows tat te N x N Fourier matrix F wit complex etries is ortogoal i te sese (A H = Hermitia cojugate of A: traspose ad complex cojugate) F H F = ci Wat is te value of c?

4 CSC Nada DN55 Sprig - Differetial Equatios II JOp p 4 (7) 3. Vo Neuma aalysis Cosider ow a umerical solutio U j wic approximates te exact solutio U to a liear PDE at te poits (t,x j ). If te growt of te solutio is bouded i te sese made precise below, ad if te umerical sceme is a cosistet approximatio to a differetial equatio, oe ca coclude tat it coverges as, Δx -> 0 to te solutio of te PDE. Te sceme is called stable, if tere exists a C(T) suc tat, for all 0 < t < T ad sufficietly small Δx, U (., t ) C( T ) U0(.,0) (STAB) It may be ecessary to require a relatio bewee Δx ad Δx for (STAB) to old. For istace, te classical time-explicit, cetral space differece sceme for te eat equatio (HEAT) requires D Δx ad we sall see repeatedly te coditio a Δx for time-explicit scemes for te advectio equatio (ADVECT). Te vo Neuma aalysis uses Fourier aalysis to fid suc relatios for periodic or Caucy problems for liear, costat coefficiet PDE. By liearizatio ad freezig of coefficiets it ca be applied also to o-liear PDE to give ecessary coditios (but ot sufficie. Te alterative metod is te eergy metod, wic as more restricted applicability but ca treat oter ids of boudary coditios. Te vo Neuma aalysis produces te growt factor G(,,) of te Fourier coefficiets over oe step U (, t ) = GU (, t ) (For a system of S equatios G is of course a S x S matrix.) IF G is uiformly (i,, ) bouded by + K, te te sceme is vo Neuma-stable. For te we ave Kt U (., t ) U (., t ) ( K U (., t 0 ) e Kt U (., t0 ) e = + Δ = U0 Example 4 Upwid sceme for ADVECT a U j = U j ( U j U j ) Represetig te U by Fourier series, we ave i x j i x j a i x j i x j U ω t e = U ω ω ω ( ) (, ) (, t )( e ( e e )) = i x j a i = U ω ω (, t ) e ( ( e )) G(,, ) Sice te exp. fuctios are liearly idepedet, we ca separate te equatios for eac wave umber, ad obtai U iθ (, t = GU + ) (, t ), G = σ ( e ) a σ = Courat umber = θ = pase sift per grid cell = ω

5 CSC Nada DN55 Sprig - Differetial Equatios II JOp p 5 (7) Tere remais to ivestigate te boud o G. We θ assumes values betwee π ad π, G traces a circle i te complex plae, wit ceter i -σ ad radius σ. If tis circle is cotaied i te uit circle, we certaily ave G <=. Tat requires 0 < σ <, i.e., te velocity a must be positive, ad te time-step small eoug. For te sceme wit te space differece tured te oter way, a U j = U j ( U j + U j ) iθ G = + σ σe Te plots of G below. Left, Uj Uj-, stable for 0 < σ <, rigt, Uj+ Uj, stable for < σ < 0 Te stability coditio is well uderstood by cosiderig te exact solutio of te PDE, U(x, = U 0 (x a: We a > 0, te solutio at (x*,t+) is iflueced by te solutio for x < x*, ad vice versa for a < 0. Terefore, te proper differece directio is called upstream or upwid. Example 5 HEAT D U j = U j + ( U ) j + U j + U j D iθ iθ D θ G = + ( e + e ) = 4 si D D We ave 4 G, so G <= for all θ requires tat 4 or D Exercise Compute G for te FTCS -sceme (Forward Time, Cetral Space) a U ( j = U j U j + U ) j ad commet o te coice of to mae G < + K uiformly i wave umber for some K.

6 CSC Nada DN55 Sprig - Differetial Equatios II JOp p 6 (7) 3. Spectral metods I fiite differece scemes, te spatial derivatives at x j are approximated by a few eigbor poits. Usig te property 3, we ca compute derivatives usig all te poits very accurately, at least for smoot grid fuctios: U Te Fourier coefficiet umber of is ( ) U ( ) x We defie te Fourier trasform operatio by FF: FF( U ) = U ad te iverse FFI, U = FFI(U ). FF is (i priciple) a matrix multiplicatio by F ad FFI by F -. However, tey are computed by te Fast Fourier Trasformatio algoritm, i N logn aritmetical operatios istead of te N required by te matrix multiply. Suc spectral differetiatio as trucatio error wic decreases expoetially wit icreasig N, if te differetiated fuctio is sufficietly smoot, wereas te differece formulas ave errors wic decay lie powers of (/N). Example 6 a o-liear PDE U t = ( U x ) + DU xx may be writte as U t = ( FFI( Ua) + D FFI(( ) Ua Uat = FF( U ) so te evaluatio of te time derivative at all N gridpoits ca be carried out wit spectral accuracy by oe applicatio of FFT ad two of ifft. Tis opes te way to usig ay explicit time-steppig sceme suc as Euler or Ruge-Kutta. Here is te sceme for Euler time steppig, iomega = fftsift(i*[-n/:n/-]*pi/a); % iomega = d/dx, cosiderig te defiitio % of MATLABs FFT Uat = FFT(U); RHS = -(IFFT(iomega.*Ua).^ + IFFT(D*(iomega.^).*Ua; Uew = U + dt*rhs; Exercise Wat is te time-step limit for Euler time steppig of te HEAT equatio wit spectral differetiatio for space derivatives? Hit te sortest wave o te grid si(x) as = π Te timestep restrictio for te HEAT equatio is proibitive, so cosider istead te IMPLICIT Euler sceme, U U = D FFI(( ) FF( U )) FF( U U ) D ( ) FF( U ) = 0 ( D ( ) ) FF( U ) = FF( U ), U = FFI(( D ( ) ) FF( U )) iomega = fftsift(i*[-n/:n/-]*pi/a); Uat = FFT(U); Uew = IFFT(Uat./(-D*dt*(iomega).^));

7 CSC Nada DN55 Sprig - Differetial Equatios II JOp p 7 (7) 3.3 Modified Equatios Cosider te upwid sceme for te ADVECT equatio, a > 0, U j U j U j U j + a = 0 (UP) wic gives a approximate solutio. Its properties may be more clarified if we fid a differetial equatio to wic it is a better approximatio ta te ADVECT equatio. Let V(x, be a smoot fuctio, te gridpoit values of wic are V ( x j, t ) = U j. Te V ( x j, t + ) V ( x j, t ) V ( x j, t ) V ( x j, t ) 0 = + a = ( Vt + Vtt + Vttt +...) + a( Vx Vxx + Vxxx +...) 6 6 Vt + avx = ( avxx Vtt ) + O( ) Vtx + avxx = O( ) Vtt = a Vxx + O( ) Vtt + avxt = O( ) Fially, a Vt + avx = ( avxx a Vxx ) + O( ) = ( σ ) Vxx + O( ) were σ = a/ is te Courat umber. Tis sows tat te upwid sceme is modeled by a a advectio-diffusio equatio wit diffusio coefficiet ( σ ) wic is positive for σ <. Te geeral rule is tat odd-order accurate differece scemes are dissipative, ad eve-order scemes are dispersive. First order scemes are TOO dissipative, ad secod order scemes TOO dispersive.